Exploratory Data Analysis - NIST eHandbook of Statistical Methods-out.pdf

Exploratory Data Analysis
NIST/SEMATECH e-Handbook of Statistical
Methods
http://www.itl.nist.gov/div898/handbook
Retrieved: 8th August 2024
This chapter presents the assumptions, principles, and techniques necessary to
gain insight into data via EDA--exploratory data analysis.
1. EDA Introduction
1. What is EDA?
2. EDA vs Classical & Bayesian
3. EDA vs Summary
4. EDA Goals
5. The Role of Graphics
6. An EDA/Graphics Example
7. General Problem Categories
2. EDAAssumptions
1. Underlying Assumptions
2. Importance
3. Techniques for Testing
Assumptions
4. Interpretation of 4-Plot
5. Consequences
3. EDA Techniques
1. Introduction
2. Analysis Questions
3. Graphical Techniques:
Alphabetical
4. Graphical Techniques: By
Problem Category
5. Quantitative Techniques
6. Probability Distributions
4. EDA Case Studies
1. Introduction
2. By Problem Category
Detailed Chapter Table of Contents
References
Dataplot Commands for EDA Techniques
EDA Introduction
Summary What is exploratory data analysis? How did it begin? How and
where did it originate? How is it differentiated from other data
analysis approaches, such as classical and Bayesian? Is EDA
the same as statistical graphics? What role does statistical
graphics play in EDA? Is statistical graphics identical to EDA?
These questions and related questions are dealt with in this
section. This section answers these questions and provides the
necessary frame of reference for EDA assumptions, principles,
and techniques.
Table of
Contents
for Section
1
1. What is EDA?
2. EDA versus Classical and Bayesian
1. Models
2. Focus
3. Techniques
4. Rigor
5. Data Treatment
6. Assumptions
3. EDA vs Summary
4. EDA Goals
5. The Role of Graphics
6. An EDA/Graphics Example

7. General Problem Categories
What is EDA?
Approach Exploratory Data Analysis (EDA) is an approach/philosophy
for data analysis that employs a variety of techniques (mostly
graphical) to
1. maximize insight into a data set;
2. uncover underlying structure;
3. extract important variables;
4. detect outliers and anomalies;
5. test underlying assumptions;
6. develop parsimonious models; and
7. determine optimal factor settings.
Focus The EDA approach is precisely that--an approach--not a set of
techniques, but an attitude/philosophy about how a data
analysis should be carried out.
Philosophy EDA is not identical to statistical graphics although the two
terms are used almost interchangeably. Statistical graphics is a
collection of techniques--all graphically based and all focusing
on one data characterization aspect. EDA encompasses a larger
venue; EDA is an approach to data analysis that postpones the
usual assumptions about what kind of model the data follow
with the more direct approach of allowing the data itself to
reveal its underlying structure and model. EDA is not a mere
collection of techniques; EDA is a philosophy as to how we
dissect a data set; what we look for; how we look; and how we
interpret. It is true that EDA heavily uses the collection of
techniques that we call "statistical graphics", but it is not
identical to statistical graphics per se.
History The seminal work in EDA is Exploratory Data Analysis, Tukey,
(1977). Over the years it has benefitted from other noteworthy
publications such as Data Analysis and Regression, Mosteller
and Tukey (1977), Interactive Data Analysis, Hoaglin (1977),
The ABC's of EDA, Velleman and Hoaglin (1981) and has
gained a large following as "the" way to analyze a data set.
Techniques Most EDA techniques are graphical in nature with a few
quantitative techniques. The reason for the heavy reliance on
graphics is that by its very nature the main role of EDA is to
open-mindedly explore, and graphics gives the analysts
unparalleled power to do so, enticing the data to reveal its
structural secrets, and being always ready to gain some new,
often unsuspected, insight into the data. In combination with
the natural pattern-recognition capabilities that we all possess,
graphics provides, of course, unparalleled power to carry this
out.
The particular graphical techniques employed in EDA are often
quite simple, consisting of various techniques of:
1. Plotting the raw data (such as data traces, histograms,
bihistograms, probability plots, lag plots, block plots,
and Youden plots.
2. Plotting simple statistics such as mean plots, standard
deviation plots, box plots, and main effects plots of the
raw data.
3. Positioning such plots so as to maximize our natural
pattern-recognition abilities, such as using multiple plots
per page.

How Does Exploratory Data Analysis differ
from Classical Data Analysis?
Data
Analysis
Approaches
EDA is a data analysis approach. What other data analysis
approaches exist and how does EDA differ from these other
approaches? Three popular data analysis approaches are:
1. Classical
2. Exploratory (EDA)
3. Bayesian
Paradigms
for Analysis
Techniques
These three approaches are similar in that they all start with a
general science/engineering problem and all yield
science/engineering conclusions. The difference is the
sequence and focus of the intermediate steps.
For classical analysis, the sequence is
Problem => Data => Model => Analysis =>
Conclusions
For EDA, the sequence is
Problem => Data => Analysis => Model =>
Conclusions
For Bayesian, the sequence is
Problem => Data => Model => Prior Distribution =>
Analysis => Conclusions
Method of
dealing with
underlying
model for
the data
distinguishes
the 3
approaches
Thus for classical analysis, the data collection is followed by
the imposition of a model (normality, linearity, etc.) and the
analysis, estimation, and testing that follows are focused on
the parameters of that model. For EDA, the data collection is
not followed by a model imposition; rather it is followed
immediately by analysis with a goal of inferring what model
would be appropriate. Finally, for a Bayesian analysis, the
analyst attempts to incorporate scientific/engineering
knowledge/expertise into the analysis by imposing a data-
independent distribution on the parameters of the selected
model; the analysis thus consists of formally combining both
the prior distribution on the parameters and the collected data
to jointly make inferences and/or test assumptions about the
model parameters.
In the real world, data analysts freely mix elements of all of
the above three approaches (and other approaches). The
above distinctions were made to emphasize the major
differences among the three approaches.
Further
discussion of
the
distinction
between the
classical
and EDA
approaches
Focusing on EDA versus classical, these two approaches
differ as follows:
1. Models
2. Focus
3. Techniques
4. Rigor
5. Data Treatment
6. Assumptions
Classical Data Analysis?
Model

Classical The classical approach imposes models (both deterministic
and probabilistic) on the data. Deterministic models include,
for example, regression models and analysis of variance
(ANOVA) models. The most common probabilistic model
assumes that the errors about the deterministic model are
normally distributed--this assumption affects the validity of
the ANOVA F tests.
Exploratory The Exploratory Data Analysis approach does not impose
deterministic or probabilistic models on the data. On the
contrary, the EDA approach allows the data to suggest
admissible models that best fit the data.
Focus
Classical The two approaches differ substantially in focus. For classical
analysis, the focus is on the model--estimating parameters of
the model and generating predicted values from the model.
Exploratory For exploratory data analysis, the focus is on the data--its
structure, outliers, and models suggested by the data.
Techniques
Classical Classical techniques are generally quantitative in nature. They
include ANOVA, t tests, chi-squared tests, and F tests.
Exploratory EDA techniques are generally graphical. They include scatter
plots, character plots, box plots, histograms, bihistograms,
probability plots, residual plots, and mean plots.
Rigor
Classical Classical techniques serve as the probabilistic foundation of
science and engineering; the most important characteristic of
classical techniques is that they are rigorous, formal, and
"objective".
Exploratory EDA techniques do not share in that rigor or formality. EDA
techniques make up for that lack of rigor by being very
suggestive, indicative, and insightful about what the
appropriate model should be.
EDA techniques are subjective and depend on interpretation
which may differ from analyst to analyst, although
experienced analysts commonly arrive at identical
conclusions.
Data Treatment
Classical Classical estimation techniques have the characteristic of
taking all of the data and mapping the data into a few numbers
("estimates"). This is both a virtue and a vice. The virtue is
that these few numbers focus on important characteristics

(location, variation, etc.) of the population. The vice is that
concentrating on these few characteristics can filter out other
characteristics (skewness, tail length, autocorrelation, etc.) of
the same population. In this sense there is a loss of
information due to this "filtering" process.
Exploratory The EDA approach, on the other hand, often makes use of
(and shows) all of the available data. In this sense there is no
corresponding loss of information.
Assumptions
Classical The "good news" of the classical approach is that tests based
on classical techniques are usually very sensitive--that is, if a
true shift in location, say, has occurred, such tests frequently
have the power to detect such a shift and to conclude that such
a shift is "statistically significant". The "bad news" is that
classical tests depend on underlying assumptions (e.g.,
normality), and hence the validity of the test conclusions
becomes dependent on the validity of the underlying
assumptions. Worse yet, the exact underlying assumptions
may be unknown to the analyst, or if known, untested. Thus
the validity of the scientific conclusions becomes intrinsically
linked to the validity of the underlying assumptions. In
practice, if such assumptions are unknown or untested, the
validity of the scientific conclusions becomes suspect.
Exploratory Many EDA techniques make little or no assumptions--they
present and show the data--all of the data--as is, with fewer
encumbering assumptions.
How Does Exploratory Data Analysis Differ
from Summary Analysis?
Summary A summary analysis is simply a numeric reduction of a
historical data set. It is quite passive. Its focus is in the past.
Quite commonly, its purpose is to simply arrive at a few key
statistics (for example, mean and standard deviation) which
may then either replace the data set or be added to the data set
in the form of a summary table.
Exploratory In contrast, EDA has as its broadest goal the desire to gain
insight into the engineering/scientific process behind the data.
Whereas summary statistics are passive and historical, EDA is
active and futuristic. In an attempt to "understand" the process
and improve it in the future, EDA uses the data as a "window"
to peer into the heart of the process that generated the data.
There is an archival role in the research and manufacturing
world for summary statistics, but there is an enormously larger
role for the EDA approach.
What are the EDA Goals?
Primary
and
Secondary
Goals
The primary goal of EDA is to maximize the analyst's insight
into a data set and into the underlying structure of a data set,
while providing all of the specific items that an analyst would
want to extract from a data set, such as:
1. a good-fitting, parsimonious model
2. a list of outliers

3. a sense of robustness of conclusions
4. estimates for parameters
5. uncertainties for those estimates
6. a ranked list of important factors
7. conclusions as to whether individual factors are
statistically significant
8. optimal settings
Insight
into the
Data
Insight implies detecting and uncovering underlying structure
in the data. Such underlying structure may not be encapsulated
in the list of items above; such items serve as the specific
targets of an analysis, but the real insight and "feel" for a data
set comes as the analyst judiciously probes and explores the
various subtleties of the data. The "feel" for the data comes
almost exclusively from the application of various graphical
techniques, the collection of which serves as the window into
the essence of the data. Graphics are irreplaceable--there are no
quantitative analogues that will give the same insight as well-
chosen graphics.
To get a "feel" for the data, it is not enough for the analyst to
know what is in the data; the analyst also must know what is
not in the data, and the only way to do that is to draw on our
own human pattern-recognition and comparative abilities in the
context of a series of judicious graphical techniques applied to
the data.
The Role of Graphics
Quantitative/
Graphical
Statistics and data analysis procedures can broadly be split
into two parts:
quantitative
graphical
Quantitative Quantitative techniques are the set of statistical procedures
that yield numeric or tabular output. Examples of quantitative
techniques include:
hypothesis testing
analysis of variance
point estimates and confidence intervals
least squares regression
These and similar techniques are all valuable and are
mainstream in terms of classical analysis.
Graphical On the other hand, there is a large collection of statistical
tools that we generally refer to as graphical techniques. These
include:
scatter plots
histograms
probability plots
residual plots
box plots
block plots
EDA
Approach
Relies
Heavily on
Graphical
Techniques
The EDA approach relies heavily on these and similar
graphical techniques. Graphical procedures are not just tools
that we could use in an EDA context, they are tools that we
must use. Such graphical tools are the shortest path to
gaining insight into a data set in terms of
testing assumptions
model selection

model validation
estimator selection
relationship identification
factor effect determination
outlier detection
If one is not using statistical graphics, then one is forfeiting
insight into one or more aspects of the underlying structure of
the data.
An EDA/Graphics Example
Anscombe
Example
A simple, classic (Anscombe) example of the central role
that graphics play in terms of providing insight into a data
set starts with the following data set:
Data X Y
10.00 8.04
8.00 6.95
13.00 7.58
9.00 8.81
11.00 8.33
14.00 9.96
6.00 7.24
4.00 4.26
12.00 10.84
7.00 4.82
5.00 5.68
Summary
Statistics
If the goal of the analysis is to compute summary statistics
plus determine the best linear fit for Y as a function of X, the
results might be given as:
N = 11
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Residual standard deviation = 1.237
Correlation = 0.816
The above quantitative analysis, although valuable, gives us
only limited insight into the data.
Scatter Plot In contrast, the following simple scatter plot of the data
suggests the following:
1. The data set "behaves like" a linear curve with some
scatter;
2. there is no justification for a more complicated model
(e.g., quadratic);
3. there are no outliers;

4. the vertical spread of the data appears to be of equal
height irrespective of the X-value; this indicates that
the data are equally-precise throughout and so a
"regular" (that is, equi-weighted) fit is appropriate.
Three
Additional
Data Sets
This kind of characterization for the data serves as the core
for getting insight/feel for the data. Such insight/feel does
not come from the quantitative statistics; on the contrary,
calculations of quantitative statistics such as intercept and
slope should be subsequent to the characterization and will
make sense only if the characterization is true. To illustrate
the loss of information that results when the graphics insight
step is skipped, consider the following three data sets
[Anscombe data sets 2, 3, and 4]:
X2 Y2 X3 Y3 X4 Y4
10.00 9.14 10.00 7.46 8.00 6.58
8.00 8.14 8.00 6.77 8.00 5.76
13.00 8.74 13.00 12.74 8.00 7.71
9.00 8.77 9.00 7.11 8.00 8.84
11.00 9.26 11.00 7.81 8.00 8.47
14.00 8.10 14.00 8.84 8.00 7.04
6.00 6.13 6.00 6.08 8.00 5.25
4.00 3.10 4.00 5.39 19.00 12.50
12.00 9.13 12.00 8.15 8.00 5.56
7.00 7.26 7.00 6.42 8.00 7.91
5.00 4.74 5.00 5.73 8.00 6.89
Quantitative
Statistics for
Data Set 2
A quantitative analysis on data set 2 yields
N = 11 Mean of X = 9.0 Mean of Y = 7.5 Intercept = 3
Slope = 0.5 Residual standard deviation = 1.237
Correlation = 0.816
which is identical to the analysis for data set 1. One might
naively assume that the two data sets are "equivalent" since
that is what the statistics tell us; but what do the statistics not
tell us?
Quantitative
Statistics for
Data Sets 3
and 4
Remarkably, a quantitative analysis on data sets 3 and 4 also
yields
N = 11 Mean of X = 9.0 Mean of Y = 7.5 Intercept = 3
Slope = 0.5 Residual standard deviation = 1.236
Correlation = 0.816 (0.817 for data set 4)
which implies that in some quantitative sense, all four of the
data sets are "equivalent". In fact, the four data sets are far
from "equivalent" and a scatter plot of each data set, which
would be step 1 of any EDA approach, would tell us that
immediately.
Scatter Plots
Interpretation
of Scatter
Plots
Conclusions from the scatter plots are:
1. data set 1 is clearly linear with some scatter.
2. data set 2 is clearly quadratic.

3. data set 3 clearly has an outlier.
4. data set 4 is obviously the victim of a poor
experimental design with a single point far removed
from the bulk of the data "wagging the dog".
Importance
of
Exploratory
Analysis
These points are exactly the substance that provide and
define "insight" and "feel" for a data set. They are the goals
and the fruits of an open exploratory data analysis (EDA)
approach to the data. Quantitative statistics are not wrong
per se, but they are incomplete. They are incomplete because
they are numeric summaries which in the summarization
operation do a good job of focusing on a particular aspect of
the data (e.g., location, intercept, slope, degree of
relatedness, etc.) by judiciously reducing the data to a few
numbers. Doing so also filters the data, necessarily omitting
and screening out other sometimes crucial information in the
focusing operation. Quantitative statistics focus but also
filter; and filtering is exactly what makes the quantitative
approach incomplete at best and misleading at worst.
The estimated intercepts (= 3) and slopes (= 0.5) for data
sets 2, 3, and 4 are misleading because the estimation is
done in the context of an assumed linear model and that
linearity assumption is the fatal flaw in this analysis.
The EDA approach of deliberately postponing the model
selection until further along in the analysis has many
rewards, not the least of which is the ultimate convergence
to a much-improved model and the formulation of valid and
supportable scientific and engineering conclusions.
General Problem Categories
Problem
Classification
The following table is a convenient way to classify EDA
problems.
Univariate
and Control
UNIVARIATE
Data:
A single column of
numbers, Y.
Model:
y = constant + error
Output:
1. A number (the
estimated constant in
the model).
2. An estimate of
uncertainty for the
constant.
3. An estimate of the
distribution for the
error.
Techniques:
4-Plot
Probability Plot
PPCC Plot
CONTROL
Data:
A single column of
numbers, Y.
Model:
y = constant + error
Output:
A "yes" or "no" to the
question "Is the system
out of control?".
Techniques:
Control Charts

Comparative
and
Screening
COMPARATIVE
Data:
A single response
variable and k
independent variables
(Y, X1, X2, ... , Xk),
primary focus is on
one (the primary
factor) of these
independent variables.
Model:
y = f(x1, x2, ..., xk) +
error
Output:
A "yes" or "no" to the
question "Is the
primary factor
significant?".
Techniques:
Block Plot
Scatter Plot
Box Plot
SCREENING
Data:
A single response
variable and k
(Y, X1, X2, ... , Xk).
Model:
y = f(x1, x2, ..., xk) +
error
Output:
1. A ranked list (from
most important to least
important) of factors.
2. Best settings for the
factors.
3. A good
model/prediction
equation relating Y to
the factors.
Techniques:
Block Plot
Probability Plot
Bihistogram
Optimization
and
Regression
OPTIMIZATION
Data:
A single response
variable and k
(Y, X1, X2, ... , Xk).
Model:
y = f(x1, x2, ..., xk) +
error
Output:
Best settings for the
factor variables.
Techniques:
Block Plot
Least Squares Fitting
Contour Plot
REGRESSION
Data:
A single response
variable and k
(Y, X1, X2, ... , Xk).
The independent
variables can be
continuous.
Model:
y = f(x1, x2, ..., xk) +
error
Output:
A good
model/prediction
equation relating Y to
the factors.
Techniques:
Least Squares Fitting
Scatter Plot
6-Plot
Time Series
and
Multivariate
TIME SERIES
Data:
A column of
time dependent
MULTIVARIATE
Data:

numbers, Y. In
addition, time is
an indpendent
variable. The
time variable
can be either
explicit or
implied. If the
data are not
equi-spaced, the
time variable
should be
explicitly
provided.
Model:
yt = f(t) + error
The model can
be either a time
domain based or
frequency
domain based.
Output:
A good
model/prediction
equation relating
Y to previous
values of Y.
Techniques:
Autocorrelation
Plot
Spectrum
Complex
Demodulation
Amplitude Plot
Complex
Demodulation
Phase Plot
ARIMA Models
k factor variables (X1, X2, ... ,
Xk).
Model:
The model is not explicit.
Output:
Identify underlying
correlation structure in the
data.
Techniques:
Star Plot
Scatter Plot Matrix
Conditioning Plot
Profile Plot
Principal Components
Clustering
Discrimination/Classification
Note that multivarate analysis is
only covered lightly in this
Handbook.
EDAAssumptions
Summary The gamut of scientific and engineering experimentation is
virtually limitless. In this sea of diversity is there any common
basis that allows the analyst to systematically and validly arrive
at supportable, repeatable research conclusions?
Fortunately, there is such a basis and it is rooted in the fact that
every measurement process, however complicated, has certain
underlying assumptions. This section deals with what those
assumptions are, why they are important, how to go about
testing them, and what the consequences are if the assumptions
do not hold.
Table of
Contents
for Section
2
1. Underlying Assumptions
2. Importance
3. Testing Assumptions
4. Importance of Plots
5. Consequences

Underlying Assumptions
Assumptions
Underlying a
Measurement
Process
There are four assumptions that typically underlie all
measurement processes; namely, that the data from the
process at hand "behave like":
1. random drawings;
2. from a fixed distribution;
3. with the distribution having fixed location; and
4. with the distribution having fixed variation.
Univariate or
Single
Response
Variable
The "fixed location" referred to in item 3 above differs for
different problem types. The simplest problem type is
univariate; that is, a single variable. For the univariate
problem, the general model
response = deterministic component + random
component
becomes
response = constant + error
Assumptions
for
Univariate
Model
For this case, the "fixed location" is simply the unknown
constant. We can thus imagine the process at hand to be
operating under constant conditions that produce a single
column of data with the properties that
the data are uncorrelated with one another;
the random component has a fixed distribution;
the deterministic component consists of only a
constant; and
the random component has fixed variation.
Extrapolation
to a Function
of Many
Variables
The universal power and importance of the univariate model
is that it can easily be extended to the more general case
where the deterministic component is not just a constant, but
is in fact a function of many variables, and the engineering
objective is to characterize and model the function.
Residuals
Will Behave
According to
Univariate
Assumptions
The key point is that regardless of how many factors there
are, and regardless of how complicated the function is, if the
engineer succeeds in choosing a good model, then the
differences (residuals) between the raw response data and
the predicted values from the fitted model should themselves
behave like a univariate process. Furthermore, the residuals
from this univariate process fit will behave like:
random drawings;
from a fixed distribution;
with fixed location (namely, 0 in this case); and
with fixed variation.
Validation of
Model
Thus if the residuals from the fitted model do in fact behave
like the ideal, then testing of underlying assumptions
becomes a tool for the validation and quality of fit of the
chosen model. On the other hand, if the residuals from the
chosen fitted model violate one or more of the above
univariate assumptions, then the chosen fitted model is
inadequate and an opportunity exists for arriving at an
improved model.
Importance

Predictability
and
Statistical
Control
Predictability is an all-important goal in science and
engineering. If the four underlying assumptions hold, then
we have achieved probabilistic predictability--the ability to
make probability statements not only about the process in the
past, but also about the process in the future. In short, such
processes are said to be "in statistical control".
Validity of
Engineering
Conclusions
Moreover, if the four assumptions are valid, then the process
is amenable to the generation of valid scientific and
engineering conclusions. If the four assumptions are not
valid, then the process is drifting (with respect to location,
variation, or distribution), unpredictable, and out of control.
A simple characterization of such processes by a location
estimate, a variation estimate, or a distribution "estimate"
inevitably leads to engineering conclusions that are not valid,
are not supportable (scientifically or legally), and which are
not repeatable in the laboratory.
Techniques for Testing Assumptions
Testing
Underlying
Assumptions
Helps Assure the
Validity of
Scientific and
Engineering
Conclusions
Because the validity of the final scientific/engineering
conclusions is inextricably linked to the validity of the
underlying univariate assumptions, it naturally follows that
there is a real necessity that each and every one of the
above four assumptions be routinely tested.
Four Techniques
to Test
Underlying
Assumptions
The following EDA techniques are simple, efficient, and
powerful for the routine testing of underlying assumptions:
1. run sequence plot (Yi versus i)
2. lag plot (Yi versus Yi-1)
3. histogram (counts versus subgroups of Y)
4. normal probability plot (ordered Y versus theoretical
ordered Y)
Plot on a Single
Page for a
Quick
Characterization
of the Data
The four EDA plots can be juxtaposed for a quick look at
the characteristics of the data. The plots below are ordered
as follows:
1. Run sequence plot - upper left
2. Lag plot - upper right
3. Histogram - lower left
4. Normal probability plot - lower right
Sample Plot:
Assumptions
Hold
This 4-plot of 500 normal random numbers reveals a
process that has fixed location, fixed variation, is random,

apparently has a fixed approximately normal distribution,
and has no outliers.
Sample Plot:
Assumptions Do
Not Hold
If one or more of the four underlying assumptions do not
hold, then it will show up in the various plots as
demonstrated in the following example.
This 4-plot reveals a process that has fixed location, fixed
variation, is non-random (oscillatory), has a non-normal,
U-shaped distribution, and has several outliers.
Interpretation of 4-Plot
Interpretation
of EDA
Plots:
Flat and
Equi-Banded,
Random,
Bell-Shaped,
and Linear
The four EDA plots discussed on the previous page are used
to test the underlying assumptions:
1. Fixed Location:
If the fixed location assumption holds, then the run
sequence plot will be flat and non-drifting.
2. Fixed Variation:
If the fixed variation assumption holds, then the
vertical spread in the run sequence plot will be the
approximately the same over the entire horizontal
axis.
3. Randomness:
If the randomness assumption holds, then the lag plot
will be structureless and random.
4. Fixed Distribution:
If the fixed distribution assumption holds, in particular
if the fixed normal distribution holds, then
1. the histogram will be bell-shaped, and
2. the normal probability plot will be linear.
Plots Utilized
to Test the
Assumptions
Conversely, the underlying assumptions are tested using the
EDA plots:
Run Sequence Plot:
If the run sequence plot is flat and non-drifting, the
fixed-location assumption holds. If the run sequence
plot has a vertical spread that is about the same over
the entire plot, then the fixed-variation assumption
holds.
Lag Plot:
If the lag plot is structureless, then the randomness
assumption holds.

Histogram:
If the histogram is bell-shaped, the underlying
distribution is symmetric and perhaps approximately
normal.
Normal Probability Plot:
If the normal probability plot is linear, the underlying
distribution is approximately normal.
If all four of the assumptions hold, then the process is said
definitionally to be "in statistical control".
Consequences
What If
Assumptions
Do Not Hold?
If some of the underlying assumptions do not hold, what can
be done about it? What corrective actions can be taken? The
positive way of approaching this is to view the testing of
underlying assumptions as a framework for learning about
the process. Assumption-testing promotes insight into
important aspects of the process that may not have surfaced
otherwise.
Primary Goal
is Correct
and Valid
Scientific
Conclusions
The primary goal is to have correct, validated, and complete
scientific/engineering conclusions flowing from the
analysis. This usually includes intermediate goals such as
the derivation of a good-fitting model and the computation
of realistic parameter estimates. It should always include the
ultimate goal of an understanding and a "feel" for "what
makes the process tick". There is no more powerful catalyst
for discovery than the bringing together of an
experienced/expert scientist/engineer and a data set ripe
with intriguing "anomalies" and characteristics.
Consequences
of Invalid
Assumptions
The following sections discuss in more detail the
consequences of invalid assumptions:
1. Consequences of non-randomness
2. Consequences of non-fixed location parameter
3. Consequences of non-fixed variation
4. Consequences related to distributional assumptions
Consequences of Non-Randomness
Randomness
Assumption
There are four underlying assumptions:
1. randomness;
2. fixed location;
3. fixed variation; and
4. fixed distribution.
The randomness assumption is the most critical but the
least tested.
Consequeces of
Non-
Randomness
If the randomness assumption does not hold, then
1. All of the usual statistical tests are invalid.
2. The calculated uncertainties for commonly used
statistics become meaningless.
3. The calculated minimal sample size required for a
pre-specified tolerance becomes meaningless.
4. The simple model: y = constant + error becomes
invalid.
5. The parameter estimates become suspect and non-
supportable.

Non-
Randomness
Due to
Autocorrelation
One specific and common type of non-randomness is
autocorrelation. Autocorrelation is the correlation between
Yt and Yt-k, where k is an integer that defines the lag for
the autocorrelation. That is, autocorrelation is a time
dependent non-randomness. This means that the value of
the current point is highly dependent on the previous point
if k = 1 (or k points ago if k is not 1). Autocorrelation is
typically detected via an autocorrelation plot or a lag plot.
If the data are not random due to autocorrelation, then
1. Adjacent data values may be related.
2. There may not be n independent snapshots of the
phenomenon under study.
3. There may be undetected "junk"-outliers.
4. There may be undetected "information-rich"-
outliers.
Consequences of Non-Fixed Location
Parameter
Location
Estimate
The usual estimate of location is the mean
$$bar{Y} = frac{1}{N} sum_{i=1}^{N}{Y_i}$$
from N measurements Y1, Y2, ... , YN.
Consequences
of Non-Fixed
Location
If the run sequence plot does not support the assumption of
fixed location, then
1. The location may be drifting.
2. The single location estimate may be meaningless (if
the process is drifting).
3. The choice of location estimator (e.g., the sample
mean) may be sub-optimal.
4. The usual formula for the uncertainty of the mean:
$$s(bar{Y}) = frac{1}{sqrt{N(N-1)}}
sqrt{sum_{i=1}^{N}{(Y_i - bar{Y})^2}}$$
may be invalid and the numerical value optimistically
small.
5. The location estimate may be poor.
6. The location estimate may be biased.
Consequences of Non-Fixed Variation
Parameter
Variation
Estimate
The usual estimate of variation is the standard deviation
$$s_Y = frac{1}{sqrt{(N-1)}}
sqrt{sum_{i=1}^{N} {(Y_i - bar{Y})^2}}$$
from N measurements Y1, Y2, ... , YN.

Consequences
of Non-Fixed
Variation
If the run sequence plot does not support the assumption of
fixed variation, then
1. The variation may be drifting.
2. The single variation estimate may be meaningless (if
the process variation is drifting).
3. The variation estimate may be poor.
4. The variation estimate may be biased.
Consequences Related to Distributional
Assumptions
Distributional
Analysis
Scientists and engineers routinely use the mean (average) to
estimate the "middle" of a distribution. It is not so well
known that the variability and the noisiness of the mean as a
location estimator are intrinsically linked with the
underlying distribution of the data. For certain distributions,
the mean is a poor choice. For any given distribution, there
exists an optimal choice-- that is, the estimator with
minimum variability/noisiness. This optimal choice may be,
for example, the median, the midrange, the midmean, the
mean, or something else. The implication of this is to
"estimate" the distribution first, and then--based on the
distribution--choose the optimal estimator. The resulting
engineering parameter estimators will have less variability
than if this approach is not followed.
Case Studies The airplane glass failure case study gives an example of
determining an appropriate distribution and estimating the
parameters of that distribution. The uniform random
numbers case study gives an example of determining a more
appropriate centrality parameter for a non-normal
distribution.
Other consequences that flow from problems with
distributional assumptions are:
Distribution 1. The distribution may be changing.
2. The single distribution estimate may be meaningless
(if the process distribution is changing).
3. The distribution may be markedly non-normal.
4. The distribution may be unknown.
5. The true probability distribution for the error may
remain unknown.
Model 1. The model may be changing.
2. The single model estimate may be meaningless.
3. The default model
Y = constant + error
may be invalid.
4. If the default model is insufficient, information about
a better model may remain undetected.
5. A poor deterministic model may be fit.
6. Information about an improved model may go
undetected.
Process 1. The process may be out-of-control.
2. The process may be unpredictable.
3. The process may be un-modelable.

EDA Techniques
Summary After you have collected a set of data, how do you do an
exploratory data analysis? What techniques do you employ?
What do the various techniques focus on? What conclusions
can you expect to reach?
This section provides answers to these kinds of questions via a
gallery of EDA techniques and a detailed description of each
technique. The techniques are divided into graphical and
quantitative techniques. For exploratory data analysis, the
emphasis is primarily on the graphical techniques.
Table of
Contents
for Section
3
1. Introduction
2. Analysis Questions
3. Graphical Techniques: Alphabetical
4. Graphical Techniques: By Problem Category
5. Quantitative Techniques: Alphabetical
6. Probability Distributions
Introduction
Graphical
and
Quantitative
Techniques
This section describes many techniques that are commonly
used in exploratory and classical data analysis. This list is by
no means meant to be exhaustive. Additional techniques (both
graphical and quantitative) are discussed in the other chapters.
Specifically, the product comparisons chapter has a much
more detailed description of many classical statistical
techniques.
EDA emphasizes graphical techniques while classical
techniques emphasize quantitative techniques. In practice, an
analyst typically uses a mixture of graphical and quantitative
techniques. In this section, we have divided the descriptions
into graphical and quantitative techniques. This is for
organizational clarity and is not meant to discourage the use
of both graphical and quantitiative techniques when analyzing
data.
Use of
Techniques
Shown in
Case
Studies
This section emphasizes the techniques themselves; how the
graph or test is defined, published references, and sample
output. The use of the techniques to answer engineering
questions is demonstrated in the case studies section. The case
studies do not demonstrate all of the techniques.
Availability
in Software
The sample plots and output in this section were generated
with the Dataplot software program. Other general purpose
statistical data analysis programs can generate most of the
plots, intervals, and tests discussed here, or macros can be
written to acheive the same result.
Analysis Questions
EDA
Questions
Some common questions that exploratory data analysis is used
to answer are:
1. What is a typical value?
2. What is the uncertainty for a typical value?
3. What is a good distributional fit for a set of numbers?
4. What is a percentile?
5. Does an engineering modification have an effect?
6. Does a factor have an effect?

7. What are the most important factors?
8. Are measurements coming from different laboratories
equivalent?
9. What is the best function for relating a response
variable to a set of factor variables?
10. What are the best settings for factors?
11. Can we separate signal from noise in time dependent
data?
12. Can we extract any structure from multivariate data?
13. Does the data have outliers?
Analyst
Should
Identify
Relevant
Questions
for his
Engineering
Problem
A critical early step in any analysis is to identify (for the
engineering problem at hand) which of the above questions
are relevant. That is, we need to identify which questions we
want answered and which questions have no bearing on the
problem at hand. After collecting such a set of questions, an
equally important step, which is invaluable for maintaining
focus, is to prioritize those questions in decreasing order of
importance. EDA techniques are tied in with each of the
questions. There are some EDA techniques (e.g., the scatter
plot) that are broad-brushed and apply almost universally. On
the other hand, there are a large number of EDA techniques
that are specific and whose specificity is tied in with one of
the above questions. Clearly if one chooses not to explicitly
identify relevant questions, then one cannot take advantage of
these question-specific EDA technqiues.
EDA
Approach
Emphasizes
Graphics
Most of these questions can be addressed by techniques
discussed in this chapter. The process modeling and process
improvement chapters also address many of the questions
above. These questions are also relevant for the classical
approach to statistics. What distinguishes the EDA approach
is an emphasis on graphical techniques to gain insight as
opposed to the classical approach of quantitative tests. Most
data analysts will use a mix of graphical and classical
quantitative techniques to address these problems.
Graphical Techniques: Alphabetic
This section provides a gallery of some useful graphical
techniques. The techniques are ordered alphabetically, so this
section is not intended to be read in a sequential fashion. The
use of most of these graphical techniques is demonstrated in
the case studies in this chapter. A few of these graphical
techniques are demonstrated in later chapters.
Autocorrelation
Plot: 1.3.3.1
Bihistogram:
1.3.3.2
Block Plot:
1.3.3.3
Bootstrap Plot:
1.3.3.4
Box-Cox
Linearity Plot:
1.3.3.5
Box-Cox
Normality Plot:
1.3.3.6
Box Plot: 1.3.3.7 Complex
Demodulation
Amplitude Plot:
1.3.3.8

Complex
Demodulation
Phase Plot:
1.3.3.9
Contour Plot:
1.3.3.10
DOE Scatter
Plot: 1.3.3.11
DOE Mean Plot:
1.3.3.12
DOE Standard
Deviation Plot:
1.3.3.13
Histogram:
1.3.3.14
Lag Plot:
1.3.3.15
Linear
Correlation Plot:
1.3.3.16
Linear Intercept
Plot: 1.3.3.17
Linear Slope
Plot: 1.3.3.18
Linear Residual
Standard
Deviation Plot:
1.3.3.19
Mean Plot:
1.3.3.20
Normal
Probability Plot:
1.3.3.21
Probability Plot:
1.3.3.22
Probability Plot
Correlation
Coefficient Plot:
1.3.3.23
Quantile-
Quantile Plot:
1.3.3.24
Run Sequence
Plot: 1.3.3.25
Scatter Plot:
1.3.3.26
Spectrum:
1.3.3.27
Standard
Deviation Plot:
1.3.3.28
Star Plot:
1.3.3.29
Weibull Plot:
1.3.3.30
Youden Plot:
1.3.3.31
4-Plot: 1.3.3.32
6-Plot: 1.3.3.33
Autocorrelation Plot
Purpose:
Check
Randomness
Autocorrelation plots (Box and Jenkins, pp. 28-32) are a
commonly-used tool for checking randomness in a data
set. This randomness is ascertained by computing
autocorrelations for data values at varying time lags. If
random, such autocorrelations should be near zero for any
and all time-lag separations. If non-random, then one or
more of the autocorrelations will be significantly non-zero.
In addition, autocorrelation plots are used in the model
identification stage for Box-Jenkins autoregressive,
moving average time series models.
Autocorrelation
is Only One
Note that uncorrelated does not necessarily mean random.
Data that has significant autocorrelation is not random.

Measure of
Randomness
However, data that does not show significant
autocorrelation can still exhibit non-randomness in other
ways. Autocorrelation is just one measure of randomness.
In the context of model validation (which is the primary
type of randomness we dicuss in the Handbook), checking
for autocorrelation is typically a sufficient test of
randomness since the residuals from a poor fitting models
tend to display non-subtle randomness. However, some
applications require a more rigorous determination of
randomness. In these cases, a battery of tests, which might
include checking for autocorrelation, are applied since data
can be non-random in many different and often subtle
ways.
An example of where a more rigorous check for
randomness is needed would be in testing random number
generators.
Sample Plot:
Autocorrelations
should be near-
zero for
randomness.
Such is not the
case in this
example and
thus the
randomness
assumption fails
This sample autocorrelation plot of the FLICKER.DAT
data set shows that the time series is not random, but rather
has a high degree of autocorrelation between adjacent and
near-adjacent observations.
Definition:
r(h) versus h
Autocorrelation plots are formed by
Vertical axis: Autocorrelation coefficient
[ R_{h} = C_{h}/C_{0} ]
where Ch is the autocovariance function
[ C_{h} = frac{1}{N}sum_{t=1}^{N-h}
(Y_{t} - bar{{Y}})(Y_{t+h} - bar{{Y}}) ]
and C0 is the variance function
[ C_{0} = frac{sum_{t=1}^{N}(Y_{t} -
bar{Y})^2}{N} ]
Note that Rh is between -1 and +1.
Note that some sources may use the following
formula for the autocovariance function
[ C_{h} = frac{1}{N-h}sum_{t=1}^{N-h}
(Y_{t} - bar{{Y}})(Y_{t+h} - bar{{Y}}) ]
Although this definition has less bias, the (1/N)
formulation has some desirable statistical properties
and is the form most commonly used in the statistics
literature. See pages 20 and 49-50 in Chatfield for
details.
Horizontal axis: Time lag h (h = 1, 2, 3, ...)

The above line also contains several horizontal
reference lines. The middle line is at zero. The other
four lines are 95 % and 99 % confidence bands.
Note that there are two distinct formulas for
generating the confidence bands.
1. If the autocorrelation plot is being used to test
for randomness (i.e., there is no time
dependence in the data), the following formula
is recommended:
[ pm frac{z_{1-alpha/2}} {sqrt{N}}
]
where N is the sample size, z is the cumulative
distribution function of the standard normal
distribution and ( alpha ) is the significance
level. In this case, the confidence bands have
fixed width that depends on the sample size.
This is the formula that was used to generate
the confidence bands in the above plot.
2. Autocorrelation plots are also used in the
model identification stage for fitting ARIMA
models. In this case, a moving average model
is assumed for the data and the following
confidence bands should be generated:
[ pm z_{1-alpha/2} sqrt{frac{1}{N}
(1 + 2 sum_{i=1}^{k}{y_{i}^2})} ]
where k is the lag, N is the sample size, z is the
cumulative distribution function of the
standard normal distribution and ( alpha ) is
the significance level. In this case, the
confidence bands increase as the lag increases.
Questions The autocorrelation plot can provide answers to the
following questions:
1. Are the data random?
2. Is an observation related to an adjacent observation?
3. Is an observation related to an observation twice-
removed? (etc.)
4. Is the observed time series white noise?
5. Is the observed time series sinusoidal?
6. Is the observed time series autoregressive?
7. What is an appropriate model for the observed time
series?
8. Is the model
valid and sufficient?
9. Is the formula [ s_{bar{{Y}}} = s/sqrt{N} ]
valid?
Importance:
Ensure validity
of engineering
conclusions
Randomness (along with fixed model, fixed variation, and
fixed distribution) is one of the four assumptions that
typically underlie all measurement processes. The
randomness assumption is critically important for the
following three reasons:
1. Most standard statistical tests depend on
randomness. The validity of the test conclusions is
directly linked to the validity of the randomness
assumption.
2. Many commonly-used statistical formulae depend
on the randomness assumption, the most common

formula being the formula for determining the
standard deviation of the sample mean:
[ s_{bar{{Y}}} = s/sqrt{N} ]
where s is the standard deviation of the data.
Although heavily used, the results from using this
formula are of no value unless the randomness
assumption holds.
3. For univariate data, the default model is
If the data are not random, this model is incorrect
and invalid, and the estimates for the parameters
(such as the constant) become nonsensical and
invalid.
In short, if the analyst does not check for randomness, then
the validity of many of the statistical conclusions becomes
suspect. The autocorrelation plot is an excellent way of
checking for such randomness.
Examples Examples of the autocorrelation plot for several common
situations are given in the following pages.
1. Random (= White Noise)
2. Weak autocorrelation
3. Strong autocorrelation and autoregressive model
4. Sinusoidal model
Related
Techniques
Partial Autocorrelation Plot
Lag Plot
Spectral Plot
Seasonal Subseries Plot
Case Study The autocorrelation plot is demonstrated in the beam
deflection data case study.
Software Autocorrelation plots are available in most general purpose
statistical software programs.
Autocorrelation Plot: Random Data
Autocorrelation
Plot
The following is a sample autocorrelation plot.
Conclusions We can make the following conclusions from this plot.
1. There are no significant autocorrelations.
2. The data are random.

Discussion Note that with the exception of lag 0, which is always 1 by
definition, almost all of the autocorrelations fall within the
95% confidence limits. In addition, there is no apparent
pattern (such as the first twenty-five being positive and the
second twenty-five being negative). This is the abscence of
a pattern we expect to see if the data are in fact random.
A few lags slightly outside the 95% and 99% confidence
limits do not neccessarily indicate non-randomness. For a
95% confidence interval, we might expect about one out of
twenty lags to be statistically significant due to random
fluctuations.
There is no associative ability to infer from a current value
Yi as to what the next value Yi+1 will be. Such non-
association is the essense of randomness. In short, adjacent
observations do not "co-relate", so we call this the "no
autocorrelation" case.
Autocorrelation Plot: Moderate Autocorrelation
Autocorrelation
Plot
The following is a sample autocorrelation plot of the
FLICKER.DAT data set.
Conclusions We can make the following conclusions from this plot.
1. The data come from an underlying autoregressive
model with moderate positive autocorrelation.
Discussion The plot starts with a moderately high autocorrelation at
lag 1 (approximately 0.75) that gradually decreases. The
decreasing autocorrelation is generally linear, but with
significant noise. Such a pattern is the autocorrelation plot
signature of "moderate autocorrelation", which in turn
provides moderate predictability if modeled properly.
Recommended
Next Step
The next step would be to estimate the parameters for the
autoregressive model:
[ Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} ]
Such estimation can be performed by using least squares
linear regression or by fitting a Box-Jenkins autoregressive
(AR) model.
The randomness assumption for least squares fitting applies
to the residuals of the model. That is, even though the
original data exhibit non-randomness, the residuals after
fitting Yi against Yi-1 should result in random residuals.

Assessing whether or not the proposed model in fact
sufficiently removed the randomness is discussed in detail
in the Process Modeling chapter.
The residual standard deviation for this autoregressive
model will be much smaller than the residual standard
deviation for the default model
[ Y_{i} = A_0 + E_{i} ]
Autocorrelation Plot: Strong Autocorrelation
and Autoregressive Model
Autocorrelation
Plot for Strong
Autocorrelation
The following is a sample autocorrelation plot of a random
walk data set.
Conclusions We can make the following conclusions from the above
plot.
model with strong positive autocorrelation.
Discussion The plot starts with a high autocorrelation at lag 1 (only
slightly less than 1) that slowly declines. It continues
decreasing until it becomes negative and starts showing an
incresing negative autocorrelation. The decreasing
autocorrelation is generally linear with little noise. Such a
pattern is the autocorrelation plot signature of "strong
autocorrelation", which in turn provides high predictability
if modeled properly.
Recommended
Next Step
The next step would be to estimate the parameters for the
[ Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} ]
Such estimation can be performed by using least squares
linear regression or by fitting a Box-Jenkins autoregressive
(AR) model.
The randomness assumption for least squares fitting
applies to the residuals of the model. That is, even though
the original data exhibit non-randomness, the residuals
after fitting Yi against Yi-1 should result in random
residuals. Assessing whether or not the proposed model in
fact sufficiently removed the randomness is discussed in
detail in the Process Modeling chapter.

[ Y_{i} = A_0 + E_{i} ]
Autocorrelation Plot: Sinusoidal Model
Autocorrelation
Plot for
Sinusoidal
Model
The following is a sample autocorrelation plot of the
LEW.DAT data set.
Conclusions We can make the following conclusions from the above
plot.
1. The data come from an underlying sinusoidal model.
Discussion The plot exhibits an alternating sequence of positive and
negative spikes. These spikes are not decaying to zero.
Such a pattern is the autocorrelation plot signature of a
sinusoidal model.
Recommended
Next Step
The beam deflection case study gives an example of
modeling a sinusoidal model.
Bihistogram
Purpose:
Check for a
change in
location,
variation, or
distribution
The bihistogram is an EDA tool for assessing whether a
before-versus-after engineering modification has caused a
change in
location;
variation; or
distribution.
It is a graphical alternative to the two-sample t-test. The
bihistogram can be more powerful than the t-test in that all of
the distributional features (location, scale, skewness, outliers)
are evident on a single plot. It is also based on the common
and well-understood histogram.

Sample Plot:
This
bihistogram
reveals that
there is a
significant
difference in
ceramic
breaking
strength
between
batch 1
(above) and
batch 2
(below)
From the above bihistogram of the JAHANMI2.DAT data
set, we can see that batch 1 is centered at a ceramic strength
value of approximately 725 while batch 2 is centered at a
ceramic strength value of approximately 625. That indicates
that these batches are displaced by about 100 strength units.
Thus the batch factor has a significant effect on the location
(typical value) for strength and hence batch is said to be
"significant" or to "have an effect". We thus see graphically
and convincingly what a t-test or analysis of variance would
indicate quantitatively.
With respect to variation, note that the spread (variation) of
the above-axis batch 1 histogram does not appear to be that
much different from the below-axis batch 2 histogram. With
respect to distributional shape, note that the batch 1
histogram is skewed left while the batch 2 histogram is more
symmetric with even a hint of a slight skewness to the right.
Thus the bihistogram reveals that there is a clear difference
between the batches with respect to location and distribution,
but not in regard to variation. Comparing batch 1 and batch
2, we also note that batch 1 is the "better batch" due to its
100-unit higher average strength (around 725).
Definition:
Two
adjoined
histograms
Bihistograms are formed by vertically juxtaposing two
histograms:
Above the axis: Histogram of the response variable for
condition 1
Below the axis: Histogram of the response variable for
condition 2
Questions The bihistogram can provide answers to the following
questions:
1. Is a (2-level) factor significant?
2. Does a (2-level) factor have an effect?
3. Does the location change between the 2 subgroups?
4. Does the variation change between the 2 subgroups?
5. Does the distributional shape change between
subgroups?
6. Are there any outliers?
Importance:
Checks 3 out
of the 4
underlying
assumptions
of a
measurement
process
The bihistogram is an important EDA tool for determining if
a factor "has an effect". Since the bihistogram provides
insight into the validity of three (location, variation, and
distribution) out of the four (missing only randomness)
underlying assumptions in a measurement process, it is an
especially valuable tool. Because of the dual (above/below)
nature of the plot, the bihistogram is restricted to assessing
factors that have only two levels. However, this is very
common in the before-versus-after character of many
scientific and engineering experiments.

Related
Techniques
t test (for shift in location)
F test (for shift in variation)
Kolmogorov-Smirnov test (for shift in distribution)
Quantile-quantile plot (for shift in location and distribution)
Case Study The bihistogram is demonstrated in the ceramic strength data
case study.
Software The bihistogram is not widely available in general purpose
statistical software programs. Bihistograms can be generated
using Dataplot and R software.
Block Plot
Purpose:
Check to
determine if
a factor of
interest has
an effect
robust over
all other
factors
The block plot (Filliben 1993) is an EDA tool for assessing
whether the factor of interest (the primary factor) has a
statistically significant effect on the response, and whether
that conclusion about the primary factor effect is valid
robustly over all other nuisance or secondary factors in the
experiment.
It replaces the analysis of variance test with a less assumption-
dependent binomial test and should be routinely used
whenever we are trying to robustly decide whether a primary
factor has an effect.
Sample
Plot:
Weld
method 2 is
lower
(better)
than weld
method 1 in
10 of 12
cases
This block plot of the SHEESLE2.DAT data set reveals that in
10 of the 12 cases (bars), weld method 2 is lower (better) than
weld method 1. From a binomial point of view, weld method
is statistically significant.
Definition Block Plots are formed as follows:
Vertical axis: Response variable Y
Horizontal axis: All combinations of all levels of all
nuisance (secondary) factors X1, X2, ...
Plot Character: Levels of the primary factor XP
Discussion:
Primary
factor is
denoted by
plot
character:
within-bar
plot
character.
Average number of defective lead wires per hour from a study
with four factors,
1. weld method (2 levels)
2. plant (2 levels)
3. speed (2 levels)
4. shift (3 levels)
are shown in the plot above. Weld method is the primary
factor and the other three factors are nuisance factors. The 12
distinct positions along the horizontal axis correspond to all
possible combinations of the three nuisance factors, i.e., 12 =
2 plants x 2 speeds x 3 shifts. These 12 conditions provide the

framework for assessing whether any conclusions about the 2
levels of the primary factor (weld method) can truly be called
"general conclusions". If we find that one weld method setting
does better (smaller average defects per hour) than the other
weld method setting for all or most of these 12 nuisance factor
combinations, then the conclusion is in fact general and
robust.
Ordering
along the
horizontal
axis
In the above chart, the ordering along the horizontal axis is as
follows:
The left 6 bars are from plant 1 and the right 6 bars are
from plant 2.
The first 3 bars are from speed 1, the next 3 bars are
from speed 2, the next 3 bars are from speed 1, and the
last 3 bars are from speed 2.
Bars 1, 4, 7, and 10 are from the first shift, bars 2, 5, 8,
and 11 are from the second shift, and bars 3, 6, 9, and
12 are from the third shift.
Setting 2 is
better than
setting 1 in
10 out of 12
cases
In the block plot for the first bar (plant 1, speed 1, shift 1),
weld method 1 yields about 28 defects per hour while weld
method 2 yields about 22 defects per hour--hence the
difference for this combination is about 6 defects per hour and
weld method 2 is seen to be better (smaller number of defects
per hour).
Is "weld method 2 is better than weld method 1" a general
conclusion?
For the second bar (plant 1, speed 1, shift 2), weld method 1 is
about 37 while weld method 2 is only about 18. Thus weld
method 2 is again seen to be better than weld method 1.
Similarly for bar 3 (plant 1, speed 1, shift 3), we see weld
method 2 is smaller than weld method 1. Scanning over all of
the 12 bars, we see that weld method 2 is smaller than weld
method 1 in 10 of the 12 cases, which is highly suggestive of
a robust weld method effect.
An event
with chance
probability
of only 2%
What is the chance of 10 out of 12 happening by chance? This
is probabilistically equivalent to testing whether a coin is fair
by flipping it and getting 10 heads in 12 tosses. The chance
(from the binomial distribution) of getting 10 (or more
extreme: 11, 12) heads in 12 flips of a fair coin is about 2%.
Such low-probability events are usually rejected as untenable
and in practice we would conclude that there is a difference in
weld methods.
Advantage:
Graphical
and
binomial
The advantages of the block plot are as follows:
A quantitative procedure (analysis of variance) is
replaced by a graphical procedure.
An F-test (analysis of variance) is replaced with a
binomial test, which requires fewer assumptions.
Questions The block plot can provide answers to the following
questions:
1. Is the factor of interest significant?
2. Does the factor of interest have an effect?
3. Does the location change between levels of the primary
factor?
4. Has the process improved?
5. What is the best setting (= level) of the primary factor?
6. How much of an average improvement can we expect
with this best setting of the primary factor?
7. Is there an interaction between the primary factor and
one or more nuisance factors?
8. Does the effect of the primary factor change depending
on the setting of some nuisance factor?

Importance:
Robustly
checks the
significance
of the factor
of interest
The block plot is a graphical technique that pointedly focuses
on whether or not the primary factor conclusions are in fact
robustly general. This question is fundamentally different
from the generic multi-factor experiment question where the
analyst asks, "What factors are important and what factors are
not" (a screening problem)? Global data analysis techniques,
such as analysis of variance, can potentially be improved by
local, focused data analysis techniques that take advantage of
this difference.
Related
Techniques
t test (for shift in location for exactly 2 levels)
ANOVA (for shift in location for 2 or more levels)
Bihistogram (for shift in location, variation, and distribution
for exactly 2 levels).
Case Study The block plot is demonstrated in the ceramic strength data
case study.
Software Block plots are not currently available in most general
purpose statistical software programs. However they can be
generated using Dataplot and, with some programming, R
software.
Youden Plot
Purpose:
Interlab
Comparisons
Youden plots are a graphical technique for analyzing interlab
data when each lab has made two runs on the same product
or one run on two different products.
The Youden plot is a simple but effective method for
comparing both the within-laboratory variability and the
between-laboratory variability.
Sample Plot
This plot shows:
1. Not all labs are equivalent.
2. Lab 4 is biased low.
3. Lab 3 has within-lab variability problems.
4. Lab 5 has an outlying run.
Definition:
Response 1
Versus
Response 2
Coded by
Lab
Youden plots are formed by:
1. Vertical axis: Response variable 1 (i.e., run 1 or
product 1 response value)
2. Horizontal axis: Response variable 2 (i.e., run 2 or

In addition, the plot symbol is the lab id (typically an integer
from 1 to k where k is the number of labs). Sometimes a 45-
degree reference line is drawn. Ideally, a lab generating two
runs of the same product should produce reasonably similar
results. Departures from this reference line indicate
inconsistency from the lab. If two different products are
being tested, then a 45-degree line may not be appropriate.
However, if the labs are consistent, the points should lie near
some fitted straight line.
Questions The Youden plot can be used to answer the following
questions:
1. Are all labs equivalent?
2. What labs have between-lab problems
(reproducibility)?
3. What labs have within-lab problems (repeatability)?
4. What labs are outliers?
Importance In interlaboratory studies or in comparing two runs from the
same lab, it is useful to know if consistent results are
generated. Youden plots should be a routine plot for
analyzing this type of data.
DOE Youden
Plot
The DOE Youden plot is a specialized Youden plot used in
the design of experiments. In particular, it is useful for full
and fractional designs.
Related
Techniques
Scatter Plot
Software The Youden plot is essentially a scatter plot, so it should be
feasible to write a macro for a Youden plot in any general
purpose statistical program that supports scatter plots.
DOE Youden Plot
DOE Youden
Plot:
Introduction
The DOE (Design of Experiments) Youden plot is a specialized Youden plot used in the
analysis of full and fractional experiment designs. In particular, it is used in conjunction with
the Yates algorithm. These designs may have a low level, coded as "-1" or "-", and a high
level, coded as "+1" or "+", for each factor. In addition, there can optionally be one or more
center points. Center points are at the midpoint between the low and high levels for each
factor and are coded as "0".
The Yates agorithm and the the DOE Youden plot only use the "-1" and "+1" points. The
Yates agorithm is used to estimate factor effects. The DOE Youden plot can be used to help
determine the approriate model to based on the effect estimates from the Yates algorithm.
Construction
of DOE
Youden Plot
The following are the primary steps in the construction of the DOE Youden plot.
1. For a given factor or interaction term, compute the mean of the response variable for
the low level of the factor and for the high level of the factor. Any center points are
omitted from the computation.
2. Plot the point where the y-coordinate is the mean for the high level of the factor and
the x-coordinate is the mean for the low level of the factor. The character used for the
plot point should identify the factor or interaction term (e.g., "1" for factor 1, "13" for
the interaction between factors 1 and 3).
3. Repeat steps 1 and 2 for each factor and interaction term of the data.
The high and low values of the interaction terms are obtained by multiplying the
corresponding values of the main level factors. For example, the interaction term X13 is
obtained by multiplying the values for X1 with the corresponding values of X3. Since the
values for X1 and X3 are either "-1" or "+1", the resulting values for X13 are also either "-1"
or "+1".

In summary, the DOE Youden plot is a plot of the mean of the response variable for the high
level of a factor or interaction term against the mean of the response variable for the low
level of that factor or interaction term.
For unimportant factors and interaction terms, these mean values should be nearly the same.
For important factors and interaction terms, these mean values should be quite different. So
the interpretation of the plot is that unimportant factors should be clustered together near the
grand mean. Points that stand apart from this cluster identify important factors that should be
included in the model.
Sample DOE
Youden Plot
The following is a DOE Youden plot for the data used in the Eddy current case study. The
analysis in that case study demonstrated that X1 and X2 were the most important factors.
Interpretation
of the Sample
DOE Youden
Plot
From the above DOE Youden plot, we see that factors 1 and 2 stand out from the others. That
is, the mean response values for the low and high levels of factor 1 and factor 2 are quite
different. For factor 3 and the 2 and 3-term interactions, the mean response values for the low
and high levels are similar.
We would conclude from this plot that factors 1 and 2 are important and should be included
in our final model while the remaining factors and interactions should be omitted from the
final model.
Case Study The Eddy current case study demonstrates the use of the DOE Youden plot in the context of
the analysis of a full factorial design.
Software DOE Youden plots are not typically available as built-in plots in statistical software
programs. However, it should be relatively straightforward to write a macro to generate this
plot in most general purpose statistical software programs.
4-Plot
Purpose:
Check
Underlying
The 4-plot is a collection of 4 specific EDA graphical
techniques whose purpose is to test the assumptions that
underlie most measurement processes. A 4-plot consists of a

Statistical
Assumptions
1. run sequence plot;
2. lag plot;
3. histogram;
4. normal probability plot.
If the 4 underlying assumptions of a typical measurement
process hold, then the above 4 plots will have a
characteristic appearance (see the normal random numbers
case study below); if any of the underlying assumptions fail
to hold, then it will be revealed by an anomalous
appearance in one or more of the plots. Several commonly
encountered situations are demonstrated in the case studies
below.
Although the 4-plot has an obvious use for univariate and
time series data, its usefulness extends far beyond that.
Many statistical models of the form
[ Y_{i} = f(X_{1}, ... , X_{k}) + E_{i} ]
have the same underlying assumptions for the error term.
That is, no matter how complicated the functional fit, the
assumptions on the underlying error term are still the same.
The 4-plot can and should be routinely applied to the
residuals when fitting models regardless of whether the
model is simple or complicated.
Sample Plot:
Process Has
Fixed
Location,
Fixed
Variation,
Non-Random
(Oscillatory),
Non-Normal
U-Shaped
Distribution,
and Has 3
Outliers.
This 4-plot reveals the following:
1. the fixed location assumption is justified as shown by
the run sequence plot in the upper left corner.
2. the fixed variation assumption is justified as shown
by the run sequence plot in the upper left corner.
3. the randomness assumption is violated as shown by
the non-random (oscillatory) lag plot in the upper
right corner.
4. the assumption of a common, normal distribution is
violated as shown by the histogram in the lower left
corner and the normal probability plot in the lower
right corner. The distribution is non-normal and is a
U-shaped distribution.
5. there are several outliers apparent in the lag plot in
the upper right corner.
Definition:
1. Run
Sequence
Plot;
2. Lag Plot;
3. Histogram;
4. Normal
Probability
Plot
The 4-plot consists of the following:
1. Run sequence plot to test fixed location and variation.
Vertically: Yi
Horizontally: i
2. Lag Plot to test randomness.
Vertically: Yi
Horizontally: Yi-1
3. Histogram to test (normal) distribution.

Vertically: Counts
Horizontally: Y
4. Normal probability plot to test normal distribution.
Vertically: Ordered Yi
Horizontally: Theoretical values from a normal
N(0,1) distribution for ordered Yi
Questions 4-plots can provide answers to many questions:
1. Is the process in-control, stable, and predictable?
2. Is the process drifting with respect to location?
3. Is the process drifting with respect to variation?
6. If the data are a time series, is is white noise?
7. If the data are a time series and not white noise, is it
sinusoidal, autoregressive, etc.?
8. If the data are non-random, what is a better model?
9. Does the process follow a normal distribution?
10. If non-normal, what distribution does the process
follow?
11. Is the model
[ Y_{i} = A_0 + E_{i} ]
12. If the default model is insufficient, what is a better
model?
13. Is the formula ( s_{bar{Y}} = s/sqrt{N} )valid?
14. Is the sample mean a good estimator of the process
location?
15. If not, what would be a better estimator?
Importance:
Testing
Underlying
Assumptions
Helps Ensure
the Validity of
the Final
Scientific and
Engineering
Conclusions
There are 4 assumptions that typically underlie all
1. random drawings;
3. with that distribution having a fixed location; and
4. with that distribution having fixed variation.
engineering. If the above 4 assumptions hold, then we have
achieved probabilistic predictability--the ability to make
probability statements not only about the process in the
processes are said to be "statistically in control". If the 4
assumptions do not hold, then we have a process that is
drifting (with respect to location, variation, or distribution),
is unpredictable, and is out of control. A simple
characterization of such processes by a location estimate, a
variation estimate, or a distribution "estimate" inevitably
leads to optimistic and grossly invalid engineering
conclusions.
Inasmuch as the validity of the final scientific and
engineering conclusions is inextricably linked to the
validity of these same 4 underlying assumptions, it
naturally follows that there is a real necessity for all 4
assumptions to be routinely tested. The 4-plot (run
sequence plot, lag plot, histogram, and normal probability
plot) is seen as a simple, efficient, and powerful way of
carrying out this routine checking.
Interpretation:
Flat, Equi-
Banded,
Of the 4 underlying assumptions:

Random, Bell-
Shaped, and
Linear
1. If the fixed location assumption holds, then the run
2. If the fixed variation assumption holds, then the
vertical spread in the run sequence plot will be
axis.
3. If the randomness assumption holds, then the lag plot
4. If the fixed distribution assumption holds (in
particular, if the fixed normal distribution assumption
holds), then the histogram will be bell-shaped and the
normal probability plot will be approximatelylinear.
If all 4 of the assumptions hold, then the process is
"statistically in control". In practice, many processes fall
short of achieving this ideal.
Related
Techniques
Run Sequence Plot
Lag Plot
Histogram
Normal Probability Plot
Spectral Plot
PPCC Plot
Case Studies The 4-plot is used in most of the case studies in this
chapter:
1. Normal random numbers (the ideal)
2. Uniform random numbers
3. Random walk
4. Josephson junction cryothermometry
5. Beam deflections
6. Filter transmittance
7. Standard resistor
8. Heat flow meter 1
Software It should be feasible to write a macro for the 4-plot in any
general purpose statistical software program that supports
the capability for multiple plots per page and supports the
underlying plot techniques.
6-Plot
Purpose:
Graphical
Model
Validation
The 6-plot is a collection of 6 specific graphical techniques
whose purpose is to assess the validity of a Y versus X fit. The
fit can be a linear fit, a non-linear fit, a LOWESS (locally
weighted least squares) fit, a spline fit, or any other fit
utilizing a single independent variable.
The 6 plots are:
1. Scatter plot of the response and predicted values versus
the independent variable;
2. Scatter plot of the residuals versus the independent
variable;
3. Scatter plot of the residuals versus the predicted values;
4. Lag plot of the residuals;
5. Histogram of the residuals;
6. Normal probability plot of the residuals.

Sample Plot
This 6-plot, which followed a linear fit, shows that the linear
model is not adequate. It suggests that a quadratic model
would be a better model.
Definition:
6
Component
Plots
1. Response and predicted values
Vertical axis: Response variable, predicted values
Horizontal axis: Independent variable
2. Residuals versus independent variable
Vertical axis: Residuals
3. Residuals versus predicted values
Horizontal axis: Predicted values
4. Lag plot of residuals
Vertical axis: RES(I)
Horizontal axis: RES(I-1)
5. Histogram of residuals
Vertical axis: Counts
Horizontal axis: Residual values
6. Normal probability plot of residuals
Vertical axis: Ordered residuals
Horizontal axis: Theoretical values from a normal
N(0,1) distribution for ordered residuals
Questions The 6-plot can be used to answer the following questions:
1. Are the residuals approximately normally distributed
with a fixed location and scale?
2. Are there outliers?
3. Is the fit adequate?
4. Do the residuals suggest a better fit?
Importance:
Validating
Model
A model involving a response variable and a single
independent variable has the form:
[ Y_{i} = f(X_{i}) + E_{i} ]
where Y is the response variable, X is the independent
variable, f is the linear or non-linear fit function, and E is the
random component. For a good model, the error component
should behave like:
1. random drawings (i.e., independent);
3. with fixed location; and
4. with fixed variation.
In addition, for fitting models it is usually further assumed
that the fixed distribution is normal and the fixed location is
zero. For a good model the fixed variation should be as small
as possible. A necessary component of fitting models is to

verify these assumptions for the error component and to
assess whether the variation for the error component is
sufficiently small. The histogram, lag plot, and normal
probability plot are used to verify the fixed distribution,
location, and variation assumptions on the error component.
The plot of the response variable and the predicted values
versus the independent variable is used to assess whether the
variation is sufficiently small. The plots of the residuals
versus the independent variable and the predicted values is
used to assess the independence assumption.
Assessing the validity and quality of the fit in terms of the
above assumptions is an absolutely vital part of the model-
fitting process. No fit should be considered complete without
an adequate model validation step.
Related
Techniques
Linear Least Squares
Non-Linear Least Squares
Scatter Plot
Run Sequence Plot
Lag Plot
Histogram
Case Study The 6-plot is used in the Alaska pipeline data case study.
general purpose statistical software program that supports the
capability for multiple plots per page and supports the
Bootstrap Plot
Purpose:
Estimate
uncertainty
The bootstrap (Efron and Gong) plot is used to estimate the
uncertainty of a statistic.
Generate
subsamples
with
replacement
To generate a bootstrap uncertainty estimate for a given
statistic from a set of data, a subsample of a size less than or
equal to the size of the data set is generated from the data, and
the statistic is calculated. This subsample is generated with
replacement so that any data point can be sampled multiple
times or not sampled at all. This process is repeated for many
subsamples, typically between 500 and 1000. The computed
values for the statistic form an estimate of the sampling
distribution of the statistic.
For example, to estimate the uncertainty of the median from a
dataset with 50 elements, we generate a subsample of 50
elements and calculate the median. This is repeated at least
500 times so that we have at least 500 values for the median.
Although the number of bootstrap samples to use is somewhat
arbitrary, 500 subsamples is usually sufficient. To calculate a
90% confidence interval for the median, the sample medians
are sorted into ascending order and the value of the 25th
median (assuming exactly 500 subsamples were taken) is the
lower confidence limit while the value of the 475th median
(assuming exactly 500 subsamples were taken) is the upper
confidence limit.

Sample
Plot:
This bootstrap plot was generated from 500 uniform random
numbers. Bootstrap plots and corresponding histograms were
generated for the mean, median, and mid-range. The
histograms for the corresponding statistics clearly show that
for uniform random numbers the mid-range has the smallest
variance and is, therefore, a superior location estimator to the
mean or the median.
Definition The bootstrap plot is formed by:
Vertical axis: Computed value of the desired statistic for
a given subsample.
Horizontal axis: Subsample number.
The bootstrap plot is simply the computed value of the
statistic versus the subsample number. That is, the bootstrap
plot generates the values for the desired statistic. This is
usually immediately followed by a histogram or some other
distributional plot to show the location and variation of the
sampling distribution of the statistic.
Questions The bootstrap plot is used to answer the following questions:
What does the sampling distribution for the statistic
look like?
What is a 95% confidence interval for the statistic?
Which statistic has a sampling distribution with the
smallest variance? That is, which statistic generates the
narrowest confidence interval?
Importance The most common uncertainty calculation is generating a
confidence interval for the mean. In this case, the uncertainty
formula can be derived mathematically. However, there are
many situations in which the uncertainty formulas are
mathematically intractable. The bootstrap provides a method
for calculating the uncertainty in these cases.
Cautuion
on use of
the
bootstrap
The bootstrap is not appropriate for all distributions and
statistics (Efron and Tibrashani). For example, because of the
shape of the uniform distribution, the bootstrap is not
appropriate for estimating the distribution of statistics that are
heavily dependent on the tails, such as the range.
Related
Techniques
Histogram
Jackknife
The jacknife is a technique that is closely related to the
bootstrap. The jackknife is beyond the scope of this handbook.
See the Efron and Gong article for a discussion of the
jackknife.
Case Study The bootstrap plot is demonstrated in the uniform random
numbers case study.

Software The bootstrap is becoming more common in general purpose
statistical software programs. However, it is still not supported
in many of these programs. Both R software and Dataplot
support a bootstrap capability.
Box-Cox Linearity Plot
Purpose:
Find the
transformation
of the X
variable that
maximizes the
correlation
between a Y
and an X
variable
When performing a linear fit of Y against X, an appropriate
transformation of X can often significantly improve the fit.
The Box-Cox transformation (Box and Cox, 1964) is a
particularly useful family of transformations. It is defined
as:
[ T(X) = (X^{lambda} - 1)/lambda ]
where X is the variable being transformed and ( lambda )
is the transformation parameter. For ( lambda ) = 0, the
natural log of the data is taken instead of using the above
formula.
The Box-Cox linearity plot is a plot of the correlation
between Y and the transformed X for given values of (
lambda ). That is, ( lambda ) is the coordinate for the
horizontal axis variable and the value of the correlation
between Y and the transformed X is the coordinate for the
vertical axis of the plot. The value of ( lambda )
corresponding to the maximum correlation (or minimum
for negative correlation) on the plot is then the optimal
choice for ( lambda ).
Transforming X is used to improve the fit. The Box-Cox
transformation applied to Y can be used as the basis for
meeting the error assumptions. That case is not covered
here. See page 225 of (Draper and Smith, 1981) or page 77
of (Ryan, 1997) for a discussion of this case.
Sample Plot
The plot of the original data with the predicted values from
a linear fit indicate that a quadratic fit might be preferable.
The Box-Cox linearity plot shows a value of ( lambda ) =
2.0. The plot of the transformed data with the predicted
values from a linear fit with the transformed data shows a
better fit (verified by the significant reduction in the
residual standard deviation).
Definition Box-Cox linearity plots are formed by
Vertical axis: Correlation coefficient from the
transformed X and Y
Horizontal axis: Value for ( lambda )

Questions The Box-Cox linearity plot can provide answers to the
1. Would a suitable transformation improve my fit?
2. What is the optimal value of the transformation
parameter?
Importance:
Find a
suitable
transformation
Transformations can often significantly improve a fit. The
Box-Cox linearity plot provides a convenient way to find a
suitable transformation without engaging in a lot of trial
and error fitting.
Related
Techniques
Linear Regression
Box-Cox Normality Plot
Case Study The Box-Cox linearity plot is demonstrated in the Alaska
pipeline data case study.
Software Box-Cox linearity plots are not a standard part of most
general purpose statistical software programs. However, the
underlying technique is based on a transformation and
computing a correlation coefficient. So if a statistical
program supports these capabilities, writing a macro for a
Box-Cox linearity plot should be feasible.
Box-Cox Normality Plot
Purpose:
Find
transformation
to normalize
data
Many statistical tests and intervals are based on the
assumption of normality. The assumption of normality
often leads to tests that are simple, mathematically
tractable, and powerful compared to tests that do not make
the normality assumption. Unfortunately, many real data
sets are in fact not approximately normal. However, an
appropriate transformation of a data set can often yield a
data set that does follow approximately a normal
distribution. This increases the applicability and usefulness
of statistical techniques based on the normality assumption.
The Box-Cox transformation is a particulary useful family
of transformations. It is defined as:
[ T(Y) = (Y^{lambda} - 1)/lambda ]
where Y is the response variable and ( lambda ) is the
transformation parameter. For ( lambda ) = 0, the natural
log of the data is taken instead of using the above formula.
Given a particular transformation such as the Box-Cox
transformation defined above, it is helpful to define a
measure of the normality of the resulting transformation.
One measure is to compute the correlation coefficient of a
normal probability plot. The correlation is computed
between the vertical and horizontal axis variables of the
probability plot and is a convenient measure of the linearity
of the probability plot (the more linear the probability plot,
the better a normal distribution fits the data).
The Box-Cox normality plot is a plot of these correlation
coefficients for various values of the ( lambda )
parameter. The value of ( lambda ) corresponding to the
maximum correlation on the plot is then the optimal choice
for ( lambda ).

Sample Plot
The histogram in the upper left-hand corner shows a data
set (first column) that has significant right skewness (and
so does not follow a normal distribution). The Box-Cox
normality plot shows that the maximum value of the
correlation coefficient is at ( lambda ) = -0.3. The
histogram of the data after applying the Box-Cox
transformation with ( lambda ) = -0.3 shows a data set for
which the normality assumption is reasonable. This is
verified with a normal probability plot of the transformed
data.
Definition Box-Cox normality plots are formed by:
Vertical axis: Correlation coefficient from the normal
probability plot after applying Box-Cox
transformation
Horizontal axis: Value for ( lambda )
Questions The Box-Cox normality plot can provide answers to the
1. Is there a transformation that will normalize my data?
2. What is the optimal value of the transformation
parameter?
Importance:
Normalization
Improves
Validity of
Tests
Normality assumptions are critical for many univariate
intervals and hypothesis tests. It is important to test the
normality assumption. If the data are in fact clearly not
normal, the Box-Cox normality plot can often be used to
find a transformation that will approximately normalize the
data.
Related
Techniques
Box-Cox Linearity Plot
Software Box-Cox normality plots are not a standard part of most
underlying technique is based on a normal probability plot
and computing a correlation coefficient. So if a statistical
program supports these capabilities, writing a macro for a
Box-Cox normality plot should be feasible.
Box Plot
Purpose:
Check
location
and
variation
shifts
Box plots (Chambers 1983) are an excellent tool for
conveying location and variation information in data sets,
particularly for detecting and illustrating location and
variation changes between different groups of data.

Sample
Plot:
This box
plot reveals
that
machine
has a
significant
effect on
energy with
respect to
location
and
possibly
variation
This box plot, comparing four machines for energy output,
shows that machine has a significant effect on energy with
respect to both location and variation. Machine 3 has the
highest energy response (about 72.5); machine 4 has the least
variable energy response with about 50% of its readings being
within 1 energy unit.
Definition Box plots are formed by
Vertical axis: Response variable Horizontal axis: The
factor of interest
More specifically, we
1. Calculate the median and the quartiles (the lower
quartile is the 25th percentile and the upper quartile is
the 75th percentile).
2. Plot a symbol at the median (or draw a line) and draw a
box (hence the name--box plot) between the lower and
upper quartiles; this box represents the middle 50% of
the data--the "body" of the data.
3. Draw a line from the lower quartile to the minimum
point and another line from the upper quartile to the
maximum point. Typically a symbol is drawn at these
minimum and maximum points, although this is
optional.
Thus the box plot identifies the middle 50% of the data, the
median, and the extreme points.
Single or
multiple
box plots
can be
drawn
A single box plot can be drawn for one batch of data with no
distinct groups. Alternatively, multiple box plots can be drawn
together to compare multiple data sets or to compare groups in
a single data set. For a single box plot, the width of the box is
arbitrary. For multiple box plots, the width of the box plot can
be set proportional to the number of points in the given group
or sample (some software implementations of the box plot
simply set all the boxes to the same width).
Box plots
with fences
There is a useful variation of the box plot that more
specifically identifies outliers. To create this variation:
1. Calculate the median and the lower and upper quartiles.
2. Plot a symbol at the median and draw a box between
the lower and upper quartiles.
3. Calculate the interquartile range (the difference between
the upper and lower quartile) and call it IQ.
4. Calculate the following points:
L1 = lower quartile - 1.5*IQ L2 = lower quartile -
3.0*IQ U1 = upper quartile + 1.5*IQ U2 = upper

quartile + 3.0*IQ
5. The line from the lower quartile to the minimum is now
drawn from the lower quartile to the smallest point that
is greater than L1. Likewise, the line from the upper
quartile to the maximum is now drawn to the largest
point smaller than U1.
6. Points between L1 and L2 or between U1 and U2 are
drawn as small circles. Points less than L2 or greater
than U2 are drawn as large circles.
Questions The box plot can provide answers to the following questions:
1. Is a factor significant?
2. Does the location differ between subgroups?
3. Does the variation differ between subgroups?
Importance:
Check the
significance
of a factor
The box plot is an important EDA tool for determining if a
factor has a significant effect on the response with respect to
either location or variation.
The box plot is also an effective tool for summarizing large
quantities of information.
Related
Techniques
Mean Plot Analysis of Variance
Case Study The box plot is demonstrated in the ceramic strength data case
study.
Software Box plots are available in most general purpose statistical
software programs.
Complex Demodulation Amplitude Plot
Purpose:
Detect
Changing
Amplitude
in
Sinusoidal
Models
In the frequency analysis of time series models, a common
model is the sinusoidal model:
[ Y_{i} = C + alphasin{(2piomega t_{i} + phi)} +
E_{i} ]
In this equation, α is the amplitude, φ is the phase shift, and ω
is the dominant frequency. In the above model, α and φ are
constant, that is they do not vary with time, ti.
The complex demodulation amplitude plot (Granger, 1964) is
used to determine if the assumption of constant amplitude is
justifiable. If the slope of the complex demodulation
amplitude plot is not zero, then the above model is typically
replaced with the model:
[ Y_{i} = C + alpha_{i}sin{(2piomega t_{i} +
phi)} + E_{i} ]
where ( hat{alpha}_{i} ) is some type of linear model fit
with standard least squares. The most common case is a linear
fit, that is the model becomes
[ Y_{i} = C + (B_0 + B_1*t_{i}) sin{(2piomega
t_{i} + phi)} + E_{i} ]

Quadratic models are sometimes used. Higher order models
are relatively rare.
Sample
Plot:
This complex demodulation amplitude plot of the LEW.DAT
data set shows that:
the amplitude is fixed at approximately 390;
there is a start-up effect; and
there is a change in amplitude at around x = 160 that
should be investigated for an outlier.
Definition: The complex demodulation amplitude plot is formed by:
Vertical axis: Amplitude
Horizontal axis: Time
The mathematical computations for determining the amplitude
are beyond the scope of the Handbook. Consult Granger
(Granger, 1964) for details.
Questions The complex demodulation amplitude plot answers the
1. Does the amplitude change over time?
2. Are there any outliers that need to be investigated?
3. Is the amplitude different at the beginning of the series
(i.e., is there a start-up effect)?
Importance:
Assumption
Checking
As stated previously, in the frequency analysis of time series
models, a common model is the sinusoidal model:
E_{i} ]
In this equation, α is assumed to be constant, that is it does not
vary with time. It is important to check whether or not this
assumption is reasonable.
The complex demodulation amplitude plot can be used to
verify this assumption. If the slope of this plot is essentially
zero, then the assumption of constant amplitude is justified. If
it is not, α should be replaced with some type of time-varying
model. The most common cases are linear (B0 + B1*t) and
quadratic (B0 + B1*t + B2*t2).
Related
Techniques
Spectral Plot
Complex Demodulation Phase Plot
Non-Linear Fitting
Case Study The complex demodulation amplitude plot is demonstrated in
the beam deflection data case study.

Software Complex demodulation amplitude plots are available in some,
but not most, general purpose statistical software programs.
Purpose:
Improve
the
estimate of
frequency
in
sinusoidal
time series
models
As stated previously, in the frequency analysis of time series
models, a common model is the sinusoidal model:
E_{i} ]
In this equation, α is the amplitude, φ is the phase shift, and ω
is the dominant frequency. In the above model, α and φ are
constant, that is they do not vary with time ti.
The complex demodulation phase plot (Granger, 1964) is used
to improve the estimate of the frequency (i.e., ω) in this model.
If the complex demodulation phase plot shows lines sloping
from left to right, then the estimate of the frequency should be
increased. If it shows lines sloping right to left, then the
frequency should be decreased. If there is essentially zero
slope, then the frequency estimate does not need to be
modified.
Sample
Plot:
This complex demodulation phase plot of the LEW.DAT data
set shows that:
the specified demodulation frequency is incorrect;
the demodulation frequency should be increased.
Definition The complex demodulation phase plot is formed by:
Vertical axis: Phase
Horizontal axis: Time
The mathematical computations for the phase plot are beyond
the scope of the Handbook. Consult Granger (Granger, 1964)
for details.
Questions The complex demodulation phase plot answers the following
question:
Is the specified demodulation frequency correct?
Importance
of a Good
The non-linear fitting for the sinusoidal model:

Initial
Estimate
for the
Frequency
E_{i} ]
is usually quite sensitive to the choice of good starting values.
The initial estimate of the frequency, ω, is obtained from a
spectral plot. The complex demodulation phase plot is used to
assess whether this estimate is adequate, and if it is not,
whether it should be increased or decreased. Using the
complex demodulation phase plot with the spectral plot can
significantly improve the quality of the non-linear fits
obtained.
Related
Techniques
Spectral Plot
Non-Linear Fitting
Case Study The complex demodulation amplitude plot is demonstrated in
the beam deflection data case study.
Software Complex demodulation phase plots are available in some, but
not most, general purpose statistical software programs.
Contour Plot
Purpose:
Display 3-d
surface on
2-d plot
A contour plot is a graphical technique for representing a 3-
dimensional surface by plotting constant z slices, called
contours, on a 2-dimensional format. That is, given a value for
z, lines are drawn for connecting the (x,y) coordinates where
that z value occurs.
The contour plot is an alternative to a 3-D surface plot.
Sample
Plot:
This contour plot shows that the surface is symmetric and
peaks in the center.
Definition The contour plot is formed by:
Vertical axis: Independent variable 2
Horizontal axis: Independent variable 1
Lines: iso-response values
The independent variables are usually restricted to a regular
grid. The actual techniques for determining the correct iso-
response values are rather complex and are almost always
computer generated.
An additional variable may be required to specify the Z values
for drawing the iso-lines. Some software packages require

explicit values. Other software packages will determine them
automatically.
If the data (or function) do not form a regular grid, you
typically need to perform a 2-D interpolation to form a regular
grid.
Questions The contour plot is used to answer the question
How does Z change as a function of X and Y?
Importance:
Visualizing
3-
dimensional
data
For univariate data, a run sequence plot and a histogram are
considered necessary first steps in understanding the data. For
2-dimensional data, a scatter plot is a necessary first step in
understanding the data.
In a similar manner, 3-dimensional data should be plotted.
Small data sets, such as result from designed experiments, can
typically be represented by block plots, DOE mean plots, and
the like ("DOE" stands for "Design of Experiments"). For
large data sets, a contour plot or a 3-D surface plot should be
considered a necessary first step in understanding the data.
DOE
Contour
Plot
The DOE contour plot is a specialized contour plot used in the
design of experiments. In particular, it is useful for full and
fractional designs.
Related
Techniques
3-D Plot
Software Contour plots are available in most general purpose statistical
software programs. They are also available in many general
purpose graphics and mathematics programs. These programs
vary widely in the capabilities for the contour plots they
generate. Many provide just a basic contour plot over a
rectangular grid while others permit color filled or shaded
contours.
Most statistical software programs that support design of
experiments will provide a DOE contour plot capability.
DOE Contour Plot
DOE
Contour
Plot:
Introduction
The DOE contour plot is a specialized contour plot used in the analysis of full and fractional
experimental designs. These designs often have a low level, coded as "-1" or "-", and a high
level, coded as "+1" or "+" for each factor. In addition, there can optionally be one or more
center points. Center points are at the mid-point between the low and high level for each
The DOE contour plot is generated for two factors. Typically, this would be the two most
important factors as determined by previous analyses (e.g., through the use of the DOE mean
plots and an analysis of variance). If more than two factors are important, you may want to
generate a series of DOE contour plots, each of which is drawn for two of these factors. You
can also generate a matrix of all pairwise DOE contour plots for a number of important
factors (similar to the scatter plot matrix for scatter plots).
The typical application of the DOE contour plot is in determining settings that will maximize
(or minimize) the response variable. It can also be helpful in determining settings that result
in the response variable hitting a pre-determined target value. The DOE contour plot plays a
useful role in determining the settings for the next iteration of the experiment. That is, the
initial experiment is typically a fractional factorial design with a fairly large number of
factors. After the most important factors are determined, the DOE contour plot can be used to
help define settings for a full factorial or response surface design based on a smaller number
of factors.
Construction
of DOE
The following are the primary steps in the construction of the DOE contour plot.

Contour Plot 1. The x and y axes of the plot represent the values of the first and second factor
(independent) variables.
2. The four vertex points are drawn. The vertex points are (-1,-1), (-1,1), (1,1), (1,-1). At
each vertex point, the average of all the response values at that vertex point is printed.
3. Similarly, if there are center points, a point is drawn at (0,0) and the average of the
response values at the center points is printed.
4. The linear DOE contour plot assumes the model:
$$Y = mu + beta_1 cdot U_1 + beta_2 cdot U_2 + beta_{12} cdot U_1
cdot U_2 $$
where μ is the overall mean of the response variable. The values of β1, β2, β12, and μ
are estimated from the vertex points using least squares estimation.
In order to generate a single contour line, we need a value for Y, say Y0. Next, we solve
for U2 in terms of U1 and, after doing the algebra, we have the equation:
$$displaystyle{U_2 = frac{(Y_0 - mu) - beta_1 cdot U_1} {beta_2 +
beta_{12} cdot U_1}}$$
We generate a sequence of points for U1 in the range -2 to 2 and compute the
corresponding values of U2. These points constitute a single contour line
corresponding to Y = Y0.
The user specifies the target values for which contour lines will be generated.
The above algorithm assumes a linear model for the design. DOE contour plots can also be
generated for the case in which we assume a quadratic model for the design. The algebra for
solving for U2 in terms of U1 becomes more complicated, but the fundamental idea is the
same. Quadratic models are needed for the case when the average for the center points does
not fall in the range defined by the vertex point (i.e., there is curvature).
Sample DOE
Contour Plot
The following is a DOE contour plot for the data used in the Eddy current case study. The

Interpretation
of the Sample
DOE
Contour Plot
From the above DOE contour plot we can derive the following information.
1. Interaction significance;
2. Best (data) setting for these two dominant factors;
Interaction
Significance
Note the appearance of the contour plot. If the contour curves are linear, then that implies that
the interaction term is not significant; if the contour curves have considerable curvature, then
that implies that the interaction term is large and important. In our case, the contour curves
do not have considerable curvature, and so we conclude that the X1*X2 term is not
significant.
Best Settings To determine the best factor settings for the already-run experiment, we first must define
what "best" means. For the Eddy current data set used to generate this DOE contour plot,
"best" means to maximize (rather than minimize or hit a target) the response. Hence from the
contour plot we determine the best settings for the two dominant factors by simply scanning
the four vertices and choosing the vertex with the largest value (= average response). In this
case, it is (X1 = +1, X2 = +1).
As for factor X3, the contour plot provides no best setting information, and so we would
resort to other tools: the main effects plot, the interaction effects matrix, or the ordered data
to determine optimal X3 settings.
Case Study The Eddy current case study demonstrates the use of the DOE contour plot in the context of
Software DOE Contour plots are available in many statistical software programs that analyze data
from designed experiments.
DOE Scatter Plot
Purpose:
Determine
The DOE scatter plot shows the response values for each level of each factor (i.e.,
independent) variable. This graphically shows how the location and scale vary for both

Important
Factors with
Respect to
Location and
Scale
within a factor variable and between different factor variables. This graphically shows which
are the important factors and can help provide a ranked list of important factors from a
designed experiment. The DOE scatter plot is a complement to the traditional analyis of
variance of designed experiments.
DOE scatter plots are typically used in conjunction with the DOE mean plot and the DOE
standard deviation plot. The DOE mean plot replaces the raw response values with mean
response values while the DOE standard deviation plot replaces the raw response values with
the standard deviation of the response values. There is value in generating all 3 of these plots.
The DOE mean and standard deviation plots are useful in that the summary measures of
location and spread stand out (they can sometimes get lost with the raw plot). However, the
raw data points can reveal subtleties, such as the presence of outliers, that might get lost with
the summary statistics.
Sample Plot:
Factors 4, 2,
3, and 7 are
the Important
Factors.
Description
of the Plot
For this sample plot of the BOXBIKE2.DAT data set, there are seven factors and each factor
has two levels. For each factor, we define a distinct x coordinate for each level of the factor.
For example, for factor 1, level 1 is coded as 0.8 and level 2 is coded as 1.2. The y coordinate
is simply the value of the response variable. The solid horizontal line is drawn at the overall
mean of the response variable. The vertical dotted lines are added for clarity.
Although the plot can be drawn with an arbitrary number of levels for a factor, it is really
only useful when there are two or three levels for a factor.
Conclusions This sample DOE scatter plot shows that:
1. there does not appear to be any outliers;
2. the levels of factors 2 and 4 show distinct location differences; and
3. the levels of factor 1 show distinct scale differences.
Definition:
Response
Values Versus
Factor
Variables
DOE scatter plots are formed by:
Vertical axis: Value of the response variable
Horizontal axis: Factor variable (with each level of the factor coded with a slightly
offset x coordinate)
Questions The DOE scatter plot can be used to answer the following questions:
1. Which factors are important with respect to location and scale?

Importance:
Identify
Important
Factors with
Respect to
Location and
Scale
The goal of many designed experiments is to determine which factors are important with
respect to location and scale. A ranked list of the important factors is also often of interest.
DOE scatter, mean, and standard deviation plots show this graphically. The DOE scatter plot
additionally shows if outliers may potentially be distorting the results.
DOE scatter plots were designed primarily for analyzing designed experiments. However,
they are useful for any type of multi-factor data (i.e., a response variable with two or more
factor variables having a small number of distinct levels) whether or not the data were
generated from a designed experiment.
Extension for
Interaction
Effects
Using the concept of the scatterplot matrix, the DOE scatter plot can be extended to display
first order interaction effects.
Specifically, if there are k factors, we create a matrix of plots with k rows and k columns. On
the diagonal, the plot is simply a DOE scatter plot with a single factor. For the off-diagonal
plots, we multiply the values of Xi and Xj. For the common 2-level designs (i.e., each factor
has two levels) the values are typically coded as -1 and 1, so the multiplied values are also -1
and 1. We then generate a DOE scatter plot for this interaction variable. This plot is called a
DOE interaction effects plot and an example is shown below.
Interpretation
of the DOE
Interaction
Effects Plot
We can first examine the diagonal elements for the main effects. These diagonal plots show a
great deal of overlap between the levels for all three factors. This indicates that location and
scale effects will be relatively small.
We can then examine the off-diagonal plots for the first order interaction effects. For
example, the plot in the first row and second column is the interaction between factors X1
and X2. As with the main effect plots, no clear patterns are evident.
Related
Techniques
DOE mean plot
DOE standard deviation plot
Block plot
Box plot
Analysis of variance
Case Study The DOE scatter plot is demonstrated in the ceramic strength data case study.
Software DOE scatter plots are available in some general purpose statistical software programs,
although the format may vary somewhat between these programs. They are essentially just
scatter plots with the X variable defined in a particular way, so it should be feasible to write
macros for DOE scatter plots in most statistical software programs.
DOE Mean Plot
Purpose:
Detect
Important
Factors
With
Respect to
Location
The DOE mean plot is appropriate for analyzing data from a designed
experiment, with respect to important factors, where the factors are at two or
more levels. The plot shows mean values for the two or more levels of each
factor plotted by factor. The means for a single factor are connected by a
straight line. The DOE mean plot is a complement to the traditional analysis
of variance of designed experiments.

This plot is typically generated for the mean. However, it can be generated
for other location statistics such as the median.
Sample
Plot:
Factors 4,
2, and 1 Are
the Most
Important
Factors
This sample DOE mean plot of the BOXBIKE2.DAT data set shows that:
1. factor 4 is the most important;
2. factor 2 is the second most important;
3. factor 1 is the third most important;
4. factor 7 is the fourth most important;
5. factor 6 is the fifth most important;
6. factors 3 and 5 are relatively unimportant.
In summary, factors 4, 2, and 1 seem to be clearly important, factors 3 and 5
seem to be clearly unimportant, and factors 6 and 7 are borderline factors
whose inclusion in any subsequent models will be determined by further
analyses.
Definition:
Mean
Response
Versus
Factor
Variables
DOE mean plots are formed by:
Vertical axis: Mean of the response variable for each level of the factor
Horizontal axis: Factor variable
Questions The DOE mean plot can be used to answer the following questions:
1. Which factors are important? The DOE mean plot does not provide a
definitive answer to this question, but it does help categorize factors as
"clearly important", "clearly not important", and "borderline
importance".
2. What is the ranking list of the important factors?
Importance:
Determine
Significant
Factors
The goal of many designed experiments is to determine which factors are
significant. A ranked order listing of the important factors is also often of
interest. The DOE mean plot is ideally suited for answering these types of
questions and we recommend its routine use in analyzing designed
experiments.
Extension
for
Interaction
Effects
Using the concept of the scatter plot matrix, the DOE mean plot can be
extended to display first-order interaction effects.

Specifically, if there are k factors, we create a matrix of plots with k rows and
k columns. On the diagonal, the plot is simply a DOE mean plot with a single
factor. For the off-diagonal plots, measurements at each level of the
interaction are plotted versus level, where level is Xi times Xj and Xi is the
code for the ith main effect level and Xj is the code for the jth main effect. For
the common 2-level designs (i.e., each factor has two levels) the values are
typically coded as -1 and 1, so the multiplied values are also -1 and 1. We
then generate a DOE mean plot for this interaction variable. This plot is
called a DOE interaction effects plot and an example is shown below.
DOE
Interaction
Effects Plot
This plot shows that the most significant factor is X1 and the most significant
interaction is between X1 and X3.
Related
Techniques
DOE scatter plot
DOE standard deviation plot
Block plot
Box plot
Case Study The DOE mean plot and the DOE interaction effects plot are demonstrated in
the ceramic strength data case study.
Software DOE mean plots are available in some general purpose statistical software
programs, although the format may vary somewhat between these programs.
It may be feasible to write macros for DOE mean plots in some statistical
software programs that do not support this plot directly.
DOE Standard Deviation Plot
Purpose:
Detect
Important
Factors
With
Respect to
Scale
The DOE standard deviation plot is appropriate for analyzing data from a
designed experiment, with respect to important factors, where the factors are
at two or more levels and there are repeated values at each level. The plot
shows standard deviation values for the two or more levels of each factor
plotted by factor. The standard deviations for a single factor are connected by
a straight line. The DOE standard deviation plot is a complement to the
traditional analysis of variance of designed experiments.
This plot is typically generated for the standard deviation. However, it can
also be generated for other scale statistics such as the range, the median
absolute deviation, or the average absolute deviation.

Sample Plot
This sample DOE standard deviation plot of the BOXBIKE2.DAT data set
shows that:
1. factor 1 has the greatest difference in standard deviations between
factor levels;
2. factor 4 has a significantly lower average standard deviation than the
average standard deviations of other factors (but the level 1 standard
deviation for factor 1 is about the same as the level 1 standard deviation
for factor 4);
3. for all factors, the level 1 standard deviation is smaller than the level 2
standard deviation.
Definition:
Response
Standard
Deviations
Versus
Factor
Variables
DOE standard deviation plots are formed by:
Vertical axis: Standard deviation of the response variable for each level
of the factor
Horizontal axis: Factor variable
Questions The DOE standard deviation plot can be used to answer the following
questions:
1. How do the standard deviations vary across factors?
2. How do the standard deviations vary within a factor?
3. Which are the most important factors with respect to scale?
4. What is the ranked list of the important factors with respect to scale?
Importance:
Assess
Variability
The goal with many designed experiments is to determine which factors are
significant. This is usually determined from the means of the factor levels
(which can be conveniently shown with a DOE mean plot). A secondary goal
is to assess the variability of the responses both within a factor and between
factors. The DOE standard deviation plot is a convenient way to do this.
Related
Techniques
DOE scatter plot
DOE mean plot
Block plot
Box plot

Case Study The DOE standard deviation plot is demonstrated in the ceramic strength data
case study.
Software DOE standard deviation plots are not available in most general purpose
statistical software programs. It may be feasible to write macros for DOE
standard deviation plots in some statistical software programs that do not
support them directly.
Histogram
Purpose:
Summarize
a
Univariate
Data Set
The purpose of a histogram (Chambers) is to graphically
summarize the distribution of a univariate data set.
The histogram graphically shows the following:
1. center (i.e., the location) of the data;
2. spread (i.e., the scale) of the data;
3. skewness of the data;
4. presence of outliers; and
5. presence of multiple modes in the data.
These features provide strong indications of the proper
distributional model for the data. The probability plot or a
goodness-of-fit test can be used to verify the distributional
model.
The examples section shows the appearance of a number of
common features revealed by histograms.
Sample
Plot
The above plot is a histogram of the Michelson speed of light
data set.
Definition The most common form of the histogram is obtained by
splitting the range of the data into equal-sized bins (called
classes). Then for each bin, the number of points from the data
set that fall into each bin are counted. That is
Vertical axis: Frequency (i.e., counts for each bin)
Horizontal axis: Response variable
The classes can either be defined arbitrarily by the user or via
some systematic rule. A number of theoretically derived rules
have been proposed by Scott (Scott 1992).
The cumulative histogram is a variation of the histogram in
which the vertical axis gives not just the counts for a single bin,
but rather gives the counts for that bin plus all bins for smaller
values of the response variable.
Both the histogram and cumulative histogram have an
additional variant whereby the counts are replaced by the

normalized counts. The names for these variants are the
relative histogram and the relative cumulative histogram.
There are two common ways to normalize the counts.
1. The normalized count is the count in a class divided by
the total number of observations. In this case the relative
counts are normalized to sum to one (or 100 if a
percentage scale is used). This is the intuitive case where
the height of the histogram bar represents the proportion
of the data in each class.
2. The normalized count is the count in the class divided by
the number of observations times the class width. For
this normalization, the area (or integral) under the
histogram is equal to one. From a probabilistic point of
view, this normalization results in a relative histogram
that is most akin to the probability density function and a
relative cumulative histogram that is most akin to the
cumulative distribution function. If you want to overlay a
probability density or cumulative distribution function on
top of the histogram, use this normalization. Although
this normalization is less intuitive (relative frequencies
greater than 1 are quite permissible), it is the appropriate
normalization if you are using the histogram to model a
probability density function.
Questions The histogram can be used to answer the following questions:
1. What kind of population distribution do the data come
from?
2. Where are the data located?
3. How spread out are the data?
4. Are the data symmetric or skewed?
5. Are there outliers in the data?
Examples 1. Normal
2. Symmetric, Non-Normal, Short-Tailed
3. Symmetric, Non-Normal, Long-Tailed
4. Symmetric and Bimodal
5. Bimodal Mixture of 2 Normals
6. Skewed (Non-Symmetric) Right
7. Skewed (Non-Symmetric) Left
8. Symmetric with Outlier
Related
Techniques
Box plot
Probability plot
The techniques below are not discussed in the Handbook.
However, they are similar in purpose to the histogram.
Additional information on them is contained in the Chambers
and Scott references.
Frequency Plot
Stem and Leaf Plot
Density Trace
Case Study The histogram is demonstrated in the heat flow meter data case
study.
Software Histograms are available in most general purpose statistical
software programs. They are also supported in most general
purpose charting, spreadsheet, and business graphics programs.
Histogram Interpretation: Normal

Symmetric,
Moderate-
Tailed
Histogram
The above is a histogram of the ZARR13.DAT data set.
Note the classical bell-shaped, symmetric histogram with
most of the frequency counts bunched in the middle and
with the counts dying off out in the tails. From a physical
science/engineering point of view, the normal distribution is
that distribution which occurs most often in nature (due in
part to the central limit theorem).
Recommended
Next Step
If the histogram indicates a symmetric, moderate tailed
distribution, then the recommended next step is to do a
normal probability plot to confirm approximate normality.
If the normal probability plot is linear, then the normal
distribution is a good model for the data.
Histogram Interpretation: Symmetric, Non-
Normal, Short-Tailed
Symmetric,
Short-Tailed
Histogram
Description of
What Short-
Tailed Means
The above is a histogram of the first 100 rows of the
TUKLAMB.DAT data set.
For a symmetric distribution, the "body" of a distribution
refers to the "center" of the distribution--commonly that
region of the distribution where most of the probability
resides--the "fat" part of the distribution. The "tail" of a
distribution refers to the extreme regions of the distribution-
-both left and right. The "tail length" of a distribution is a
term that indicates how fast these extremes approach zero.
For a short-tailed distribution, the tails approach zero very
fast. Such distributions commonly have a truncated
("sawed-off") look. The classical short-tailed distribution is
the uniform (rectangular) distribution in which the
probability is constant over a given range and then drops to

zero everywhere else--we would speak of this as having no
tails, or extremely short tails.
For a moderate-tailed distribution, the tails decline to zero
in a moderate fashion. The classical moderate-tailed
distribution is the normal (Gaussian) distribution.
For a long-tailed distribution, the tails decline to zero very
slowly--and hence one is apt to see probability a long way
from the body of the distribution. The classical long-tailed
distribution is the Cauchy distribution.
In terms of tail length, the histogram shown above would be
characteristic of a "short-tailed" distribution.
The optimal (unbiased and most precise) estimator for
location for the center of a distribution is heavily dependent
on the tail length of the distribution. The common choice of
taking N observations and using the calculated sample mean
as the best estimate for the center of the distribution is a
good choice for the normal distribution (moderate tailed), a
poor choice for the uniform distribution (short tailed), and a
horrible choice for the Cauchy distribution (long tailed).
Although for the normal distribution the sample mean is as
precise an estimator as we can get, for the uniform and
Cauchy distributions, the sample mean is not the best
estimator.
For the uniform distribution, the midrange
midrange = (smallest + largest) / 2
is the best estimator of location. For a Cauchy distribution,
the median is the best estimator of location.
Recommended
Next Step
If the histogram indicates a symmetric, short-tailed
distribution, the recommended next step is to generate a
uniform probability plot. If the uniform probability plot is
linear, then the uniform distribution is an appropriate model
for the data.
Histogram Interpretation: Symmetric, Non-
Normal, Long-Tailed
Symmetric,
Long-Tailed
Histogram
Description of
Long-Tailed
The above is a histogram of the DZIUBA1.DAT data set.
The previous example contains a discussion of the
distinction between short-tailed, moderate-tailed, and long-
tailed distributions.
In terms of tail length, the histogram shown above would be
characteristic of a "long-tailed" distribution.

Recommended
Next Step
If the histogram indicates a symmetric, long tailed
distribution, the recommended next step is to do a Cauchy
probability plot. If the Cauchy probability plot is linear,
then the Cauchy distribution is an appropriate model for the
data. Alternatively, a Tukey Lambda PPCC plot may
provide insight into a suitable distributional model for the
data.
Histogram Interpretation: Symmetric and
Bimodal
Symmetric,
Bimodal
Histogram
Description of
Bimodal
The above is a histogram of the LEW.DAT data set.
The mode of a distribution is that value which is most
frequently occurring or has the largest probability of
occurrence. The sample mode occurs at the peak of the
histogram.
For many phenomena, it is quite common for the
distribution of the response values to cluster around a single
mode (unimodal) and then distribute themselves with lesser
frequency out into the tails. The normal distribution is the
classic example of a unimodal distribution.
The histogram shown above illustrates data from a bimodal
(2 peak) distribution. The histogram serves as a tool for
diagnosing problems such as bimodality. Questioning the
underlying reason for distributional non-unimodality
frequently leads to greater insight and improved
deterministic modeling of the phenomenon under study. For
example, for the data presented above, the bimodal
histogram is caused by sinusoidality in the data.
Recommended
Next Step
If the histogram indicates a symmetric, bimodal
distribution, the recommended next steps are to:
1. Do a run sequence plot or a scatter plot to check for
sinusoidality.
2. Do a lag plot to check for sinusoidality. If the lag plot
is elliptical, then the data are sinusoidal.
3. If the data are sinusoidal, then a spectral plot is used
to graphically estimate the underlying sinusoidal
frequency.
4. If the data are not sinusoidal, then a Tukey Lambda
PPCC plot may determine the best-fit symmetric
distribution for the data.
5. The data may be fit with a mixture of two
distributions. A common approach to this case is to fit
a mixture of 2 normal or lognormal distributions.
Further discussion of fitting mixtures of distributions
is beyond the scope of this Handbook.

Histogram Interpretation: Bimodal Mixture of
2 Normals
Histogram
from Mixture
of 2 Normal
Distributions
Discussion of
Unimodal and
Bimodal
The above is a histogram of the ZARR14.DAT data set.
The histogram shown above illustrates data from a bimodal
(2 peak) distribution.
In contrast to the previous example, this example illustrates
bimodality due not to an underlying deterministic model,
but bimodality due to a mixture of probability models. In
this case, each of the modes appears to have a rough bell-
shaped component. One could easily imagine the above
histogram being generated by a process consisting of two
normal distributions with the same standard deviation but
with two different locations (one centered at approximately
9.17 and the other centered at approximately 9.26). If this is
the case, then the research challenge is to determine
physically why there are two similar but separate sub-
processes.
Recommended
Next Steps
If the histogram indicates that the data might be
appropriately fit with a mixture of two normal distributions,
the recommended next step is:
Fit the normal mixture model using either least squares or
maximum likelihood. The general normal mixing model is
[ M = pphi_{1} + (1-p)phi_{2} ]
where p is the mixing proportion (between 0 and 1) and (
phi_1 ) and ( phi_2 ) are normal probability density
functions with location and scale parameters μ1, σ1, μ2, and
σ2, respectively. That is, there are 5 parameters to estimate
in the fit.
Whether maximum likelihood or least squares is used, the
quality of the fit is sensitive to good starting values. For the
mixture of two normals, the histogram can be used to
provide initial estimates for the location and scale
parameters of the two normal distributions.
Both Dataplot code and R code can be used to fit a mixture
of two normals.

Histogram Interpretation: Skewed (Non-
Normal) Right
Right-Skewed
Histogram
Discussion of
Skewness
The above is a histogram of the SUNSPOT.DAT data set.
A symmetric distribution is one in which the 2 "halves" of
the histogram appear as mirror-images of one another. A
skewed (non-symmetric) distribution is a distribution in
which there is no such mirror-imaging.
For skewed distributions, it is quite common to have one
tail of the distribution considerably longer or drawn out
relative to the other tail. A "skewed right" distribution is one
in which the tail is on the right side. A "skewed left"
distribution is one in which the tail is on the left side. The
above histogram is for a distribution that is skewed right.
Skewed distributions bring a certain philosophical
complexity to the very process of estimating a "typical
value" for the distribution. To be specific, suppose that the
analyst has a collection of 100 values randomly drawn from
a distribution, and wishes to summarize these 100
observations by a "typical value". What does typical value
mean? If the distribution is symmetric, the typical value is
unambiguous-- it is a well-defined center of the distribution.
For example, for a bell-shaped symmetric distribution, a
center point is identical to that value at the peak of the
distribution.
For a skewed distribution, however, there is no "center" in
the usual sense of the word. Be that as it may, several
"typical value" metrics are often used for skewed
distributions. The first metric is the mode of the
distribution. Unfortunately, for severely-skewed
distributions, the mode may be at or near the left or right
tail of the data and so it seems not to be a good
representative of the center of the distribution. As a second
choice, one could conceptually argue that the mean (the
point on the horizontal axis where the distributiuon would
balance) would serve well as the typical value. As a third
choice, others may argue that the median (that value on the
horizontal axis which has exactly 50% of the data to the left
(and also to the right) would serve as a good typical value.
For symmetric distributions, the conceptual problem
disappears because at the population level the mode, mean,
and median are identical. For skewed distributions,
however, these 3 metrics are markedly different. In practice,
for skewed distributions the most commonly reported
typical value is the mean; the next most common is the
median; the least common is the mode. Because each of
these 3 metrics reflects a different aspect of "centerness", it
is recommended that the analyst report at least 2 (mean and

median), and preferably all 3 (mean, median, and mode) in
summarizing and characterizing a data set.
Some Causes
for Skewed
Data
Skewed data often occur due to lower or upper bounds on
the data. That is, data that have a lower bound are often
skewed right while data that have an upper bound are often
skewed left. Skewness can also result from start-up effects.
For example, in reliability applications some processes may
have a large number of initial failures that could cause left
skewness. On the other hand, a reliability process could
have a long start-up period where failures are rare resulting
in right-skewed data.
Data collected in scientific and engineering applications
often have a lower bound of zero. For example, failure data
must be non-negative. Many measurement processes
generate only positive data. Time to occurence and size are
common measurements that cannot be less than zero.
Recommended
Next Steps
If the histogram indicates a right-skewed data set, the
recommended next steps are to:
1. Quantitatively summarize the data by computing and
reporting the sample mean, the sample median, and
the sample mode.
2. Determine the best-fit distribution (skewed-right)
from the
Weibull family (for the maximum)
Gamma family
Chi-square family
Lognormal family
Power lognormal family
3. Consider a normalizing transformation such as the
Box-Cox transformation.
Histogram Interpretation: Skewed (Non-
Symmetric) Left
Skewed
Left
Histogram
The above is a histogram of the MORALES.DAT data set.
The issues for skewed left data are similar to those for skewed
right data.
Histogram Interpretation: Symmetric with
Outlier

Symmetric
Histogram
with Outlier
Discussion of
Outliers
The above is a histogram of the ZARR13.DAT data set with
four values of 9.45 added.
A symmetric distribution is one in which the 2 "halves" of
the histogram appear as mirror-images of one another. The
above example is symmetric with the exception of outlying
data near Y = 4.5.
An outlier is a data point that comes from a distribution
different (in location, scale, or distributional form) from the
bulk of the data. In the real world, outliers have a range of
causes, from as simple as
1. operator blunders
2. equipment failures
3. day-to-day effects
4. batch-to-batch differences
5. anomalous input conditions
6. warm-up effects
to more subtle causes such as
1. A change in settings of factors that (knowingly or
unknowingly) affect the response.
2. Nature is trying to tell us something.
Outliers
Should be
Investigated
All outliers should be taken seriously and should be
investigated thoroughly for explanations. Automatic outlier-
rejection schemes (such as throw out all data beyond 4
sample standard deviations from the sample mean) are
particularly dangerous.
The classic case of automatic outlier rejection becoming
automatic information rejection was the South Pole ozone
depletion problem. Ozone depletion over the South Pole
would have been detected years earlier except for the fact
that the satellite data recording the low ozone readings had
outlier-rejection code that automatically screened out the
"outliers" (that is, the low ozone readings) before the
analysis was conducted. Such inadvertent (and incorrect)
purging went on for years. It was not until ground-based
South Pole readings started detecting low ozone readings
that someone decided to double-check as to why the
satellite had not picked up this fact--it had, but it had gotten
thrown out!
The best attitude is that outliers are our "friends", outliers
are trying to tell us something, and we should not stop until
we are comfortable in the explanation for each outlier.
Recommended
Next Steps
If the histogram shows the presence of outliers, the
recommended next steps are:

1. Graphically check for outliers (in the commonly
encountered normal case) by generating a box plot. In
general, box plots are a much better graphical tool for
detecting outliers than are histograms.
2. Quantitatively check for outliers (in the commonly
encountered normal case) by carrying out Grubbs test
which indicates how many sample standard
deviations away from the sample mean are the data in
question. Large values indicate outliers.
Lag Plot
Purpose:
Check for
randomness
A lag plot checks whether a data set or time series is random
or not. Random data should not exhibit any identifiable
structure in the lag plot. Non-random structure in the lag plot
indicates that the underlying data are not random. Several
common patterns for lag plots are shown in the examples
below.
Sample
Plot
This sample lag plot of the MAVRO.DAT data set exhibits a
linear pattern. This shows that the data are strongly non-
random and further suggests that an autoregressive model
might be appropriate.
Definition A lag is a fixed time displacement. For example, given a data
set Y1, Y2 ..., Yn, Y2 and Y7 have lag 5 since 7 - 2 = 5. Lag
plots can be generated for any arbitrary lag, although the most
commonly used lag is 1.
A plot of lag 1 is a plot of the values of Yi versus Yi-1
Vertical axis: Yi for all i
Horizontal axis: Yi-1 for all i
Questions Lag plots can provide answers to the following questions:
2. Is there serial correlation in the data?
3. What is a suitable model for the data?
Importance Inasmuch as randomness is an underlying assumption for most
statistical estimation and testing techniques, the lag plot
should be a routine tool for researchers.
Examples Random (White Noise)

Weak autocorrelation
Strong autocorrelation and autoregressive model
Sinusoidal model and outliers
Related
Techniques
Spectrum
Runs Test
Case Study The lag plot is demonstrated in the beam deflection data case
study.
Software Lag plots are not directly available in most general purpose
statistical software programs. Since the lag plot is essentially a
scatter plot with the 2 variables properly lagged, it should be
feasible to write a macro for the lag plot in most statistical
programs.
Lag Plot: Random Data
Lag Plot
Conclusions We can make the following conclusions based on the above
plot of 200 normal random numbers.
2. The data exhibit no autocorrelation.
3. The data contain no outliers.
Discussion The lag plot shown above is for lag = 1. Note the absence of
structure. One cannot infer, from a current value Yi-1, the next
value Yi. Thus for a known value Yi-1 on the horizontal axis
(say, Yi-1 = +0.5), the Yi-th value could be virtually anything
(from Yi = -2.5 to Yi = +1.5). Such non-association is the
essence of randomness.
Lag Plot: Moderate Autocorrelation
Lag Plot

Conclusions We can make the conclusions based on the above plot of the
FLICKER.DAT data set.
1. The data are from an underlying autoregressive
model with moderate positive autocorrelation
Discussion In the plot above for lag = 1, note how the points tend to
cluster (albeit noisily) along the diagonal. Such clustering is
the lag plot signature of moderate autocorrelation.
If the process were completely random, knowledge of a
current observation (say Yi-1 = 0) would yield virtually no
knowledge about the next observation Yi. If the process has
moderate autocorrelation, as above, and if Yi-1 = 0, then the
range of possible values for Yi is seen to be restricted to a
smaller range (.01 to +.01). This suggests prediction is
possible using an autoregressive model.
Recommended
Next Step
Estimate the parameters for the autoregressive model:
[ Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} ]
Since Yi and Yi-1 are precisely the axes of the lag plot, such
estimation is a linear regression straight from the lag plot.
The residual standard deviation for the autoregressive
[ Y_{i} = A_0 + E_{i} ]
Lag Plot: Strong Autocorrelation and
Autoregressive Model

Lag Plot
plot of the random walk data set.
model with strong positive autocorrelation
Discussion Note the tight clustering of points along the diagonal. This is
the lag plot signature of a process with strong positive
autocorrelation. Such processes are highly non-random--there
is strong association between an observation and a succeeding
observation. In short, if you know Yi-1 you can make a strong
guess as to what Yi will be.
If the above process were completely random, the plot would
have a shotgun pattern, and knowledge of a current
observation (say Yi-1 = 3) would yield virtually no knowledge
about the next observation Yi (it could here be anywhere from
-2 to +8). On the other hand, if the process had strong
autocorrelation, as seen above, and if Yi-1 = 3, then the range
of possible values for Yi is seen to be restricted to a smaller
range (2 to 4)--still wide, but an improvement nonetheless
(relative to -2 to +8) in predictive power.
Recommended
Next Step
When the lag plot shows a strongly autoregressive pattern and
only successive observations appear to be correlated, the next
steps are to:
1. Extimate the parameters for the autoregressive model:
[ Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} ]
Since Yi and Yi-1 are precisely the axes of the lag plot,
such estimation is a linear regression straight from the
lag plot.
[ Y_{i} = A_0 + E_{i} ]
2. Reexamine the system to arrive at an explanation for
the strong autocorrelation. Is it due to the
1. phenomenon under study; or
2. drifting in the environment; or
3. contamination from the data acquisition system?
Sometimes the source of the problem is contamination
and carry-over from the data acquisition system where

the system does not have time to electronically recover
before collecting the next data point. If this is the case,
then consider slowing down the sampling rate to
achieve randomness.
Lag Plot: Sinusoidal Models and Outliers
Lag Plot
plot of the LEW.DAT data set.
1. The data come from an underlying single-cycle
sinusoidal model.
2. The data contain three outliers.
Discussion In the plot above for lag = 1, note the tight elliptical
clustering of points. Processes with a single-cycle
sinusoidal model will have such elliptical lag plots.
Consequences
of Ignoring
Cyclical
Pattern
If one were to naively assume that the above process came
from the null model
[ Y_{i} = A_0 + E_{i} ]
and then estimate the constant by the sample mean, then the
analysis would suffer because
1. the sample mean would be biased and meaningless;
2. the confidence limits would be meaningless and
optimistically small.
The proper model
[ Y_{i} = C + alphasin{(2piomega t_{i} + phi)}
+ E_{i} ]
(where α is the amplitude, ω is the frequency--between 0
and .5 cycles per observation--, and ( phi ) is the phase)
can be fit by standard non-linear least squares, to estimate
the coefficients and their uncertainties.
The lag plot is also of value in outlier detection. Note in the
above plot that there appears to be 4 points lying off the
ellipse. However, in a lag plot, each point in the original
data set Y shows up twice in the lag plot--once as Yi and
once as Yi-1. Hence the outlier in the upper left at Yi = 300
is the same raw data value that appears on the far right at Yi-
1 = 300. Thus (-500,300) and (300,200) are due to the same
outlier, namely the 158th data point: 300. The correct value
for this 158th point should be approximately -300 and so it

appears that a sign got dropped in the data collection. The
other two points lying off the ellipse, at roughly (100,100)
and at (0,-50), are caused by two faulty data values: the
third data point of -15 should be about +125 and the fourth
data point of +141 should be about -50, respectively. Hence
the 4 apparent lag plot outliers are traceable to 3 actual
outliers in the original run sequence: at points 4 (-15), 5
(141) and 158 (300). In retrospect, only one of these (point
158 (= 300)) is an obvious outlier in the run sequence plot.
Unexpected
Value of EDA
Frequently a technique (e.g., the lag plot) is constructed to
check one aspect (e.g., randomness) which it does well.
Along the way, the technique also highlights some other
anomaly of the data (namely, that there are 3 outliers). Such
outlier identification and removal is extremely important for
detecting irregularities in the data collection system, and
also for arriving at a "purified" data set for modeling. The
lag plot plays an important role in such outlier
identification.
Recommended
Next Step
When the lag plot indicates a sinusoidal model with
possible outliers, the recommended next steps are:
1. Do a spectral plot to obtain an initial estimate of the
frequency of the underlying cycle. This will be
helpful as a starting value for the subsequent non-
linear fitting.
2. Omit the outliers.
3. Carry out a non-linear fit of the model to the 197
points.
[ Y_{i} = C + alphasin{(2piomega t_{i} +
phi)} + E_{i} ]
Linear Correlation Plot
Purpose:
Detect
changes in
correlation
between
groups
Linear correlation plots are used to assess whether or not
correlations are consistent across groups. That is, if your data
is in groups, you may want to know if a single correlation
can be used across all the groups or whether separate
correlations are required for each group.
Linear correlation plots are often used in conjunction with
linear slope, linear intercept, and linear residual standard
deviation plots. A linear correlation plot could be generated
intially to see if linear fitting would be a fruitful direction. If
the correlations are high, this implies it is worthwhile to
continue with the linear slope, intercept, and residual
standard deviation plots. If the correlations are weak, a
different model needs to be pursued.
In some cases, you might not have groups. Instead you may
have different data sets and you want to know if the same
correlation can be adequately applied to each of the data sets.
In this case, simply think of each distinct data set as a group
and apply the linear slope plot as for groups.

Sample Plot
This linear correlation plot of the HSU12.DAT data set
shows that the correlations are high for all groups. This
implies that linear fits could provide a good model for each
of these groups.
Definition:
Group
Correlations
Versus
Group ID
Linear correlation plots are formed by:
Vertical axis: Group correlations
Horizontal axis: Group identifier
A reference line is plotted at the correlation between the full
data sets.
Questions The linear correlation plot can be used to answer the
following questions.
1. Are there linear relationships across groups?
2. Are the strength of the linear relationships relatively
constant across the groups?
Importance:
Checking
Group
Homogeneity
For grouped data, it may be important to know whether the
different groups are homogeneous (i.e., similar) or
heterogeneous (i.e., different). Linear correlation plots help
answer this question in the context of linear fitting.
Related
Techniques
Linear Intercept Plot
Linear Slope Plot
Linear Residual Standard Deviation Plot
Linear Fitting
Case Study The linear correlation plot is demonstrated in the Alaska
Software Most general purpose statistical software programs do not
support a linear correlation plot. However, if the statistical
program can generate correlations over a group, it should be
feasible to write a macro to generate this plot.
Purpose:
Detect
changes in
linear
intercepts
between
groups
Linear intercept plots are used to graphically assess whether
or not linear fits are consistent across groups. That is, if your
data have groups, you may want to know if a single fit can be
used across all the groups or whether separate fits are
required for each group.
Linear intercept plots are typically used in conjunction with
linear slope and linear residual standard deviation plots.

In some cases you might not have groups. Instead, you have
different data sets and you want to know if the same fit can
be adequately applied to each of the data sets. In this case,
simply think of each distinct data set as a group and apply the
linear intercept plot as for groups.
Sample Plot
This linear intercept plot of the HSU12.DAT data set shows
that there is a shift in intercepts. Specifically, the first three
intercepts are lower than the intercepts for the other groups.
Note that these are small differences in the intercepts.
Definition:
Group
Intercepts
Versus
Group ID
Linear intercept plots are formed by:
Vertical axis: Group intercepts from linear fits
A reference line is plotted at the intercept from a linear fit
using all the data.
Questions The linear intercept plot can be used to answer the following
questions.
1. Is the intercept from linear fits relatively constant
across groups?
2. If the intercepts vary across groups, is there a
discernible pattern?
Importance:
Checking
Group
Homogeneity
heterogeneous (i.e., different). Linear intercept plots help
Related
Techniques
Linear Slope Plot
Linear Fitting
Case Study The linear intercept plot is demonstrated in the Alaska
support a linear intercept plot. However, if the statistical
program can generate linear fits over a group, it should be
Linear Slope Plot
Purpose:
Detect
changes in
Linear slope plots are used to graphically assess whether or
not linear fits are consistent across groups. That is, if your
data have groups, you may want to know if a single fit can be

linear slopes
between
groups
used across all the groups or whether separate fits are
required for each group.
Linear slope plots are typically used in conjunction with
linear intercept and linear residual standard deviation plots.
linear slope plot as for groups.
Sample Plot
This linear slope plot of the HSU12.DAT data set shows that
the slopes are about 0.174 (plus or minus 0.002) for all
groups. There does not appear to be a pattern in the variation
of the slopes. This implies that a single fit may be adequate.
Definition:
Group
Slopes
Versus
Group ID
Linear slope plots are formed by:
Vertical axis: Group slopes from linear fits
A reference line is plotted at the slope from a linear fit using
all the data.
Questions The linear slope plot can be used to answer the following
questions.
1. Do you get the same slope across groups for linear
fits?
2. If the slopes differ, is there a discernible pattern in the
slopes?
Importance:
Checking
Group
Homogeneity
heterogeneous (i.e., different). Linear slope plots help answer
this question in the context of linear fitting.
Related
Techniques
Linear Fitting
Case Study The linear slope plot is demonstrated in the Alaska pipeline
data case study.
support a linear slope plot. However, if the statistical
program can generate linear fits over a group, it should be

Purpose:
Detect
Changes in
Linear
Residual
Standard
Deviation
Between
Groups
Linear residual standard deviation (RESSD) plots are used to
graphically assess whether or not linear fits are consistent
across groups. That is, if your data have groups, you may
want to know if a single fit can be used across all the groups
or whether separate fits are required for each group.
The residual standard deviation is a goodness-of-fit measure.
That is, the smaller the residual standard deviation, the closer
is the fit to the data.
Linear RESSD plots are typically used in conjunction with
linear intercept and linear slope plots. The linear intercept
and slope plots convey whether or not the fits are consistent
across groups while the linear RESSD plot conveys whether
the adequacy of the fit is consistent across groups.
linear RESSD plot as for groups.
Sample Plot
This linear RESSD plot of the HSU12.DAT data set shows
that the residual standard deviations from a linear fit are
about 0.0025 for all the groups.
Definition:
Group
Residual
Standard
Deviation
Versus
Group ID
Linear RESSD plots are formed by:
Vertical axis: Group residual standard deviations from
linear fits
A reference line is plotted at the residual standard deviation
from a linear fit using all the data. This reference line will
typically be much greater than any of the individual residual
standard deviations.
Questions The linear RESSD plot can be used to answer the following
questions.
1. Is the residual standard deviation from a linear fit
constant across groups?
2. If the residual standard deviations vary, is there a
discernible pattern across the groups?
Importance:
Checking
Group
Homogeneity
heterogeneous (i.e., different). Linear RESSD plots help

Related
Techniques
Linear Slope Plot
Linear Fitting
Case Study The linear residual standard deviation plot is demonstrated in
the Alaska pipeline data case study.
support a linear residual standard deviation plot. However, if
the statistical program can generate linear fits over a group, it
should be feasible to write a macro to generate this plot.
Mean Plot
Purpose:
Detect
changes in
location
between
groups
Mean plots are used to see if the mean varies between
different groups of the data. The grouping is determined by
the analyst. In most cases, the data set contains a specific
grouping variable. For example, the groups may be the levels
of a factor variable. In the sample plot below, the months of
the year provide the grouping.
Mean plots can be used with ungrouped data to determine if
the mean is changing over time. In this case, the data are split
into an arbitrary number of equal-sized groups. For example,
a data series with 400 points can be divided into 10 groups of
40 points each. A mean plot can then be generated with these
groups to see if the mean is increasing or decreasing over
time.
Although the mean is the most commonly used measure of
location, the same concept applies to other measures of
location. For example, instead of plotting the mean of each
group, the median or the trimmed mean might be plotted
instead. This might be done if there were significant outliers
in the data and a more robust measure of location than the
mean was desired.
Mean plots are typically used in conjunction with standard
deviation plots. The mean plot checks for shifts in location
while the standard deviation plot checks for shifts in scale.
Sample Plot
This sample mean plot of the DRAFT69.DAT data set shows
a shift of location after the 6th month.
Definition:
Group
Means
Versus
Group ID
Mean plots are formed by:
Vertical axis: Group mean
A reference line is plotted at the overall mean.

Questions The mean plot can be used to answer the following questions.
1. Are there any shifts in location?
2. What is the magnitude of the shifts in location?
3. Is there a distinct pattern in the shifts in location?
Importance:
Checking
Assumptions
A common assumption in 1-factor analyses is that of constant
location. That is, the location is the same for different levels
of the factor variable. The mean plot provides a graphical
check for that assumption. A common assumption for
univariate data is that the location is constant. By grouping
the data into equal intervals, the mean plot can provide a
graphical test of this assumption.
Related
Techniques
Standard Deviation Plot
DOE Mean Plot
Box Plot
support a mean plot. However, if the statistical program can
generate the mean over a group, it should be feasible to write
a macro to generate this plot.
Purpose:
Check If Data
Are
Approximately
Normally
Distributed
The normal probability plot (Chambers et al., 1983) is a
graphical technique for assessing whether or not a data set
is approximately normally distributed.
The data are plotted against a theoretical normal distribution
in such a way that the points should form an approximate
straight line. Departures from this straight line indicate
departures from normality.
The normal probability plot is a special case of the
probability plot. We cover the normal probability plot
separately due to its importance in many applications.
Sample Plot
The points on this normal probablity plot of 100 normal
random numbers form a nearly linear pattern, which
indicates that the normal distribution is a good model for
this data set.
Definition:
Ordered
Response
Values Versus
Normal Order
Statistic
Medians
The normal probability plot is formed by:
Vertical axis: Ordered response values
Horizontal axis: Normal order statistic medians
The observations are plotted as a function of the
corresponding normal order statistic medians which are

defined as:
Ni = G(Ui)
where Ui are the uniform order statistic medians (defined
below) and G is the percent point function of the normal
distribution. The percent point function is the inverse of the
cumulative distribution function (probability that x is less
than or equal to some value). That is, given a probability,
we want the corresponding x of the cumulative distribution
function.
The uniform order statistic medians (see Filliben 1975) can
be approximated by:
Ui = 1 - Un for i = 1
Ui = (i - 0.3175)/(n + 0.365) for i = 2, 3, ..., n-1
Ui = 0.5(1/n) for i = n
In addition, a straight line can be fit to the points and added
as a reference line. The further the points vary from this
line, the greater the indication of departures from normality.
Probability plots for distributions other than the normal are
computed in exactly the same way. The normal percent
point function (the G) is simply replaced by the percent
point function of the desired distribution. That is, a
probability plot can easily be generated for any distribution
for which you have the percent point function.
One advantage of this method of computing probability
plots is that the intercept and slope estimates of the fitted
line are in fact estimates for the location and scale
parameters of the distribution. Although this is not too
important for the normal distribution since the location and
scale are estimated by the mean and standard deviation,
respectively, it can be useful for many other distributions.
The correlation coefficient of the points on the normal
probability plot can be compared to a table of critical values
to provide a formal test of the hypothesis that the data come
from a normal distribution.
Questions The normal probability plot is used to answer the following
questions.
1. Are the data normally distributed?
2. What is the nature of the departure from normality
(data skewed, shorter than expected tails, longer than
expected tails)?
Importance:
Check
Normality
Assumption
The underlying assumptions for a measurement process are
that the data should behave like:
1. random drawings;
3. with fixed location;
4. with fixed scale.
Probability plots are used to assess the assumption of a
fixed distribution. In particular, most statistical models are
of the form:
response = deterministic + random
where the deterministic part is the fit and the random part is
error. This error component in most common statistical
models is specifically assumed to be normally distributed
with fixed location and scale. This is the most frequent
application of normal probability plots. That is, a model is

fit and a normal probability plot is generated for the
residuals from the fitted model. If the residuals from the
fitted model are not normally distributed, then one of the
major assumptions of the model has been violated.
Examples 1. Data are normally distributed
2. Data have short tails
3. Data have fat tails
4. Data are skewed right
Related
Techniques
Histogram
Probability plots for other distributions (e.g., Weibull)
Probability plot correlation coefficient plot (PPCC plot)
Anderson-Darling Goodness-of-Fit Test
Chi-Square Goodness-of-Fit Test
Kolmogorov-Smirnov Goodness-of-Fit Test
Case Study The normal probability plot is demonstrated in the heat flow
meter data case study.
Software Most general purpose statistical software programs can
generate a normal probability plot.
Normal Probability Plot: Normally Distributed
Data
Normal
Probability
Plot
The following normal probability plot is from the heat flow
meter data.
Conclusions We can make the following conclusions from the above plot.
1. The normal probability plot shows a strongly linear
pattern. There are only minor deviations from the line
fit to the points on the probability plot.
2. The normal distribution appears to be a good model for
these data.
Discussion Visually, the probability plot shows a strongly linear pattern.
This is verified by the correlation coefficient of 0.9989 of the
line fit to the probability plot. The fact that the points in the
lower and upper extremes of the plot do not deviate
significantly from the straight-line pattern indicates that there
are not any significant outliers (relative to a normal
distribution).
In this case, we can quite reasonably conclude that the normal
distribution provides an excellent model for the data. The
intercept and slope of the fitted line give estimates of 9.26 and

0.023 for the location and scale parameters of the fitted
normal distribution.
Normal Probability Plot: Data Have Short Tails
Normal
Probability
Plot for
Data with
Short Tails
The following is a normal probability plot for 500 random
numbers generated from a Tukey-Lambda distribution with
the parameter equal to 1.1.
1. The normal probability plot shows a non-linear pattern.
2. The normal distribution is not a good model for these
data.
Discussion For data with short tails relative to the normal distribution, the
non-linearity of the normal probability plot shows up in two
ways. First, the middle of the data shows an S-like pattern.
This is common for both short and long tails. Second, the first
few and the last few points show a marked departure from the
reference fitted line. In comparing this plot to the long tail
example in the next section, the important difference is the
direction of the departure from the fitted line for the first few
and last few points. For short tails, the first few points show
increasing departure from the fitted line above the line and
last few points show increasing departure from the fitted line
below the line. For long tails, this pattern is reversed.
In this case, we can reasonably conclude that the normal
distribution does not provide an adequate fit for this data set.
For probability plots that indicate short-tailed distributions,
the next step might be to generate a Tukey Lambda PPCC
plot. The Tukey Lambda PPCC plot can often be helpful in
identifying an appropriate distributional family.
Normal Probability Plot: Data Have Long Tails
Normal
Probability
Plot for
Data with
Long Tails
The following is a normal probability plot of 500 numbers
generated from a double exponential distribution. The double
exponential distribution is symmetric, but relative to the
normal it declines rapidly and has longer tails.

1. The normal probability plot shows a reasonably linear
pattern in the center of the data. However, the tails,
particularly the lower tail, show departures from the
fitted line.
2. A distribution other than the normal distribution would
be a good model for these data.
Discussion For data with long tails relative to the normal distribution, the
non-linearity of the normal probability plot can show up in
two ways. First, the middle of the data may show an S-like
pattern. This is common for both short and long tails. In this
particular case, the S pattern in the middle is fairly mild.
Second, the first few and the last few points show marked
departure from the reference fitted line. In the plot above, this
is most noticeable for the first few data points. In comparing
this plot to the short-tail example in the previous section, the
important difference is the direction of the departure from the
fitted line for the first few and the last few points. For long
tails, the first few points show increasing departure from the
fitted line below the line and last few points show increasing
departure from the fitted line above the line. For short tails,
this pattern is reversed.
In this case we can reasonably conclude that the normal
distribution can be improved upon as a model for these data.
For probability plots that indicate long-tailed distributions, the
next step might be to generate a Tukey Lambda PPCC plot.
The Tukey Lambda PPCC plot can often be helpful in
identifying an appropriate distributional family.
Normal Probability Plot: Data are Skewed
Right
Normal
Probability
Plot for
Data that
are Skewed
Right

Conclusions We can make the following conclusions from the above plot
of the SUNSPOT.DAT data set.
1. The normal probability plot shows a strongly non-linear
pattern. Specifically, it shows a quadratic pattern in
which all the points are below a reference line drawn
between the first and last points.
2. The normal distribution is not a good model for these
data.
Discussion This quadratic pattern in the normal probability plot is the
signature of a significantly right-skewed data set. Similarly, if
all the points on the normal probability plot fell above the
reference line connecting the first and last points, that would
be the signature pattern for a significantly left-skewed data
set.
In this case we can quite reasonably conclude that we need to
model these data with a right skewed distribution such as the
Weibull or lognormal.
Probability Plot
Purpose:
Check If
Data Follow
a Given
Distribution
The probability plot (Chambers et al., 1983) is a graphical
technique for assessing whether or not a data set follows a
given distribution such as the normal or Weibull.
The data are plotted against a theoretical distribution in such
a way that the points should form approximately a straight
line. Departures from this straight line indicate departures
from the specified distribution.
The correlation coefficient associated with the linear fit to
the data in the probability plot is a measure of the goodness
of the fit. Estimates of the location and scale parameters of
the distribution are given by the intercept and slope.
Probability plots can be generated for several competing
distributions to see which provides the best fit, and the
probability plot generating the highest correlation coefficient
is the best choice since it generates the straightest probability
plot.
For distributions with shape parameters (not counting
location and scale parameters), the shape parameters must be
known in order to generate the probability plot. For
distributions with a single shape parameter, the probability
plot correlation coefficient (PPCC) plot provides an
excellent method for estimating the shape parameter.
We cover the special case of the normal probability plot
separately due to its importance in many statistical
applications.

Sample Plot
This data is a set of 500 Weibull random numbers with a
shape parameter = 2, location parameter = 0, and scale
parameter = 1. The Weibull probability plot indicates that the
Weibull distribution does in fact fit these data well.
Definition:
Ordered
Response
Values
Versus Order
Statistic
Medians for
the Given
Distribution
The probability plot is formed by:
Vertical axis: Ordered response values
Horizontal axis: Order statistic medians for the given
distribution
The order statistic medians (see Filliben 1975) can be
approximated by:
Ni = G(Ui)
where Ui are the uniform order statistic medians (defined
below) and G is the percent point function for the desired
distribution. The percent point function is the inverse of the
cumulative distribution function (probability that x is less
than or equal to some value). That is, given a probability, we
want the corresponding x of the cumulative distribution
function.
The uniform order statistic medians are defined as:
mi = 1 - mn for i = 1
mi = (i - 0.3175)/(n + 0.365) for i = 2, 3, ..., n-1
mi = 0.5(1/n) for i = n
In addition, a straight line can be fit to the points and added
as a reference line. The further the points vary from this line,
the greater the indication of a departure from the specified
distribution.
This definition implies that a probability plot can be easily
generated for any distribution for which the percent point
function can be computed.
One advantage of this method of computing proability plots
is that the intercept and slope estimates of the fitted line are
in fact estimates for the location and scale parameters of the
distribution. Although this is not too important for the
normal distribution (the location and scale are estimated by
the mean and standard deviation, respectively), it can be
useful for many other distributions.
Questions The probability plot is used to answer the following
questions:
Does a given distribution, such as the Weibull, provide
a good fit to my data?
What distribution best fits my data?

What are good estimates for the location and scale
parameters of the chosen distribution?
Importance:
Check
distributional
assumption
The discussion for the normal probability plot covers the use
of probability plots for checking the fixed distribution
assumption.
Some statistical models assume data have come from a
population with a specific type of distribution. For example,
in reliability applications, the Weibull, lognormal, and
exponential are commonly used distributional models.
Probability plots can be useful for checking this
distributional assumption.
Related
Techniques
Histogram
Probability Plot Correlation Coefficient (PPCC) Plot
Hazard Plot
Quantile-Quantile Plot
Anderson-Darling Goodness of Fit
Chi-Square Goodness of Fit
Kolmogorov-Smirnov Goodness of Fit
Case Study The probability plot is demonstrated in the uniform random
numbers case study.
Software Most general purpose statistical software programs support
probability plots for at least a few common distributions.
Probability Plot Correlation Coefficient Plot
Purpose:
Graphical
Technique for
Finding the
Shape
Parameter of
a
Distributional
Family that
Best Fits a
Data Set
The probability plot correlation coefficient (PPCC) plot
(Filliben 1975) is a graphical technique for identifying the
shape parameter for a distributional family that best
describes the data set. This technique is appropriate for
families, such as the Weibull, that are defined by a single
shape parameter and location and scale parameters, and it is
not appropriate for distributions, such as the normal, that are
defined only by location and scale parameters.
The PPCC plot is generated as follows. For a series of
values for the shape parameter, the correlation coefficient is
computed for the probability plot associated with a given
value of the shape parameter. These correlation coefficients
are plotted against their corresponding shape parameters.
The maximum correlation coefficient corresponds to the
optimal value of the shape parameter. For better precision,
two iterations of the PPCC plot can be generated; the first is
for finding the right neighborhood and the second is for fine
tuning the estimate.
The PPCC plot is used first to find a good value of the shape
parameter. The probability plot is then generated to find
estimates of the location and scale parameters and in
addition to provide a graphical assessment of the adequacy
of the distributional fit.
Compare
Distributions
In addition to finding a good choice for estimating the shape
parameter of a given distribution, the PPCC plot can be
useful in deciding which distributional family is most
appropriate. For example, given a set of reliabilty data, you
might generate PPCC plots for a Weibull, lognormal,
gamma, and inverse Gaussian distributions, and possibly
others, on a single page. This one page would show the best
value for the shape parameter for several distributions and
would additionally indicate which of these distributional

families provides the best fit (as measured by the maximum
probability plot correlation coefficient). That is, if the
maximum PPCC value for the Weibull is 0.99 and only 0.94
for the lognormal, then we could reasonably conclude that
the Weibull family is the better choice.
Tukey-
Lambda
PPCC Plot
for Symmetric
Distributions
The Tukey Lambda PPCC plot, with shape parameter λ, is
particularly useful for symmetric distributions. It indicates
whether a distribution is short or long tailed and it can
further indicate several common distributions. Specifically,
1. λ = -1: distribution is approximately Cauchy
2. λ = 0: distribution is exactly logistic
3. λ = 0.14: distribution is approximately normal
4. λ = 0.5: distribution is U-shaped
5. λ = 1: distribution is exactly uniform
If the Tukey Lambda PPCC plot gives a maximum value
near 0.14, we can reasonably conclude that the normal
distribution is a good model for the data. If the maximum
value is less than 0.14, a long-tailed distribution such as the
double exponential or logistic would be a better choice. If
the maximum value is near -1, this implies the selection of
very long-tailed distribution, such as the Cauchy. If the
maximum value is greater than 0.14, this implies a short-
tailed distribution such as the Beta or uniform.
The Tukey-Lambda PPCC plot is used to suggest an
appropriate distribution. You should follow-up with PPCC
and probability plots of the appropriate alternatives.
Use
Judgement
When
Selecting An
Appropriate
Distributional
Family
When comparing distributional models, do not simply
choose the one with the maximum PPCC value. In many
cases, several distributional fits provide comparable PPCC
values. For example, a lognormal and Weibull may both fit a
given set of reliability data quite well. Typically, we would
consider the complexity of the distribution. That is, a
simpler distribution with a marginally smaller PPCC value
may be preferred over a more complex distribution.
Likewise, there may be theoretical justification in terms of
the underlying scientific model for preferring a distribution
with a marginally smaller PPCC value in some cases. In
other cases, we may not need to know if the distributional
model is optimal, only that it is adequate for our purposes.
That is, we may be able to use techniques designed for
normally distributed data even if other distributions fit the
data somewhat better.
Sample Plot The following is a PPCC plot of 100 normal random
numbers. The maximum value of the correlation coefficient
= 0.997 at λ = 0.099.
This PPCC plot shows that:
1. the best-fit symmetric distribution is nearly normal;

2. the data are not long tailed;
3. the sample mean would be an appropriate estimator of
location.
We can follow-up this PPCC plot with a normal probability
plot to verify the normality model for the data.
Definition: The PPCC plot is formed by:
Vertical axis: Probability plot correlation coefficient;
Horizontal axis: Value of shape parameter.
Questions The PPCC plot answers the following questions:
1. What is the best-fit member within a distributional
family?
2. Does the best-fit member provide a good fit (in terms
of generating a probability plot with a high correlation
coefficient)?
3. Does this distributional family provide a good fit
compared to other distributions?
4. How sensitive is the choice of the shape parameter?
Importance Many statistical analyses are based on distributional
assumptions about the population from which the data have
been obtained. However, distributional families can have
radically different shapes depending on the value of the
shape parameter. Therefore, finding a reasonable choice for
the shape parameter is a necessary step in the analysis. In
many analyses, finding a good distributional model for the
data is the primary focus of the analysis. In both of these
cases, the PPCC plot is a valuable tool.
Related
Techniques
Probability Plot
Maximum Likelihood Estimation
Least Squares Estimation
Method of Moments Estimation
Software PPCC plots are currently not available in most common
underlying technique is based on probability plots and
correlation coefficients, so it should be possible to write
macros for PPCC plots in statistical programs that support
these capabilities. Dataplot supports PPCC plots.
Purpose:
Check If
Two Data
Sets Can Be
Fit With the
Same
Distribution
The quantile-quantile (q-q) plot is a graphical technique for
determining if two data sets come from populations with a
common distribution.
A q-q plot is a plot of the quantiles of the first data set against
the quantiles of the second data set. By a quantile, we mean
the fraction (or percent) of points below the given value. That
is, the 0.3 (or 30%) quantile is the point at which 30% percent
of the data fall below and 70% fall above that value.
A 45-degree reference line is also plotted. If the two sets come
from a population with the same distribution, the points
should fall approximately along this reference line. The
greater the departure from this reference line, the greater the
evidence for the conclusion that the two data sets have come
from populations with different distributions.

The advantages of the q-q plot are:
1. The sample sizes do not need to be equal.
2. Many distributional aspects can be simultaneously
tested. For example, shifts in location, shifts in scale,
changes in symmetry, and the presence of outliers can
all be detected from this plot. For example, if the two
data sets come from populations whose distributions
differ only by a shift in location, the points should lie
along a straight line that is displaced either up or down
from the 45-degree reference line.
The q-q plot is similar to a probability plot. For a probability
plot, the quantiles for one of the data samples are replaced
with the quantiles of a theoretical distribution.
Sample Plot
This q-q plot of the JAHANMI2.DAT data set shows that
1. These 2 batches do not appear to have come from
populations with a common distribution.
2. The batch 1 values are significantly higher than the
corresponding batch 2 values.
3. The differences are increasing from values 525 to 625.
Then the values for the 2 batches get closer again.
Definition:
Quantiles
for Data Set
1 Versus
Quantiles of
Data Set 2
The q-q plot is formed by:
Vertical axis: Estimated quantiles from data set 1
Horizontal axis: Estimated quantiles from data set 2
Both axes are in units of their respective data sets. That is, the
actual quantile level is not plotted. For a given point on the q-
q plot, we know that the quantile level is the same for both
points, but not what that quantile level actually is.
If the data sets have the same size, the q-q plot is essentially a
plot of sorted data set 1 against sorted data set 2. If the data
sets are not of equal size, the quantiles are usually picked to
correspond to the sorted values from the smaller data set and
then the quantiles for the larger data set are interpolated.
Questions The q-q plot is used to answer the following questions:
Do two data sets come from populations with a
common distribution?
Do two data sets have common location and scale?
Do two data sets have similar distributional shapes?
Do two data sets have similar tail behavior?
Importance:
Check for
Common
Distribution
When there are two data samples, it is often desirable to know
if the assumption of a common distribution is justified. If so,
then location and scale estimators can pool both data sets to
obtain estimates of the common location and scale. If two

samples do differ, it is also useful to gain some understanding
of the differences. The q-q plot can provide more insight into
the nature of the difference than analytical methods such as
the chi-square and Kolmogorov-Smirnov 2-sample tests.
Related
Techniques
Bihistogram
T Test
F Test
2-Sample Chi-Square Test
2-Sample Kolmogorov-Smirnov Test
Case Study The quantile-quantile plot is demonstrated in the ceramic
strength data case study.
Software Q-Q plots are available in some general purpose statistical
software programs. If the number of data points in the two
samples are equal, it should be relatively easy to write a
macro in statistical programs that do not support the q-q plot.
If the number of points are not equal, writing a macro for a q-
q plot may be difficult.
Run-Sequence Plot
Purpose:
Check for
Shifts in
Location
and Scale
and Outliers
Run sequence plots (Chambers 1983) are an easy way to
graphically summarize a univariate data set. A common
assumption of univariate data sets is that they behave like:
1. random drawings;
3. with a common location; and
4. with a common scale.
With run sequence plots, shifts in location and scale are
typically quite evident. Also, outliers can easily be detected.
Sample
Plot:
Last Third
of Data
Shows a
Shift of
Location
This sample run sequence plot of the MAVRO.DAT data set
shows that the location shifts up for the last third of the data.
Definition:
y(i) Versus i
Run sequence plots are formed by:
Vertical axis: Response variable Yi
Horizontal axis: Index i (i = 1, 2, 3, ... )
Questions The run sequence plot can be used to answer the following
questions
1. Are there any shifts in location?
2. Are there any shifts in variation?

The run sequence plot can also give the analyst an excellent
feel for the data.
Importance:
Check
Univariate
Assumptions
For univariate data, the default model is
where the error is assumed to be random, from a fixed
distribution, and with constant location and scale. The validity
of this model depends on the validity of these assumptions.
The run sequence plot is useful for checking for constant
location and scale.
Even for more complex models, the assumptions on the error
term are still often the same. That is, a run sequence plot of
the residuals (even from very complex models) is still vital
for checking for outliers and for detecting shifts in location
and scale.
Related
Techniques
Scatter Plot
Histogram
Lag Plot
Case Study The run sequence plot is demonstrated in the Filter
transmittance data case study.
Software Run sequence plots are available in most general purpose
Scatter Plot
Purpose:
Check for
Relationship
A scatter plot (Chambers 1983) reveals relationships or
association between two variables. Such relationships
manifest themselves by any non-random structure in the plot.
Various common types of patterns are demonstrated in the
examples.
Sample
Plot:
Linear
Relationship
Between
Variables Y
and X
This sample plot of the Alaska pipeline data reveals a linear
relationship between the two variables indicating that a linear
regression model might be appropriate.
Definition:
Y Versus X
A scatter plot is a plot of the values of Y versus the
corresponding values of X:
Vertical axis: variable Y--usually the response variable
Horizontal axis: variable X--usually some variable we
suspect may ber related to the response

Questions Scatter plots can provide answers to the following questions:
1. Are variables X and Y related?
2. Are variables X and Y linearly related?
3. Are variables X and Y non-linearly related?
4. Does the variation in Y change depending on X?
Examples 1. No relationship
2. Strong linear (positive correlation)
3. Strong linear (negative correlation)
4. Exact linear (positive correlation)
5. Quadratic relationship
6. Exponential relationship
7. Sinusoidal relationship (damped)
8. Variation of Y doesn't depend on X (homoscedastic)
9. Variation of Y does depend on X (heteroscedastic)
10. Outlier
Combining
Scatter
Plots
Scatter plots can also be combined in multiple plots per page
to help understand higher-level structure in data sets with
more than two variables.
The scatterplot matrix generates all pairwise scatter plots on a
single page. The conditioning plot, also called a co-plot or
subset plot, generates scatter plots of Y versus X dependent on
the value of a third variable.
Causality Is
Not Proved
By
Association
The scatter plot uncovers relationships in data.
"Relationships" means that there is some structured
association (linear, quadratic, etc.) between X and Y. Note,
however, that even though
causality implies association
association does NOT imply causality.
Scatter plots are a useful diagnostic tool for determining
association, but if such association exists, the plot may or may
not suggest an underlying cause-and-effect mechanism. A
scatter plot can never "prove" cause and effect--it is
ultimately only the researcher (relying on the underlying
science/engineering) who can conclude that causality actually
exists.
Appearance The most popular rendition of a scatter plot is
1. some plot character (e.g., X) at the data points, and
2. no line connecting data points.
Other scatter plot format variants include
1. an optional plot character (e.g, X) at the data points, but
2. a solid line connecting data points.
In both cases, the resulting plot is referred to as a scatter plot,
although the former (discrete and disconnected) is the author's
personal preference since nothing makes it onto the screen
except the data--there are no interpolative artifacts to bias the
interpretation.
Related
Techniques
Run Sequence Plot
Box Plot
Block Plot
Case Study The scatter plot is demonstrated in the load cell calibration
data case study.
Software Scatter plots are a fundamental technique that should be
available in any general purpose statistical software program.

Scatter plots are also available in most graphics and
spreadsheet programs as well.
Scatter Plot: No Relationship
Scatter Plot
with No
Relationship
Discussion Note in the plot above how for a given value of X (say X =
0.5), the corresponding values of Y range all over the place
from Y = -2 to Y = +2. The same is true for other values of X.
This lack of predictablility in determining Y from a given
value of X, and the associated amorphous, non-structured
appearance of the scatter plot leads to the summary
conclusion: no relationship.
Scatter Plot: Strong Linear (positive
correlation) Relationship
Scatter
Plot
Showing
Strong
Positive
Linear
Correlation
Discussion Note in the plot above of the LEW3.DAT data set how a
straight line comfortably fits through the data; hence a linear
relationship exists. The scatter about the line is quite small, so
there is a strong linear relationship. The slope of the line is
positive (small values of X correspond to small values of Y;
large values of X correspond to large values of Y), so there is a
positive co-relation (that is, a positive correlation) between X
and Y.
Scatter Plot: Strong Linear (negative
correlation) Relationship

Scatter
Plot
Showing a
Strong
Negative
Correlation
Discussion Note in the plot above how a straight line comfortably fits
through the data; hence there is a linear relationship. The
scatter about the line is quite small, so there is a strong linear
relationship. The slope of the line is negative (small values of
X correspond to large values of Y; large values of X correspond
to small values of Y), so there is a negative co-relation (that is,
a negative correlation) between X and Y.
Scatter Plot: Exact Linear (positive correlation)
Relationship
Scatter Plot
Showing an
Exact
Linear
Relationship
Discussion Note in the plot above how a straight line comfortably fits
through the data; hence there is a linear relationship. The
scatter about the line is zero--there is perfect predictability
between X and Y), so there is an exact linear relationship. The
slope of the line is positive (small values of X correspond to
small values of Y; large values of X correspond to large values
of Y), so there is a positive co-relation (that is, a positive
correlation) between X and Y.
Scatter Plot: Quadratic Relationship

Scatter Plot
Showing
Quadratic
Relationship
Discussion Note in the plot above how no imaginable simple straight line
could ever adequately describe the relationship between X and
Y--a curved (or curvilinear, or non-linear) function is needed.
The simplest such curvilinear function is a quadratic model
[ Y_{i} = A + BX_{i} + CX_{i}^{2} + E_{i} ]
for some A, B, and C. Many other curvilinear functions are
possible, but the data analysis principle of parsimony suggests
that we try fitting a quadratic function first.
Scatter Plot: Exponential Relationship
Scatter Plot
Showing
Exponential
Relationship
Discussion Note that a simple straight line is grossly inadequate in
describing the relationship between X and Y in this plot of the
CHWIRUT2.DAT data set. A quadratic model would prove
lacking, especially for large values of X. In this example, the
large values of X correspond to nearly constant values of Y,
and so a non-linear function beyond the quadratic is needed.
Among the many other non-linear functions available, one of
the simpler ones is the exponential model
[ Y_{i} = A + B e^{CX_{i}} + E_{i} ]
for some A, B, and C. In this case, an exponential function
would, in fact, fit well, and so one is led to the summary
conclusion of an exponential relationship.
Scatter Plot: Sinusoidal Relationship (damped)

Scatter Plot
Showing a
Sinusoidal
Relationship
Discussion The complex relationship between X and Y in this plot of the
LUTHER.DAT data set appears to be basically oscillatory,
and so one is naturally drawn to the trigonometric sinusoidal
model:
E_{i} ]
Closer inspection of the scatter plot reveals that the amount of
swing (the amplitude α in the model) does not appear to be
constant but rather is decreasing (damping) as X gets large.
We thus would be led to the conclusion: damped sinusoidal
relationship, with the simplest corresponding model being
[ Y_{i} = C + (B_0 + B_1*t_{i})sin{(2piomega
t_{i} + phi)} + E_{i} ]
Scatter Plot: Variation of Y Does Not Depend
on X (homoscedastic)
Scatter Plot
Showing
Homoscedastic
Variability
Discussion This scatter plot reveals a linear relationship between X and
Y: for a given value of X, the predicted value of Y will fall
on a line. The plot further reveals that the variation in Y
about the predicted value is about the same (+- 10 units),
regardless of the value of X. Statistically, this is referred to
as homoscedasticity. Such homoscedasticity is very
important as it is an underlying assumption for regression,
and its violation leads to parameter estimates with inflated
variances. If the data are homoscedastic, then the usual
regression estimates can be used. If the data are not
homoscedastic, then the estimates can be improved using
weighting procedures as shown in the next example.

Scatter Plot: Variation of Y Does Depend on X
(heteroscedastic)
Scatter Plot
Showing
Heteroscedastic
Variability
Discussion This scatter plot of the Alaska pipeline data reveals an
approximate linear relationship between X and Y, but more
importantly, it reveals a statistical condition referred to as
heteroscedasticity (that is, nonconstant variation in Y over
the values of X). For a heteroscedastic data set, the
variation in Y differs depending on the value of X. In this
example, small values of X yield small scatter in Y while
large values of X result in large scatter in Y.
Heteroscedasticity complicates the analysis somewhat, but
its effects can be overcome by:
1. proper weighting of the data with noisier data being
weighted less, or by
2. performing a Y variable transformation to achieve
homoscedasticity. The Box-Cox normality plot can
help determine a suitable transformation.
Impact of
Ignoring
Unequal
Variability in
the Data
Fortunately, unweighted regression analyses on
heteroscedastic data produce estimates of the coefficients
that are unbiased. However, the coefficients will not be as
precise as they would be with proper weighting.
Note further that if heteroscedasticity does exist, it is
frequently useful to plot and model the local variation (
mbox{var}(Y_i | X_i) ) as a function of X, as in (
mbox{var}(Y_i | X_i) = g(X_i) ) This modeling has two
advantages:
1. it provides additional insight and understanding as to
how the response Y relates to X; and
2. it provides a convenient means of forming weights
for a weighted regression by simply using
[ w_i = W(Y_i | X_i) = frac{1} {mbox{Var}
(Y_i | X_i)} = frac{1} {g(X_i)} ]
The topic of non-constant variation is discussed in some
detail in the process modeling chapter.
Scatter Plot: Outlier

Scatter
Plot
Showing
Outliers
Discussion The scatter plot here reveals
1. a basic linear relationship between X and Y for most of
the data, and
2. a single outlier (at X = 375).
An outlier is defined as a data point that emanates from a
different model than do the rest of the data. The data here
appear to come from a linear model with a given slope and
variation except for the outlier which appears to have been
generated from some other model.
Outlier detection is important for effective modeling. Outliers
should be excluded from such model fitting. If all the data here
are included in a linear regression, then the fitted model will be
poor virtually everywhere. If the outlier is omitted from the
fitting process, then the resulting fit will be excellent almost
everywhere (for all points except the outlying point).
Scatterplot Matrix
Purpose:
Check
Pairwise
Relationships
Between
Variables
Given a set of variables X1, X2, ... , Xk, the scatterplot matrix
contains all the pairwise scatter plots of the variables on a
single page in a matrix format. That is, if there are k
variables, the scatterplot matrix will have k rows and k
columns and the ith row and jth column of this matrix is a
plot of Xi versus Xj.
Although the basic concept of the scatterplot matrix is
simple, there are numerous alternatives in the details of the
plots.
1. The diagonal plot is simply a 45-degree line since we
are plotting Xi versus Xi. Although this has some
usefulness in terms of showing the univariate
distribution of the variable, other alternatives are
common. Some users prefer to use the diagonal to
print the variable label. Another alternative is to plot
the univariate histogram on the diagonal. Alternatively,
we could simply leave the diagonal blank.
2. Since Xi versus Xj is equivalent to Xj versus Xi with
the axes reversed, some prefer to omit the plots below
the diagonal.
3. It can be helpful to overlay some type of fitted curve
on the scatter plot. Although a linear or quadratic fit
can be used, the most common alternative is to overlay
a lowess curve.

4. Due to the potentially large number of plots, it can be
somewhat tricky to provide the axes labels in a way
that is both informative and visually pleasing. One
alternative that seems to work well is to provide axis
labels on alternating rows and columns. That is, row
one will have tic marks and axis labels on the left
vertical axis for the first plot only while row two will
have the tic marks and axis labels for the right vertical
axis for the last plot in the row only. This alternating
pattern continues for the remaining rows. A similar
pattern is used for the columns and the horizontal axes
labels. Another alternative is to put the minimum and
maximum scale value in the diagonal plot with the
variable name.
5. Some analysts prefer to connect the scatter plots.
Others prefer to leave a little gap between each plot.
6. Although this plot type is most commonly used for
scatter plots, the basic concept is both simple and
powerful and extends easily to other plot formats that
involve pairwise plots such as the quantile-quantile
plot and the bihistogram.
Sample Plot
This sample plot was generated from pollution data collected
by NIST chemist Lloyd Currie.
There are a number of ways to view this plot. If we are
primarily interested in a particular variable, we can scan the
row and column for that variable. If we are interested in
finding the strongest relationship, we can scan all the plots
and then determine which variables are related.
Definition Given k variables, scatter plot matrices are formed by
creating k rows and k columns. Each row and column defines
a single scatter plot
The individual plot for row i and column j is defined as
Vertical axis: Variable Xi
Horizontal axis: Variable Xj
Questions The scatterplot matrix can provide answers to the following
questions:
1. Are there pairwise relationships between the
variables?
2. If there are relationships, what is the nature of these
relationships?
4. Is there clustering by groups in the data?

Linking and
Brushing
The scatterplot matrix serves as the foundation for the
concepts of linking and brushing.
By linking, we mean showing how a point, or set of points,
behaves in each of the plots. This is accomplished by
highlighting these points in some fashion. For example, the
highlighted points could be drawn as a filled circle while the
remaining points could be drawn as unfilled circles. A typical
application of this would be to show how an outlier shows
up in each of the individual pairwise plots. Brushing extends
this concept a bit further. In brushing, the points to be
highlighted are interactively selected by a mouse and the
scatterplot matrix is dynamically updated (ideally in real
time). That is, we can select a rectangular region of points in
one plot and see how those points are reflected in the other
plots. Brushing is discussed in detail by Becker, Cleveland,
and Wilks in the paper "Dynamic Graphics for Data
Analysis" (Cleveland and McGill, 1988).
Related
Techniques
Star plot
Scatter plot
Conditioning plot
Locally weighted least squares
Software Scatterplot matrices are becoming increasingly common in
general purpose statistical software programs. If a software
program does not generate scatterplot matrices, but it does
provide multiple plots per page and scatter plots, it should be
possible to write a macro to generate a scatterplot matrix.
Brushing is available in a few of the general purpose
statistical software programs that emphasize graphical
approaches.
Conditioning Plot
Purpose:
Check
pairwise
relationship
between
two
variables
conditional
on a third
variable
A conditioning plot, also known as a coplot or subset plot, is a
plot of two variables conditional on the value of a third
variable (called the conditioning variable). The conditioning
variable may be either a variable that takes on only a few
discrete values or a continuous variable that is divided into a
limited number of subsets.
One limitation of the scatterplot matrix is that it cannot show
interaction effects with another variable. This is the strength of
the conditioning plot. It is also useful for displaying scatter
plots for groups in the data. Although these groups can also be
plotted on a single plot with different plot symbols, it can
often be visually easier to distinguish the groups using the
conditioning plot.
Although the basic concept of the conditioning plot matrix is
simple, there are numerous alternatives in the details of the
plots.
1. It can be helpful to overlay some type of fitted curve on
the scatter plot. Although a linear or quadratic fit can be
used, the most common alternative is to overlay a
lowess curve.
2. Due to the potentially large number of plots, it can be
somewhat tricky to provide the axis labels in a way that
is both informative and visually pleasing. One
alternative that seems to work well is to provide axis
labels on alternating rows and columns. That is, row one
will have tic marks and axis labels on the left vertical
axis for the first plot only while row two will have the
tic marks and axis labels for the right vertical axis for
the last plot in the row only. This alternating pattern

continues for the remaining rows. A similar pattern is
used for the columns and the horizontal axis labels.
Note that this approach only works if the axes limits are
fixed to common values for all of the plots.
3. Some analysts prefer to connect the scatter plots. Others
prefer to leave a little gap between each plot.
Alternatively, each plot can have its own labeling with
the plots not connected.
4. Although this plot type is most commonly used for
scatter plots, the basic concept is both simple and
powerful and extends easily to other plot formats.
Sample
Plot
In this plot of of the PR1.DAT data set, temperature has six
distinct values. We plot torque versus time for each of these
temperatures. This example is discussed in more detail in the
process modeling chapter.
Definition Given the variables X, Y, and Z, the conditioning plot is
formed by dividing the values of Z into k groups. There are
several ways that these groups may be formed. There may be a
natural grouping of the data, the data may be divided into
several equal sized groups, the grouping may be determined
by clusters in the data, and so on. The page will be divided
into n rows and c columns where nc ≥ k. Each row and column
defines a single scatter plot.
The individual plot for row i and column j is defined as
Vertical axis: Variable Y
Horizontal axis: Variable X
where only the points in the group corresponding to the ith
row and jth column are used.
Questions The conditioning plot can provide answers to the following
questions:
1. Is there a relationship between two variables?
2. If there is a relationship, does the nature of the
relationship depend on the value of a third variable?
3. Are groups in the data similar?
Related
Techniques
Scatter plot
Scatterplot matrix
Locally weighted least squares
Software Conditioning plots are becoming increasingly common in
general purpose statistical software programs, including R and
Dataplot. If a software program does not generate conditioning
plots, but it does provide multiple plots per page and scatter

plots, it should be possible to write a macro to generate a
conditioning plot.
Spectral Plot
Purpose:
Examine
Cyclic
Structure
A spectral plot ( Jenkins and Watts 1968 or Bloomfield 1976)
is a graphical technique for examining cyclic structure in the
frequency domain. It is a smoothed Fourier transform of the
autocovariance function.
The frequency is measured in cycles per unit time where unit
time is defined to be the distance between 2 points. A
frequency of 0 corresponds to an infinite cycle while a
frequency of 0.5 corresponds to a cycle of 2 data points. Equi-
spaced time series are inherently limited to detecting
frequencies between 0 and 0.5.
Trends should typically be removed from the time series before
applying the spectral plot. Trends can be detected from a run
sequence plot. Trends are typically removed by differencing
the series or by fitting a straight line (or some other polynomial
curve) and applying the spectral analysis to the residuals.
Spectral plots are often used to find a starting value for the
frequency, ω, in the sinusoidal model
E_{i} ]
See the beam deflection case study for an example of this.
Sample
Plot
This spectral plot of the LEW.DAT data set shows one
dominant frequency of approximately 0.3 cycles per
observation.
Definition:
Variance
Versus
Frequency
The spectral plot is formed by:
Vertical axis: Smoothed variance (power)
Horizontal axis: Frequency (cycles per observation)
The computations for generating the smoothed variances can
be involved and are not discussed further here. The details can
be found in the Jenkins and Bloomfield references and in most
texts that discuss the frequency analysis of time series.
Questions The spectral plot can be used to answer the following
questions:
1. How many cyclic components are there?
2. Is there a dominant cyclic frequency?
3. If there is a dominant cyclic frequency, what is it?

Importance
Check
Cyclic
Behavior
of Time
Series
The spectral plot is the primary technique for assessing the
cyclic nature of univariate time series in the frequency domain.
It is almost always the second plot (after a run sequence plot)
generated in a frequency domain analysis of a time series.
Examples 1. Random (= White Noise)
2. Strong autocorrelation and autoregressive model
3. Sinusoidal model
Related
Techniques
Complex Demodulation Amplitude Plot
Case Study The spectral plot is demonstrated in the beam deflection data
case study.
Software Spectral plots are a fundamental technique in the frequency
analysis of time series. They are available in many general
purpose statistical software programs.
Spectral Plot: Random Data
Spectral
Plot of 200
Normal
Random
Numbers
of 200 normal random numbers.
1. There are no dominant peaks.
2. There is no identifiable pattern in the spectrum.
Discussion For random data, the spectral plot should show no dominant
peaks or distinct pattern in the spectrum. For the sample plot
above, there are no clearly dominant peaks and the peaks
seem to fluctuate at random. This type of appearance of the
spectral plot indicates that there are no significant cyclic
patterns in the data.
Spectral Plot: Strong Autocorrelation and
Autoregressive Model

Spectral Plot
for Random
Walk Data
of the FLICKER.DAT data set.
1. Strong dominant peak near zero.
2. Peak decays rapidly towards zero.
3. An autoregressive model is an appropriate model.
Discussion This spectral plot starts with a dominant peak near zero and
rapidly decays to zero. This is the spectral plot signature of
a process with strong positive autocorrelation. Such
processes are highly non-random in that there is high
association between an observation and a succeeding
observation. In short, if you know Yi you can make a strong
guess as to what Yi+1 will be.
Recommended
Next Step
The next step would be to determine the parameters for the
[ Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} ]
Such estimation can be done by linear regression or by
fitting a Box-Jenkins autoregressive (AR) model.
[ Y_{i} = A_0 + E_{i} ]
Then the system should be reexamined to find an
explanation for the strong autocorrelation. Is it due to the
1. phenomenon under study; or
2. drifting in the environment; or
3. contamination from the data acquisition system
(DAS)?
Oftentimes the source of the problem is item (3) above
where contamination and carry-over from the data
acquisition system result because the DAS does not have
time to electronically recover before collecting the next data
point. If this is the case, then consider slowing down the
sampling rate to re-achieve randomness.
Spectral Plot: Sinusoidal Model

Spectral Plot
for Sinusoidal
Model
of the LEW.DAT data set.
1. There is a single dominant peak at approximately 0.3.
2. There is an underlying single-cycle sinusoidal model.
Discussion This spectral plot shows a single dominant frequency. This
indicates that a single-cycle sinusoidal model might be
appropriate.
If one were to naively assume that the data represented by
the graph could be fit by the model
[ Y_{i} = A_0 + E_{i} ]
and then estimate the constant by the sample mean, the
analysis would be incorrect because
the sample mean is biased;
the confidence interval for the mean, which is valid
only for random data, is meaningless and too small.
On the other hand, the choice of the proper model
[ Y_{i} = C + alphasin{(2piomega t_{i} + phi)}
+ E_{i} ]
where α is the amplitude, ω is the frequency (between 0 and
.5 cycles per observation), and ( phi ) is the phase, can be
fit by non-linear least squares. The beam deflection data
case study demonstrates fitting this type of model.
Recommended
Next Steps
The recommended next steps are to:
1. Estimate the frequency from the spectral plot. This
will be helpful as a starting value for the subsequent
non-linear fitting. A complex demodulation phase
plot can be used to fine tune the estimate of the
frequency before performing the non-linear fit.
2. Do a complex demodulation amplitude plot to obtain
an initial estimate of the amplitude and to determine
if a constant amplitude is justified.
3. Carry out a non-linear fit of the model
[ Y_{i} = C + alphasin{(2piomega t_{i} +
phi)} + E_{i} ]

Purpose:
Detect
Changes in
Scale
Between
Groups
Standard deviation plots are used to see if the standard
deviation varies between different groups of the data. The
grouping is determined by the analyst. In most cases, the data
provide a specific grouping variable. For example, the groups
may be the levels of a factor variable. In the sample plot
below, the months of the year provide the grouping.
Standard deviation plots can be used with ungrouped data to
determine if the standard deviation is changing over time. In
this case, the data are broken into an arbitrary number of
equal-sized groups. For example, a data series with 400 points
can be divided into 10 groups of 40 points each. A standard
deviation plot can then be generated with these groups to see
if the standard deviation is increasing or decreasing over time.
Although the standard deviation is the most commonly used
measure of scale, the same concept applies to other measures
of scale. For example, instead of plotting the standard
deviation of each group, the median absolute deviation or the
average absolute deviation might be plotted instead. This
might be done if there were significant outliers in the data and
a more robust measure of scale than the standard deviation
was desired.
Standard deviation plots are typically used in conjunction
with mean plots. The mean plot would be used to check for
shifts in location while the standard deviation plot would be
used to check for shifts in scale.
Sample Plot
This sample standard deviation plot of the PBF11.DAT data
set shows
1. there is a shift in variation;
2. greatest variation is during the summer months.
Definition:
Group
Standard
Deviations
Versus
Group ID
Standard deviation plots are formed by:
Vertical axis: Group standard deviations
A reference line is plotted at the overall standard deviation.
Questions The standard deviation plot can be used to answer the
1. Are there any shifts in variation?
2. What is the magnitude of the shifts in variation?
3. Is there a distinct pattern in the shifts in variation?
Importance:
Checking
Assumptions
A common assumption in 1-factor analyses is that of equal
variances. That is, the variance is the same for different levels
of the factor variable. The standard deviation plot provides a
graphical check for that assumption. A common assumption

for univariate data is that the variance is constant. By
grouping the data into equi-sized intervals, the standard
deviation plot can provide a graphical test of this assumption.
Related
Techniques
Mean Plot
DOE Standard Deviation Plot
support a standard deviation plot. However, if the statistical
program can generate the standard deviation for a group, it
should be feasible to write a macro to generate this plot.
Star Plot
Purpose:
Display
Multivariate
Data
The star plot (Chambers 1983) is a method of displaying
multivariate data. Each star represents a single observation.
Typically, star plots are generated in a multi-plot format with
many stars on each page and each star representing one
observation.
Star plots are used to examine the relative values for a single
data point (e.g., point 3 is large for variables 2 and 4, small
for variables 1, 3, 5, and 6) and to locate similar points or
dissimilar points.
Sample Plot The plot below contains the star plots of 16 cars. The data file
actually contains 74 cars, but we restrict the plot to what can
reasonably be shown on one page. The variable list for the
sample star plot is
1. Price
2. Mileage (MPG)
3. 1978 Repair Record (1 = Worst, 5 = Best)
4. 1977 Repair Record (1 = Worst, 5 = Best)
5. Headroom
6. Rear Seat Room
7. Trunk Space
8. Weight
9. 9
10. Length
We can look at these plots individually or we can use them to
identify clusters of cars with similar features. For example,
we can look at the star plot of the Cadillac Seville and see that
it is one of the most expensive cars, gets below average (but
not among the worst) gas mileage, has an average repair
record, and has average-to-above-average roominess and size.
We can then compare the Cadillac models (the last three
plots) with the AMC models (the first three plots). This
comparison shows distinct patterns. The AMC models tend to
be inexpensive, have below average gas mileage, and are

small in both height and weight and in roominess. The
Cadillac models are expensive, have poor gas mileage, and
are large in both size and roominess.
Definition The star plot consists of a sequence of equi-angular spokes,
called radii, with each spoke representing one of the variables.
The data length of a spoke is proportional to the magnitude of
the variable for the data point relative to the maximum
magnitude of the variable across all data points. A line is
drawn connecting the data values for each spoke. This gives
the plot a star-like appearance and the origin of the name of
this plot.
Questions The star plot can be used to answer the following questions:
1. What variables are dominant for a given observation?
2. Which observations are most similar, i.e., are there
clusters of observations?
Weakness in
Technique
Star plots are helpful for small-to-moderate-sized multivariate
data sets. Their primary weakness is that their effectiveness is
limited to data sets with less than a few hundred points. After
that, they tend to be overwhelming.
Graphical techniques suited for large data sets are discussed
by Scott.
Related
Techniques
Alternative ways to plot multivariate data are discussed in
Chambers, du Toit, and Everitt.
Software Star plots are available in some general purpose statistical
software progams.
Weibull Plot
Purpose:
Graphical
Check To See
If Data Come
From a
Population
That Would
Be Fit by a
Weibull
Distribution
The Weibull plot (Nelson 1982) is a graphical technique for
determining if a data set comes from a population that
would logically be fit by a 2-parameter Weibull distribution
(the location is assumed to be zero).
The Weibull plot has special scales that are designed so that
if the data do in fact follow a Weibull distribution, the points
will be linear (or nearly linear). The least squares fit of this
line yields estimates for the shape and scale parameters of
the Weibull distribution (the location is assumed to be zero).
Specifically, the shape parameter is the reciprocal of the
slope of the fitted line and the scale parameter is the
exponent of the intercept of the fitted line.
The Weibull distribution also has the property that the scale
parameter falls at the 63.2% point irrespective of the value
of the shape parameter. The plot shows a horizontal line at
this 63.2% point and a vertical line where the horizontal line
intersects the least squares fitted line. This vertical line
shows the value of scale parameter.

Sample Plot
This Weibull plot of the FULLER2.DAT data set shows that:
1. the assumption of a Weibull distribution is reasonable;
2. the scale parameter estimate is computed to be 33.32;
3. the shape parameter estimate is computed to be 5.28;
and
4. there are no outliers.
Note that the values on the x-axis ("0", "1", and "2") are the
exponents. These actually denote the value 100 = 1, 101 =
10, and 102 = 100.
Definition:
Weibull
Cumulative
Probability
Versus
LN(Ordered
Response)
The Weibull plot is formed by:
Vertical axis: Weibull cumulative probability
expressed as a percentage
Horizontal axis: ordered failure times (in a LOG10
scale)
The vertical scale is ln(-ln(1-p)) where p=(i-0.3)/(n+0.4) and
i is the rank of the observation. This scale is chosen in order
to linearize the resulting plot for Weibull data.
Questions The Weibull plot can be used to answer the following
questions:
1. Do the data follow a 2-parameter Weibull
distribution?
2. What is the best estimate of the shape parameter for
the 2-parameter Weibull distribution?
3. What is the best estimate of the scale (= variation)
parameter for the 2-parameter Weibull distribution?
Importance:
Check
Distributional
Assumptions
Many statistical analyses, particularly in the field of
reliability, are based on the assumption that the data follow a
Weibull distribution. If the analysis assumes the data follow
a Weibull distribution, it is important to verify this
assumption and, if verified, find good estimates of the
Weibull parameters.
Related
Techniques
Weibull Probability Plot
Weibull PPCC Plot
Weibull Hazard Plot
The Weibull probability plot (in conjunction with the
Weibull PPCC plot), the Weibull hazard plot, and the
Weibull plot are all similar techniques that can be used for
assessing the adequacy of the Weibull distribution as a
model for the data, and additionally providing estimation for
the shape, scale, or location parameters.
The Weibull hazard plot and Weibull plot are designed to
handle censored data (which the Weibull probability plot
does not).

Case Study The Weibull plot is demonstrated in the fatigue life of
aluminum alloy specimens case study.
Software Weibull plots are generally available in statistical software
programs that are designed to analyze reliability data.
Purpose:
Interlab
Comparisons
Youden plots are a graphical technique for analyzing interlab
data when each lab has made two runs on the same product
or one run on two different products.
The Youden plot is a simple but effective method for
comparing both the within-laboratory variability and the
between-laboratory variability.
Sample Plot
This Youden plot of the UGIANSKY.DAT data set shows:
1. Not all labs are equivalent.
2. Lab 4 is biased low.
3. Lab 3 has within-lab variability problems.
4. Lab 5 has an outlying run.
Definition:
Response 1
Versus
Response 2
Coded by
Lab
Youden plots are formed by:
1. Vertical axis: Response variable 1 (i.e., run 1 or
2. Horizontal axis: Response variable 2 (i.e., run 2 or
In addition, the plot symbol is the lab id (typically an integer
from 1 to k where k is the number of labs). Sometimes a 45-
degree reference line is drawn. Ideally, a lab generating two
runs of the same product should produce reasonably similar
results. Departures from this reference line indicate
inconsistency from the lab. If two different products are
being tested, then a 45-degree line may not be appropriate.
However, if the labs are consistent, the points should lie near
some fitted straight line.
Questions The Youden plot can be used to answer the following
questions:
1. Are all labs equivalent?
2. What labs have between-lab problems
(reproducibility)?
3. What labs have within-lab problems (repeatability)?
4. What labs are outliers?
Importance In interlaboratory studies or in comparing two runs from the
same lab, it is useful to know if consistent results are

generated. Youden plots should be a routine plot for
analyzing this type of data.
DOE Youden
Plot
The DOE Youden plot is a specialized Youden plot used in
the design of experiments. In particular, it is useful for full
and fractional designs.
Related
Techniques
Scatter Plot
Software The Youden plot is essentially a scatter plot, so it should be
feasible to write a macro for a Youden plot in any general
purpose statistical program that supports scatter plots.
DOE Youden
Plot:
Introduction
The DOE (Design of Experiments) Youden plot is a specialized Youden plot used in the
analysis of full and fractional experiment designs. In particular, it is used in support of an
analysis of variance. These designs may have a low level, coded as "-1" or "-", and a high
level, coded as "+1" or "+", for each factor. In addition, there can optionally be one or more
center points. Center points are at the midpoint between the low and high levels for each
The DOE Youden plot only uses the "-1" and "+1" points and can be used to help determine
the appropriate model.
Construction
of DOE
Youden Plot
The following are the primary steps in the construction of the DOE Youden plot.
1. For a given factor or interaction term, compute the mean of the response variable for
the low level of the factor and for the high level of the factor. Any center points are
omitted from the computation.
2. Plot the point where the y-coordinate is the mean for the high level of the factor and
the x-coordinate is the mean for the low level of the factor. The character used for the
plot point should identify the factor or interaction term (e.g., "1" for factor 1, "13" for
the interaction between factors 1 and 3).
3. Repeat steps 1 and 2 for each factor and interaction term of the data.
The high and low values of the interaction terms are obtained by multiplying the
corresponding values of the main level factors. For example, the interaction term X13 is
obtained by multiplying the values for X1 with the corresponding values of X3. Since the
values for X1 and X3 are either "-1" or "+1", the resulting values for X13 are also either "-1"
or "+1".
In summary, the DOE Youden plot is a plot of the mean of the response variable for the high
level of a factor or interaction term against the mean of the response variable for the low
level of that factor or interaction term.
For unimportant factors and interaction terms, these mean values should be nearly the same.
For important factors and interaction terms, these mean values should be quite different. So
the interpretation of the plot is that unimportant factors should be clustered together near the
grand mean. Points that stand apart from this cluster identify important factors that should be
included in the model.
Sample DOE
Youden Plot
The following is a DOE Youden plot for the data used in the Eddy current case study. The

Interpretation
of the Sample
DOE Youden
Plot
From the above DOE Youden plot, we see that factors 1 and 2 stand out from the others. That
is, the mean response values for the low and high levels of factor 1 and factor 2 are quite
different. For factor 3 and the 2 and 3-term interactions, the mean response values for the low
and high levels are similar.
We would conclude from this plot that factors 1 and 2 are important and should be included
in our final model while the remaining factors and interactions should be omitted from the
final model.
Case Study The Eddy current case study demonstrates the use of the DOE Youden plot in the context of
Software DOE Youden plots are not typically available as built-in plots in statistical software
programs. However, it should be relatively straightforward to write a macro to generate this
plot in most general purpose statistical software programs.
Purpose:
Check
Underlying
Statistical
Assumptions
The 4-plot is a collection of 4 specific EDA graphical
techniques whose purpose is to test the assumptions that
underlie most measurement processes. A 4-plot consists of a
1. run sequence plot;
2. lag plot;
3. histogram;
4. normal probability plot.
If the 4 underlying assumptions of a typical measurement
process hold, then the above 4 plots will have a
characteristic appearance (see the normal random numbers
case study below); if any of the underlying assumptions fail
to hold, then it will be revealed by an anomalous
appearance in one or more of the plots. Several commonly
encountered situations are demonstrated in the case studies
below.

Although the 4-plot has an obvious use for univariate and
time series data, its usefulness extends far beyond that.
Many statistical models of the form
[ Y_{i} = f(X_{1}, ... , X_{k}) + E_{i} ]
have the same underlying assumptions for the error term.
That is, no matter how complicated the functional fit, the
assumptions on the underlying error term are still the same.
The 4-plot can and should be routinely applied to the
residuals when fitting models regardless of whether the
model is simple or complicated.
Sample Plot:
Process Has
Fixed
Location,
Fixed
Variation,
Non-Random
(Oscillatory),
Non-Normal
U-Shaped
Distribution,
and Has 3
Outliers.
This 4-plot of the LEW.DAT data set reveals the following:
1. the fixed location assumption is justified as shown by
the run sequence plot in the upper left corner.
2. the fixed variation assumption is justified as shown
by the run sequence plot in the upper left corner.
3. the randomness assumption is violated as shown by
the non-random (oscillatory) lag plot in the upper
right corner.
4. the assumption of a common, normal distribution is
violated as shown by the histogram in the lower left
corner and the normal probability plot in the lower
right corner. The distribution is non-normal and is a
U-shaped distribution.
5. there are several outliers apparent in the lag plot in
the upper right corner.
Definition:
1. Run
Sequence
Plot;
2. Lag Plot;
3. Histogram;
4. Normal
Probability
Plot
1. Run sequence plot to test fixed location and variation.
Vertically: Yi
Horizontally: i
2. Lag Plot to test randomness.
Vertically: Yi
Horizontally: Yi-1
3. Histogram to test (normal) distribution.
Vertically: Counts
Horizontally: Y
4. Normal probability plot to test normal distribution.
Vertically: Ordered Yi
Horizontally: Theoretical values from a normal
N(0,1) distribution for ordered Yi
Questions 4-plots can provide answers to many questions:
1. Is the process in-control, stable, and predictable?
2. Is the process drifting with respect to location?
3. Is the process drifting with respect to variation?

6. If the data are a time series, is white noise?
7. If the data are a time series and not white noise, is it
sinusoidal, autoregressive, etc.?
8. If the data are non-random, what is a better model?
9. Does the process follow a normal distribution?
10. If non-normal, what distribution does the process
follow?
11. Is the model
[ Y_{i} = A_0 + E_{i} ]
12. If the default model is insufficient, what is a better
model?
13. Is the formula [ s_{bar{Y}} = s/sqrt{N} ] valid?
14. Is the sample mean a good estimator of the process
location?
15. If not, what would be a better estimator?
Importance:
Testing
Underlying
Assumptions
Helps Ensure
the Validity of
the Final
Scientific and
Engineering
Conclusions
There are 4 assumptions that typically underlie all
1. random drawings;
3. with that distribution having a fixed location; and
4. with that distribution having fixed variation.
engineering. If the above 4 assumptions hold, then we have
achieved probabilistic predictability--the ability to make
probability statements not only about the process in the
processes are said to be "statistically in control". If the 4
assumptions do not hold, then we have a process that is
drifting (with respect to location, variation, or distribution),
is unpredictable, and is out of control. A simple
characterization of such processes by a location estimate, a
variation estimate, or a distribution "estimate" inevitably
leads to optimistic and grossly invalid engineering
conclusions.
Inasmuch as the validity of the final scientific and
engineering conclusions is inextricably linked to the
validity of these same 4 underlying assumptions, it
naturally follows that there is a real necessity for all 4
assumptions to be routinely tested. The 4-plot (run
sequence plot, lag plot, histogram, and normal probability
plot) is seen as a simple, efficient, and powerful way of
carrying out this routine checking.
Interpretation:
Flat, Equi-
Banded,
Random, Bell-
Shaped, and
Linear
Of the 4 underlying assumptions:
1. If the fixed location assumption holds, then the run
2. If the fixed variation assumption holds, then the
vertical spread in the run sequence plot will be
axis.
3. If the randomness assumption holds, then the lag plot
4. If the fixed distribution assumption holds (in
particular, if the fixed normal distribution assumption
holds), then the histogram will be bell-shaped and the
normal probability plot will be approximatelylinear.
If all 4 of the assumptions hold, then the process is
"statistically in control". In practice, many processes fall

short of achieving this ideal.
Related
Techniques
Run Sequence Plot
Lag Plot
Histogram
Spectral Plot
PPCC Plot
Case Studies The 4-plot is used in most of the case studies in this
chapter:
1. Normal random numbers (the ideal)
2. Uniform random numbers
3. Random walk
4. Josephson junction cryothermometry
5. Beam deflections
6. Filter transmittance
7. Standard resistor
8. Heat flow meter 1
general purpose statistical software program that supports
the capability for multiple plots per page and supports the
Purpose:
Graphical
Model
Validation
The 6-plot is a collection of 6 specific graphical techniques
whose purpose is to assess the validity of a Y versus X fit. The
fit can be a linear fit, a non-linear fit, a LOWESS (locally
weighted least squares) fit, a spline fit, or any other fit
utilizing a single independent variable.
The 6 plots are:
1. Scatter plot of the response and predicted values versus
the independent variable;
2. Scatter plot of the residuals versus the independent
variable;
3. Scatter plot of the residuals versus the predicted values;
4. Lag plot of the residuals;
5. Histogram of the residuals;
6. Normal probability plot of the residuals.
Sample Plot
This 6-plot of the PONTIUS.DAT data set, which followed a
linear fit, shows that the linear model is not adequate. It
suggests that a quadratic model would be a better model.

Definition:
6
Component
Plots
1. Response and predicted values
Vertical axis: Response variable, predicted values
2. Residuals versus independent variable
3. Residuals versus predicted values
Horizontal axis: Predicted values
4. Lag plot of residuals
Vertical axis: RES(I)
Horizontal axis: RES(I-1)
5. Histogram of residuals
Vertical axis: Counts
Horizontal axis: Residual values
6. Normal probability plot of residuals
Vertical axis: Ordered residuals
Horizontal axis: Theoretical values from a normal
N(0,1) distribution for ordered residuals
Questions The 6-plot can be used to answer the following questions:
1. Are the residuals approximately normally distributed
with a fixed location and scale?
3. Is the fit adequate?
4. Do the residuals suggest a better fit?
Importance:
Validating
Model
A model involving a response variable and a single
independent variable has the form:
[ Y_{i} = f(X_{i}) + E_{i} ]
where Y is the response variable, X is the independent
variable, f is the linear or non-linear fit function, and E is the
random component. For a good model, the error component
should behave like:
1. random drawings (i.e., independent);
3. with fixed location; and
4. with fixed variation.
In addition, for fitting models it is usually further assumed
that the fixed distribution is normal and the fixed location is
zero. For a good model the fixed variation should be as small
as possible. A necessary component of fitting models is to
verify these assumptions for the error component and to
assess whether the variation for the error component is
sufficiently small. The histogram, lag plot, and normal
probability plot are used to verify the fixed distribution,
location, and variation assumptions on the error component.
The plot of the response variable and the predicted values
versus the independent variable is used to assess whether the
variation is sufficiently small. The plots of the residuals
versus the independent variable and the predicted values is
used to assess the independence assumption.
Assessing the validity and quality of the fit in terms of the
above assumptions is an absolutely vital part of the model-
fitting process. No fit should be considered complete without
an adequate model validation step.
Related
Techniques
Linear Least Squares
Non-Linear Least Squares
Scatter Plot
Run Sequence Plot
Lag Plot

Histogram
Case Study The 6-plot is used in the Alaska pipeline data case study.
general purpose statistical software program that supports the
capability for multiple plots per page and supports the
Graphical Techniques:
By Problem Category
Univariate
y = c + e
Run Sequence
Plot: 1.3.3.25
Lag Plot:
1.3.3.15
Histogram:
1.3.3.14
Normal
Probability
Plot: 1.3.3.21
4-Plot: 1.3.3.32 PPCC Plot:
1.3.3.23
Weibull Plot:
1.3.3.30
Probability
Plot: 1.3.3.22
Box-Cox
Linearity Plot:
1.3.3.5
Box-Cox
Normality Plot:
1.3.3.6
Bootstrap Plot:
1.3.3.4
Time Series
y = f(t) + e
Run Sequence
Plot: 1.3.3.25
Spectral Plot:
1.3.3.27
Autocorrelation
Plot: 1.3.3.1
Complex
Demodulation
Amplitude
Plot: 1.3.3.8
Complex
Demodulation
Phase Plot:
1.3.3.9

1 Factor
y = f(x) + e
Scatter Plot:
1.3.3.26
Box Plot:
1.3.3.7
Bihistogram:
1.3.3.2
Quantile-
Quantile Plot:
1.3.3.24
Mean Plot:
1.3.3.20
Standard
Deviation Plot:
1.3.3.28
Multi-
Factor/Comparative
y =
f(xp, x1,x2,...,xk) +
e Block Plot:
1.3.3.3
Multi-
Factor/Screening
y =
f(x1,x2,x3,...,xk) +
e DOE Scatter
Plot: 1.3.3.11
DOE Mean
Plot: 1.3.3.12
DOE Standard
Deviation Plot:
1.3.3.13
Contour Plot:
1.3.3.10
Regression
y =
f(x1,x2,x3,...,xk) +
e
Scatter Plot:
1.3.3.26
6-Plot: 1.3.3.33 Linear
Correlation
Plot: 1.3.3.16
Linear
Intercept Plot:
1.3.3.17
Linear Slope
Plot: 1.3.3.18
Linear Residual
Standard
Deviation
Plot:1.3.3.19
Interlab
(y1,y2) = f(x) + e
Youden Plot:
1.3.3.31

Multivariate
(y1,y2,...,yp)
Star Plot:
1.3.3.29
Quantitative Techniques
Confirmatory
Statistics
The techniques discussed in this section are classical
statistical methods as opposed to EDA techniques. EDA and
classical techniques are not mutually exclusive and can be
used in a complementary fashion. For example, the analysis
can start with some simple graphical techniques such as the
4-plot followed by the classical confirmatory methods
discussed herein to provide more rigorous statements about
the conclusions. If the classical methods yield different
conclusions than the graphical analysis, then some effort
should be invested to explain why. Often this is an indication
that some of the assumptions of the classical techniques are
violated.
Many of the quantitative techniques fall into two broad
categories:
1. Interval estimation
2. Hypothesis tests
Interval
Estimates
It is common in statistics to estimate a parameter from a
sample of data. The value of the parameter using all of the
possible data, not just the sample data, is called the
population parameter or true value of the parameter. An
estimate of the true parameter value is made using the
sample data. This is called a point estimate or a sample
estimate.
For example, the most commonly used measure of location
is the mean. The population, or true, mean is the sum of all
the members of the given population divided by the number
of members in the population. As it is typically impractical
to measure every member of the population, a random
sample is drawn from the population. The sample mean is
calculated by summing the values in the sample and dividing
by the number of values in the sample. This sample mean is
then used as the point estimate of the population mean.
Interval estimates expand on point estimates by
incorporating the uncertainty of the point estimate. In the
example for the mean above, different samples from the
same population will generate different values for the sample
mean. An interval estimate quantifies this uncertainty in the
sample estimate by computing lower and upper values of an
interval which will, with a given level of confidence (i.e.,
probability), contain the population parameter.
Hypothesis
Tests
Hypothesis tests also address the uncertainty of the sample
estimate. However, instead of providing an interval, a
hypothesis test attempts to refute a specific claim about a
population parameter based on the sample data. For example,
the hypothesis might be one of the following:
the population mean is equal to 10
the population standard deviation is equal to 5
the means from two populations are equal
the standard deviations from 5 populations are equal
To reject a hypothesis is to conclude that it is false. However,
to accept a hypothesis does not mean that it is true, only that

we do not have evidence to believe otherwise. Thus
hypothesis tests are usually stated in terms of both a
condition that is doubted (null hypothesis) and a condition
that is believed (alternative hypothesis).
A common format for a hypothesis test is:
H0: A statement of the null hypothesis, e.g., two
population means are equal.
Ha: A statement of the alternative hypothesis, e.g.,
two population means are not equal.
Test
Statistic:
The test statistic is based on the specific
hypothesis test.
Significance
Level:
The significance level, α, defines the
sensitivity of the test. A value of α = 0.05
means that we inadvertently reject the null
hypothesis 5% of the time when it is in fact
true. This is also called the type I error. The
choice of α is somewhat arbitrary, although in
practice values of 0.1, 0.05, and 0.01 are
commonly used.
The probability of rejecting the null
hypothesis when it is in fact false is called the
power of the test and is denoted by 1 - β. Its
complement, the probability of accepting the
null hypothesis when the alternative
hypothesis is, in fact, true (type II error), is
called β and can only be computed for a
specific alternative hypothesis.
Critical
Region:
The critical region encompasses those values
of the test statistic that lead to a rejection of
the null hypothesis. Based on the distribution
of the test statistic and the significance level, a
cut-off value for the test statistic is computed.
Values either above or below or both
(depending on the direction of the test) this
cut-off define the critical region.
Practical
Versus
Statistical
Significance
It is important to distinguish between statistical significance
and practical significance. Statistical significance simply
means that we reject the null hypothesis. The ability of the
test to detect differences that lead to rejection of the null
hypothesis depends on the sample size. For example, for a
particularly large sample, the test may reject the null
hypothesis that two process means are equivalent. However,
in practice the difference between the two means may be
relatively small to the point of having no real engineering
significance. Similarly, if the sample size is small, a
difference that is large in engineering terms may not lead to
rejection of the null hypothesis. The analyst should not just
blindly apply the tests, but should combine engineering
judgement with statistical analysis.
Bootstrap
Uncertainty
Estimates
In some cases, it is possible to mathematically derive
appropriate uncertainty intervals. This is particularly true for
intervals based on the assumption of a normal distribution.
However, there are many cases in which it is not possible to
mathematically derive the uncertainty. In these cases, the
bootstrap provides a method for empirically determining an
appropriate interval.
Table of
Contents
Some of the more common classical quantitative techniques
are listed below. This list of quantitative techniques is by no
means meant to be exhaustive. Additional discussions of
classical statistical techniques are contained in the product
comparisons chapter.
Location
1. Measures of Location

2. Confidence Limits for the Mean and One
Sample t-Test
3. Two Sample t-Test for Equal Means
4. One Factor Analysis of Variance
5. Multi-Factor Analysis of Variance
Scale (or variability or spread)
1. Measures of Scale
2. Bartlett's Test
3. Chi-Square Test
4. F-Test
5. Levene Test
Skewness and Kurtosis
1. Measures of Skewness and Kurtosis
Randomness
1. Autocorrelation
2. Runs Test
Distributional Measures
1. Anderson-Darling Test
2. Chi-Square Goodness-of-Fit Test
3. Kolmogorov-Smirnov Test
Outliers
1. Detection of Outliers
2. Grubbs Test
3. Tietjen-Moore Test
4. Generalized Extreme Deviate Test
2-Level Factorial Designs
1. Yates Algorithm
Measures of Location
Location A fundamental task in many statistical analyses is to estimate
a location parameter for the distribution; i.e., to find a typical
or central value that best describes the data.
Definition
of Location
The first step is to define what we mean by a typical value.
For univariate data, there are three common definitions:
1. mean - the mean is the sum of the data points divided by
the number of data points. That is,
[ bar{Y} = sum_{i=1}^{N}Y_{i}/N ]
The mean is that value that is most commonly referred
to as the average. We will use the term average as a
synonym for the mean and the term typical value to
refer generically to measures of location.
2. median - the median is the value of the point which has
half the data smaller than that point and half the data
larger than that point. That is, if X1, X2, ... ,XN is a
random sample sorted from smallest value to largest
value, then the median is defined as:
[ tilde{Y} = Y_{(N+1)/2} ;;;;;; mbox{if
$N$ is odd} ]
[ tilde{Y} = (Y_{N/2} + Y_{(N/2)+1})/2
;;;;;; mbox{if $N$ is even} ]
3. mode - the mode is the value of the random sample that
occurs with the greatest frequency. It is not necessarily
unique. The mode is typically used in a qualitative
fashion. For example, there may be a single dominant
hump in the data perhaps two or more smaller humps in
the data. This is usually evident from a histogram of the
data.

When taking samples from continuous populations, we
need to be somewhat careful in how we define the
mode. That is, any specific value may not occur more
than once if the data are continuous. What may be a
more meaningful, if less exact measure, is the midpoint
of the class interval of the histogram with the highest
peak.
Why
Different
Measures
A natural question is why we have more than one measure of
the typical value. The following example helps to explain why
these alternative definitions are useful and necessary.
This plot shows histograms for 10,000 random numbers
generated from a normal, an exponential, a Cauchy, and a
lognormal distribution.
Normal
Distribution
The first histogram is a sample from a normal distribution.
The mean is 0.005, the median is -0.010, and the mode is
-0.144 (the mode is computed as the midpoint of the
histogram interval with the highest peak).
The normal distribution is a symmetric distribution with well-
behaved tails and a single peak at the center of the
distribution. By symmetric, we mean that the distribution can
be folded about an axis so that the 2 sides coincide. That is, it
behaves the same to the left and right of some center point.
For a normal distribution, the mean, median, and mode are
actually equivalent. The histogram above generates similar
estimates for the mean, median, and mode. Therefore, if a
histogram or normal probability plot indicates that your data
are approximated well by a normal distribution, then it is
reasonable to use the mean as the location estimator.
Exponential
Distribution
The second histogram is a sample from an exponential
distribution. The mean is 1.001, the median is 0.684, and the
mode is 0.254 (the mode is computed as the midpoint of the
The exponential distribution is a skewed, i. e., not symmetric,
distribution. For skewed distributions, the mean and median
are not the same. The mean will be pulled in the direction of
the skewness. That is, if the right tail is heavier than the left
tail, the mean will be greater than the median. Likewise, if the
left tail is heavier than the right tail, the mean will be less than
the median.
For skewed distributions, it is not at all obvious whether the
mean, the median, or the mode is the more meaningful
measure of the typical value. In this case, all three measures
are useful.
Cauchy
Distribution
The third histogram is a sample from a Cauchy distribution.
The mean is 3.70, the median is -0.016, and the mode is

-0.362 (the mode is computed as the midpoint of the
For better visual comparison with the other data sets, we
restricted the histogram of the Cauchy distribution to values
between -10 and 10. The full Cauchy data set in fact has a
minimum of approximately -29,000 and a maximum of
approximately 89,000.
The Cauchy distribution is a symmetric distribution with
heavy tails and a single peak at the center of the distribution.
The Cauchy distribution has the interesting property that
collecting more data does not provide a more accurate
estimate of the mean. That is, the sampling distribution of the
mean is equivalent to the sampling distribution of the original
data. This means that for the Cauchy distribution the mean is
useless as a measure of the typical value. For this histogram,
the mean of 3.7 is well above the vast majority of the data.
This is caused by a few very extreme values in the tail.
However, the median does provide a useful measure for the
typical value.
Although the Cauchy distribution is an extreme case, it does
illustrate the importance of heavy tails in measuring the mean.
Extreme values in the tails distort the mean. However, these
extreme values do not distort the median since the median is
based on ranks. In general, for data with extreme values in the
tails, the median provides a better estimate of location than
does the mean.
Lognormal
Distribution
The fourth histogram is a sample from a lognormal
distribution. The mean is 1.677, the median is 0.989, and the
mode is 0.680 (the mode is computed as the midpoint of the
The lognormal is also a skewed distribution. Therefore the
mean and median do not provide similar estimates for the
location. As with the exponential distribution, there is no
obvious answer to the question of which is the more
meaningful measure of location.
Robustness There are various alternatives to the mean and median for
measuring location. These alternatives were developed to
address non-normal data since the mean is an optimal
estimator if in fact your data are normal.
Tukey and Mosteller defined two types of robustness where
robustness is a lack of susceptibility to the effects of
nonnormality.
1. Robustness of validity means that the confidence
intervals for the population location have a 95% chance
of covering the population location regardless of what
the underlying distribution is.
2. Robustness of efficiency refers to high effectiveness in
the face of non-normal tails. That is, confidence
intervals for the population location tend to be almost as
narrow as the best that could be done if we knew the
true shape of the distributuion.
The mean is an example of an estimator that is the best we can
do if the underlying distribution is normal. However, it lacks
robustness of validity. That is, confidence intervals based on
the mean tend not to be precise if the underlying distribution
is in fact not normal.
The median is an example of a an estimator that tends to have
robustness of validity but not robustness of efficiency.
The alternative measures of location try to balance these two
concepts of robustness. That is, the confidence intervals for

the case when the data are normal should be almost as narrow
as the confidence intervals based on the mean. However, they
should maintain their validity even if the underlying data are
not normal. In particular, these alternatives address the
problem of heavy-tailed distributions.
Alternative
Measures of
Location
A few of the more common alternative location measures are:
1. Mid-Mean - computes a mean using the data between
the 25th and 75th percentiles.
2. Trimmed Mean - similar to the mid-mean except
different percentile values are used. A common choice
is to trim 5% of the points in both the lower and upper
tails, i.e., calculate the mean for data between the 5th
and 95th percentiles.
3. Winsorized Mean - similar to the trimmed mean.
However, instead of trimming the points, they are set to
the lowest (or highest) value. For example, all data
below the 5th percentile are set equal to the value of the
5th percentile and all data greater than the 95th
percentile are set equal to the 95th percentile.
4. Mid-range = (smallest + largest)/2.
The first three alternative location estimators defined above
have the advantage of the median in the sense that they are not
unduly affected by extremes in the tails. However, they
generate estimates that are closer to the mean for data that are
normal (or nearly so).
The mid-range, since it is based on the two most extreme
points, is not robust. Its use is typically restricted to situations
in which the behavior at the extreme points is relevant.
Case Study The uniform random numbers case study compares the
performance of several different location estimators for a
particular non-normal distribution.
compute at least some of the measures of location discussed
above.
Confidence Limits for the Mean
Purpose:
Interval
Estimate
for Mean
Confidence limits for the mean (Snedecor and Cochran, 1989) are an interval estimate for the
mean. Interval estimates are often desirable because the estimate of the mean varies from
sample to sample. Instead of a single estimate for the mean, a confidence interval generates a
lower and upper limit for the mean. The interval estimate gives an indication of how much
uncertainty there is in our estimate of the true mean. The narrower the interval, the more
precise is our estimate.
Confidence limits are expressed in terms of a confidence coefficient. Although the choice of
confidence coefficient is somewhat arbitrary, in practice 90 %, 95 %, and 99 % intervals are
often used, with 95 % being the most commonly used.
As a technical note, a 95 % confidence interval does not mean that there is a 95 % probability
that the interval contains the true mean. The interval computed from a given sample either
contains the true mean or it does not. Instead, the level of confidence is associated with the
method of calculating the interval. The confidence coefficient is simply the proportion of
samples of a given size that may be expected to contain the true mean. That is, for a 95 %
confidence interval, if many samples are collected and the confidence interval computed, in the
long run about 95 % of these intervals would contain the true mean.
Definition:
Confidence
Confidence limits are defined as:

Interval [ bar{Y} pm t_{1 - alpha/2, , N-1} ,, frac{s}{sqrt{N}} ]
where (bar{Y}) is the sample mean, s is the sample standard deviation, N is the sample size,
α is the desired significance level, and t1-α/2, N-1 is the 100(1-α/2) percentile of the t
distribution with N - 1 degrees of freedom. Note that the confidence coefficient is 1 - α.
From the formula, it is clear that the width of the interval is controlled by two factors:
1. As N increases, the interval gets narrower from the (sqrt{N}) term.
That is, one way to obtain more precise estimates for the mean is to increase the sample
size.
2. The larger the sample standard deviation, the larger the confidence interval. This simply
means that noisy data, i.e., data with a large standard deviation, are going to generate
wider intervals than data with a smaller standard deviation.
Definition:
Hypothesis
Test
To test whether the population mean has a specific value, (mu_{0}), against the two-sided
alternative that it does not have a value (mu_{0}), the confidence interval is converted to
hypothesis-test form. The test is a one-sample t-test, and it is defined as:
H0: ( mu = mu_{0} )
Ha: ( mu neq mu_{0} )
Test Statistic: ( T = (bar{Y} - mu_{0})/(s/sqrt{N}) )
where (bar{Y}), N, and s are defined as above.
Significance
Level:
α. The most commonly used value for α is 0.05.
Critical Region: Reject the null hypothesis that the mean is a specified value, (mu_{0}),
if
( T < t_{alpha/2, , N-1} )
or
( T > t_{1 - alpha/2, , N-1} )
Confidence
Interval
Example
We generated a 95 %, two-sided confidence interval for the ZARR13.DAT data set based on
the following information.
N = 195
MEAN = 9.261460
STANDARD DEVIATION = 0.022789
t1-0.025,N-1 = 1.9723
LOWER LIMIT = 9.261460 - 1.9723*0.022789/√195
UPPER LIMIT = 9.261460 + 1.9723*0.022789/√195
Thus, a 95 % confidence interval for the mean is (9.258242, 9.264679).
t-Test
Example
We performed a two-sided, one-sample t-test using the ZARR13.DAT data set to test the null
hypothesis that the population mean is equal to 5.
H0: μ = 5
Ha: μ ≠ 5
Test statistic: T = 2611.284
Degrees of freedom: ν = 194
Significance level: α = 0.05
Critical value: t1-α/2,ν = 1.9723
Critical region: Reject H0 if |T| > 1.9723
We reject the null hypotheses for our two-tailed t-test because the absolute value of the test
statistic is greater than the critical value. If we were to perform an upper, one-tailed test, the
critical value would be t1-α,ν = 1.6527, and we would still reject the null hypothesis.
The confidence interval provides an alternative to the hypothesis test. If the confidence interval
contains 5, then H0 cannot be rejected. In our example, the confidence interval (9.258242,
9.264679) does not contain 5, indicating that the population mean does not equal 5 at the 0.05
level of significance.

In general, there are three possible alternative hypotheses and rejection regions for the one-
sample t-test:
Alternative Hypothesis Rejection Region
Ha: μ ≠ μ0 |T| > t1-α/2,ν
Ha: μ > μ0 T > t1-α,ν
Ha: μ < μ0 T < tα,ν
The rejection regions for three posssible alternative hypotheses using our example data are
shown in the following graphs.
Questions Confidence limits for the mean can be used to answer the following questions:
1. What is a reasonable estimate for the mean?
2. How much variability is there in the estimate of the mean?
3. Does a given target value fall within the confidence limits?
Related
Techniques
Two-Sample t-Test
Confidence intervals for other location estimators such as the median or mid-mean tend to be
mathematically difficult or intractable. For these cases, confidence intervals can be obtained
using the bootstrap.
Case Study Heat flow meter data.
Software Confidence limits for the mean and one-sample t-tests are available in just about all general
purpose statistical software programs. Both Dataplot code and R code can be used to generate
the analyses in this section. These scripts use the ZARR13.DAT data file.

Two-Sample t-Test for Equal Means
Purpose:
Test if two
population
means are
equal
The two-sample t-test (Snedecor and Cochran, 1989) is used to determine if two population
means are equal. A common application is to test if a new process or treatment is superior to a
current process or treatment.
There are several variations on this test.
1. The data may either be paired or not paired. By paired, we mean that there is a one-to-
one correspondence between the values in the two samples. That is, if X1, X2, ..., Xn and
Y1, Y2, ... , Yn are the two samples, then Xi corresponds to Yi. For paired samples, the
difference Xi - Yi is usually calculated. For unpaired samples, the sample sizes for the
two samples may or may not be equal. The formulas for paired data are somewhat
simpler than the formulas for unpaired data.
2. The variances of the two samples may be assumed to be equal or unequal. Equal
variances yields somewhat simpler formulas, although with computers this is no longer a
significant issue.
3. In some applications, you may want to adopt a new process or treatment only if it
exceeds the current treatment by some threshold. In this case, we can state the null
hypothesis in the form that the difference between the two populations means is equal to
some constant (mu_{1} - mu_{2} = d_{0}) where the constant is the desired
threshold.
Definition The two-sample t-test for unpaired data is defined as:
H0: ( mu_{1} = mu_{2} )
Ha: ( mu_{1} neq mu_{2} )
Test Statistic: ( T = frac{bar{Y_{1}} - bar{Y_{2}}} {sqrt{{s^{2}_{1}}/N_{1} +
{s^{2}_{2}}/N_{2}}} )
where N1 and N2 are the sample sizes, ( bar{Y_{1}} ) and ( bar{Y_{2}} )
are the sample means, and ( {s^{2}_{1}} ) and ( {s^{2}_{2}} ) are the
sample variances.
If equal variances are assumed, then the formula reduces to:
( T = frac{bar{Y_{1}} - bar{Y_{2}}} {s_{p}sqrt{1/N_{1} +
1/N_{2}}} )
where
( s_{p}^{2} = frac{(N_{1}-1){s^{2}_{1}} + (N_{2}-1){s^{2}_{2}}}
{N_{1} + N_{2} - 2} )
Significance
Level:
α.
Critical
Region:
Reject the null hypothesis that the two means are equal if
|T| > t1-α/2,ν
where t1-α/2,ν is the critical value of the t distribution with ν degrees of freedom
where
( upsilon = frac{(s^{2}_{1}/N_{1} + s^{2}_{2}/N_{2})^{2}}
{(s^{2}_{1}/N_{1})^{2}/(N_{1}-1) +
(s^{2}_{2}/N_{2})^{2}/(N_{2}-1) } )
If equal variances are assumed, then ν = N1 + N2 - 2
Two-
Sample t-
Test
Example
The following two-sample t-test was generated for the AUTO83B.DAT data set. The data set
contains miles per gallon for U.S. cars (sample 1) and for Japanese cars (sample 2); the
summary statistics for each sample are shown below.
SAMPLE 1:
NUMBER OF OBSERVATIONS = 249
MEAN = 20.14458

STANDARD ERROR OF THE MEAN = 0.40652
SAMPLE 2:
MEAN = 30.48101
STANDARD ERROR OF THE MEAN = 0.68717
We are testing the hypothesis that the population means are equal for the two samples. We
assume that the variances for the two samples are equal.
H0: μ1 = μ2
Ha: μ1 ≠ μ2
Test statistic: T = -12.62059
Pooled standard deviation: sp = 6.34260
Critical value (upper tail): t1-α/2,ν = 1.9673
The absolute value of the test statistic for our example, 12.62059, is greater than the critical
value of 1.9673, so we reject the null hypothesis and conclude that the two population means
are different at the 0.05 significance level.
In general, there are three possible alternative hypotheses and rejection regions for the one-
sample t-test:
Alternative Hypothesis Rejection Region
Ha: μ1 ≠ μ2 |T| > t1-α/2,ν
Ha: μ1 > μ2 T > t1-α,ν
Ha: μ1 < μ2 T < tα,ν
For our two-tailed t-test, the critical value is t1-α/2,ν = 1.9673, where α = 0.05 and ν = 326. If
we were to perform an upper, one-tailed test, the critical value would be t1-α,ν = 1.6495. The
rejection regions for three posssible alternative hypotheses using our example data are shown
below.

Questions Two-sample t-tests can be used to answer the following questions:
1. Is process 1 equivalent to process 2?
2. Is the new process better than the current process?
3. Is the new process better than the current process by at least some pre-determined
threshold amount?
Related
Techniques
Confidence Limits for the Mean
Analysis of Variance
Case
Study
Ceramic strength data.
Software Two-sample t-tests are available in just about all general purpose statistical software programs.
Both Dataplot code and R code can be used to generate the analyses in this section. These
scripts use the AUTO83B.DAT data file.
Data Used for Two-Sample t-Test
Data Used
for Two-
Sample t-
Test
Example
The following is the data used for the two-sample t-test
example. The first column is miles per gallon for U.S. cars and
the second column is miles per gallon for Japanese cars. For
the t-test example, rows with the second column equal to -999
were deleted.
18 24
15 27
18 27
16 25
17 31
15 35
14 24
14 19

14 28
15 23
15 27
14 20
15 22
14 18
22 20
18 31
21 32
21 31
10 32
10 24
11 26
9 29
28 24
25 24
19 33
16 33
17 32
19 28
18 19
14 32
14 34
14 26
14 30
12 22
13 22
13 33
18 39
22 36
19 28
18 27
23 21
26 24
25 30
20 34
21 32
13 38
14 37
15 30
14 31
17 37
11 32
13 47
12 41
13 45
15 34
13 33
13 24
14 32
22 39
28 35
13 32
14 37
13 38
14 34
15 34
12 32
13 33
13 32
14 25
13 24
12 37
13 31
18 36
16 36
18 34
18 38
23 32
11 38
12 32
13 -999
12 -999
18 -999
21 -999
19 -999
21 -999
15 -999
16 -999
15 -999
11 -999
20 -999
21 -999
19 -999
15 -999
26 -999

25 -999
16 -999
16 -999
18 -999
16 -999
13 -999
14 -999
14 -999
14 -999
28 -999
19 -999
18 -999
15 -999
15 -999
16 -999
15 -999
16 -999
14 -999
17 -999
16 -999
15 -999
18 -999
21 -999
20 -999
13 -999
23 -999
20 -999
23 -999
18 -999
19 -999
25 -999
26 -999
18 -999
16 -999
16 -999
15 -999
22 -999
22 -999
24 -999
23 -999
29 -999
25 -999
20 -999
18 -999
19 -999
18 -999
27 -999
13 -999
17 -999
13 -999
13 -999
13 -999
30 -999
26 -999
18 -999
17 -999
16 -999
15 -999
18 -999
21 -999
19 -999
19 -999
16 -999
16 -999
16 -999
16 -999
25 -999
26 -999
31 -999
34 -999
36 -999
20 -999
19 -999
20 -999
19 -999
21 -999
20 -999
25 -999
21 -999
19 -999
21 -999
21 -999
19 -999
18 -999
19 -999
18 -999

18 -999
18 -999
30 -999
31 -999
23 -999
24 -999
22 -999
20 -999
22 -999
20 -999
21 -999
17 -999
18 -999
17 -999
18 -999
17 -999
16 -999
19 -999
19 -999
36 -999
27 -999
23 -999
24 -999
34 -999
35 -999
28 -999
29 -999
27 -999
34 -999
32 -999
28 -999
26 -999
24 -999
19 -999
28 -999
24 -999
27 -999
27 -999
26 -999
24 -999
30 -999
39 -999
35 -999
34 -999
30 -999
22 -999
27 -999
20 -999
18 -999
28 -999
27 -999
34 -999
31 -999
29 -999
27 -999
24 -999
23 -999
38 -999
36 -999
25 -999
38 -999
26 -999
22 -999
36 -999
27 -999
27 -999
32 -999
28 -999
31 -999
One-Factor ANOVA
Purpose:
Test for
Equal
Means
Across
Groups
One factor analysis of variance (Snedecor and Cochran, 1989)
is a special case of analysis of variance (ANOVA), for one
factor of interest, and a generalization of the two-sample t-test.
The two-sample t-test is used to decide whether two groups
(levels) of a factor have the same mean. One-way analysis of

variance generalizes this to levels where k, the number of
levels, is greater than or equal to 2.
For example, data collected on, say, five instruments have one
factor (instruments) at five levels. The ANOVA tests whether
instruments have a significant effect on the results.
Definition The Product and Process Comparisons chapter (chapter 7)
contains a more extensive discussion of one-factor ANOVA,
including the details for the mathematical computations of one-
way analysis of variance.
The model for the analysis of variance can be stated in two
mathematically equivalent ways. In the following discussion,
each level of each factor is called a cell. For the one-way case,
a cell and a level are equivalent since there is only one factor.
In the following, the subscript i refers to the level and the
subscript j refers to the observation within a level. For
example, Y23 refers to the third observation in the second level.
The first model is
[ Y_{ij} = mu_{i} + E_{ij} ]
This model decomposes the response into a mean for each cell
and an error term. The analysis of variance provides estimates
for each cell mean. These estimated cell means are the
predicted values of the model and the differences between the
response variable and the estimated cell means are the
residuals. That is
( hat{Y_{ij}} = hat{mu}_{i} )
( R_{ij} = Y_{ij} - hat{mu}_{i} )
The second model is
( Y_{ij} = mu + alpha_{i} + E_{ij} )
This model decomposes the response into an overall (grand)
mean, the effect of the ith factor level, and an error term. The
analysis of variance provides estimates of the grand mean and
the effect of the ith factor level. The predicted values and the
residuals of the model are
( hat{Y}_{ij} = hat{mu} + hat{alpha}_{i} )
( R_{ij} = Y_{ij} - hat{mu} - hat{alpha}_{i} )
The distinction between these models is that the second model
divides the cell mean into an overall mean and the effect of the
i-th factor level. This second model makes the factor effect
more explicit, so we will emphasize this approach.
Model
Validation
Note that the ANOVA model assumes that the error term, Eij,
should follow the assumptions for a univariate measurement
process. That is, after performing an analysis of variance, the
model should be validated by analyzing the residuals.
One-Way
ANOVA
Example
A one-way analysis of variance was generated for the GEAR.DAT data set. The
data set contains 10 measurements of gear diameter for ten different batches for a
total of 100 measurements.
DEGREES OF SUM OF MEAN
SOURCE FREEDOM SQUARES SQUARE F STATISTIC
---------------- ---------- -------- -------- -----------
BATCH 9 0.000729 0.000081 2.2969
RESIDUAL 90 0.003174 0.000035
TOTAL (CORRECTED) 99 0.003903 0.000039
RESIDUAL STANDARD DEVIATION = 0.00594

BATCH N MEAN SD(MEAN)
---------------------------------
1 10 0.99800 0.00188
2 10 0.99910 0.00188
3 10 0.99540 0.00188
4 10 0.99820 0.00188
5 10 0.99190 0.00188
6 10 0.99880 0.00188
7 10 1.00150 0.00188
8 10 1.00040 0.00188
9 10 0.99830 0.00188
10 10 0.99480 0.00188
The ANOVA table decomposes the variance into the following component sum of
squares:
Total sum of squares. The degrees of freedom for this entry is the number
of observations minus one.
Sum of squares for the factor. The degrees of freedom for this entry is the
number of levels minus one. The mean square is the sum of squares
divided by the number of degrees of freedom.
Residual sum of squares. The degrees of freedom is the total degrees of
freedom minus the factor degrees of freedom. The mean square is the sum
of squares divided by the number of degrees of freedom.
The sums of squares summarize how much of the variance in the data (total sum
of squares) is accounted for by the factor effect (batch sum of squares) and how
much is random error (residual sum of squares). Ideally, we would like most of
the variance to be explained by the factor effect.
The ANOVA table provides a formal F test for the factor effect. For our example,
we are testing the following hypothesis.
H0: All individual batch means are equal.
Ha: At least one batch mean is not equal to the others.
The F statistic is the batch mean square divided by the residual mean square. This
statistic follows an F distribution with (k-1) and (N-k) degrees of freedom. For
our example, the critical F value (upper tail) for α = 0.05, (k-1) = 9, and (N-k) =
90 is 1.9856. Since the F statistic, 2.2969, is greater than the critical value, we
conclude that there is a significant batch effect at the 0.05 level of significance.
Once we have determined that there is a significant batch effect, we might be
interested in comparing individual batch means. The batch means and the
standard errors of the batch means provide some information about the individual
batches. However, we may want to employ multiple comparison methods for a
more formal analysis. (See Box, Hunter, and Hunter for more information.)
In addition to the quantitative ANOVA output, it is recommended that any
analysis of variance be complemented with model validation. At a minimum, this
should include:
1. a run sequence plot of the residuals,
2. a normal probability plot of the residuals, and
3. a scatter plot of the predicted values against the residuals.
Question The analysis of variance can be used to answer the following question
Are means the same across groups in the data?
Importance The analysis of uncertainty depends on whether the factor significantly affects the
outcome.
Related
Techniques
Two-sample t-test
Multi-factor analysis of variance
Regression
Box plot
Software Most general purpose statistical software programs can generate an analysis of
variance. Both Dataplot code and R code can be used to generate the analyses in

this section. These scripts use the GEAR.DAT data file.
Multi-factor Analysis of Variance
Purpose:
Detect
significant
factors
The analysis of variance (ANOVA) (Neter, Wasserman, and Kutner, 1990) is used
to detect significant factors in a multi-factor model. In the multi-factor model,
there is a response (dependent) variable and one or more factor (independent)
variables. This is a common model in designed experiments where the
experimenter sets the values for each of the factor variables and then measures the
response variable.
Each factor can take on a certain number of values. These are referred to as the
levels of a factor. The number of levels can vary betweeen factors. For designed
experiments, the number of levels for a given factor tends to be small. Each factor
and level combination is a cell. Balanced designs are those in which the cells have
an equal number of observations and unbalanced designs are those in which the
number of observations varies among cells. It is customary to use balanced designs
in designed experiments.
Definition The Product and Process Comparisons chapter (chapter 7) contains a more
extensive discussion of two-factor ANOVA, including the details for the
mathematical computations.
The model for the analysis of variance can be stated in two mathematically
equivalent ways. We explain the model for a two-way ANOVA (the concepts are
the same for additional factors). In the following discussion, each combination of
factors and levels is called a cell. In the following, the subscript i refers to the level
of factor 1, j refers to the level of factor 2, and the subscript k refers to the kth
observation within the (i,j)th cell. For example, Y235 refers to the fifth observation
in the second level of factor 1 and the third level of factor 2.
The first model is
( Y_{ijk} = mu_{ij} + E_{ijk} )
This model decomposes the response into a mean for each cell and an error term.
The analysis of variance provides estimates for each cell mean. These cell means
are the predicted values of the model and the differences between the response
variable and the estimated cell means are the residuals. That is
( hat{Y}_{ijk} = hat{mu}_{ij} )
( R_{ijk} = Y_{ijk} - hat{mu}_{ij} )
The second model is
( Y_{ijk} = mu + alpha_{i} + beta_{j} + E_{ijk} )
This model decomposes the response into an overall (grand) mean, factor effects (
( hat{alpha}_{i} ) and ( hat{beta}_{j} ) represent the effects of the i-th level
of the first factor and the j-th level of the second factor, respectively), and an error
term. The analysis of variance provides estimates of the grand mean and the factor
effects. The predicted values and the residuals of the model are
( hat{Y}_{ijk} = hat{mu} + hat{alpha}_{i} + hat{beta}_{j} )
( R_{ijk} = Y_{ijk} - hat{mu} - hat{alpha}_{i} - hat{beta}_{j} )
The distinction between these models is that the second model divides the cell
mean into an overall mean and factor effects. This second model makes the factor
effect more explicit, so we will emphasize this approach.
Model
Validation
Note that the ANOVA model assumes that the error term, Eijk, should follow the
assumptions for a univariate measurement process. That is, after performing an
analysis of variance, the model should be validated by analyzing the residuals.
Multi-
Factor
An analysis of variance was performed for the JAHANMI2.DAT data set. The data
contains four, two-level factors: table speed, down feed rate, wheel grit size, and

ANOVA
Example
batch. There are 30 measurements of ceramic strength for each factor combination
for a total of 480 measurements.
SOURCE DF SUM OF SQUARES MEAN SQUARE F STATISTIC
------------------------------------------------------------------
TABLE SPEED 1 26672.726562 26672.726562 6.7080
DOWN FEED RATE 1 11524.053711 11524.053711 2.8982
WHEEL GRIT SIZE 1 14380.633789 14380.633789 3.6166
BATCH 1 727143.125000 727143.125000 182.8703
RESIDUAL 475 1888731.500000 3976.276855
TOTAL (CORRECTED) 479 2668446.000000 5570.868652
RESIDUAL STANDARD DEVIATION = 63.05772781
FACTOR LEVEL N MEAN SD(MEAN)
------------------------------------------------
TABLE SPEED -1 240 657.53168 2.87818
1 240 642.62286 2.87818
DOWN FEED RATE -1 240 645.17755 2.87818
1 240 654.97723 2.87818
WHEEL GRIT SIZE -1 240 655.55084 2.87818
1 240 644.60376 2.87818
BATCH 1 240 688.99890 2.87818
2 240 611.15594 2.87818
The ANOVA decomposes the variance into the following component sum of
squares:
Total sum of squares. The degrees of freedom for this entry is the number of
observations minus one.
Sum of squares for each of the factors. The degrees of freedom for these
entries are the number of levels for the factor minus one. The mean square is
the sum of squares divided by the number of degrees of freedom.
Residual sum of squares. The degrees of freedom is the total degrees of
freedom minus the sum of the factor degrees of freedom. The mean square is
the sum of squares divided by the number of degrees of freedom.
The analysis of variance summarizes how much of the variance in the data (total
sum of squares) is accounted for by the factor effects (factor sum of squares) and
how much is due to random error (residual sum of squares). Ideally, we would like
most of the variance to be explained by the factor effects. The ANOVA table
provides a formal F test for the factor effects. To test the overall batch effect in our
example we use the following hypotheses.
H0: All individual batch means are equal.
Ha: At least one batch mean is not equal to the others.
The F statistic is the mean square for the factor divided by the residual mean
square. This statistic follows an F distribution with (k-1) and (N-k) degrees of
freedom where k is the number of levels for the given factor. Here, we see that the
size of the "direction" effect dominates the size of the other effects. For our
example, the critical F value (upper tail) for α = 0.05, (k-1) = 1, and (N-k) = 475 is
3.86111. Thus, "table speed" and "batch" are significant at the 5 % level while
"down feed rate" and "wheel grit size" are not significant at the 5 % level.
In addition to the quantitative ANOVA output, it is recommended that any analysis
of variance be complemented with model validation. At a minimum, this should
include
1. A run sequence plot of the residuals.
2. A normal probability plot of the residuals.
3. A scatter plot of the predicted values against the residuals.
Questions The analysis of variance can be used to answer the following questions:
1. Do any of the factors have a significant effect?
2. Which is the most important factor?
3. Can we account for most of the variability in the data?
Related
Techniques
One-factor analysis of variance
Two-sample t-test
Box plot
Block plot
DOE mean plot

Case
Study
The quantitative ANOVA approach can be contrasted with the more graphical
EDA approach in the ceramic strength case study.
Software Most general purpose statistical software programs can perform multi-factor
analysis of variance. Both Dataplot code and R code can be used to generate the
analyses in this section. These scripts use the JAHANMI2.DAT data file.
Measures of Scale
Scale,
Variability,
or Spread
A fundamental task in many statistical analyses is to
characterize the spread, or variability, of a data set. Measures
of scale are simply attempts to estimate this variability.
When assessing the variability of a data set, there are two key
components:
1. How spread out are the data values near the center?
2. How spread out are the tails?
Different numerical summaries will give different weight to
these two elements. The choice of scale estimator is often
driven by which of these components you want to emphasize.
The histogram is an effective graphical technique for showing
both of these components of the spread.
Definitions
of
Variability
For univariate data, there are several common numerical
measures of the spread:
1. variance - the variance is defined as
( s^{2} = sum_{i=1}^{N}(Y_{i} -
bar{Y})^{2}/(N - 1) )
where (bar{Y}) is the mean of the data.
The variance is roughly the arithmetic average of the
squared distance from the mean. Squaring the distance
from the mean has the effect of giving greater weight to
values that are further from the mean. For example, a
point 2 units from the mean adds 4 to the above sum
while a point 10 units from the mean adds 100 to the
sum. Although the variance is intended to be an overall
measure of spread, it can be greatly affected by the tail
behavior.
2. standard deviation - the standard deviation is the square
root of the variance. That is,
( s = sqrt{sum_{i=1}^{N}(Y_{i} -
bar{Y})^{2}/(N - 1)} )
The standard deviation restores the units of the spread
to the original data units (the variance squares the
units).
3. range - the range is the largest value minus the smallest
value in a data set. Note that this measure is based only
on the lowest and highest extreme values in the sample.
The spread near the center of the data is not captured at
all.
4. average absolute deviation - the average absolute
deviation (AAD) is defined as
( AAD = sum_{i=1}^{N}(|Y_{i} - bar{Y}|)/N
)
where (bar{Y}) is the mean of the data and |Y| is the
absolute value of Y. This measure does not square the

distance from the mean, so it is less affected by extreme
observations than are the variance and standard
deviation.
5. median absolute deviation - the median absolute
deviation (MAD) is defined as
( MAD = median (|Y_{i} - tilde{Y}|) )
where (tilde{Y}) is the median of the data and |Y| is
the absolute value of Y. This is a variation of the
average absolute deviation that is even less affected by
extremes in the tail because the data in the tails have
less influence on the calculation of the median than they
do on the mean.
6. interquartile range - this is the value of the 75th
percentile minus the value of the 25th percentile. This
measure of scale attempts to measure the variability of
points near the center.
In summary, the variance, standard deviation, average
absolute deviation, and median absolute deviation measure
both aspects of the variability; that is, the variability near the
center and the variability in the tails. They differ in that the
average absolute deviation and median absolute deviation do
not give undue weight to the tail behavior. On the other hand,
the range only uses the two most extreme points and the
interquartile range only uses the middle portion of the data.
Why
Different
Measures?
The following example helps to clarify why these alternative
defintions of spread are useful and necessary.
This plot shows histograms for 10,000 random numbers
generated from a normal, a double exponential, a Cauchy, and
a Tukey-Lambda distribution.
Normal
Distribution
The standard deviation is 0.997, the median absolute deviation
is 0.681, and the range is 7.87.
behaved tails and a single peak at the center of the
distribution. By symmetric, we mean that the distribution can
be folded about an axis so that the two sides coincide. That is,
it behaves the same to the left and right of some center point.
In this case, the median absolute deviation is a bit less than the
standard deviation due to the downweighting of the tails. The
range of a little less than 8 indicates the extreme values fall
within about 4 standard deviations of the mean. If a histogram
or normal probability plot indicates that your data are
approximated well by a normal distribution, then it is
reasonable to use the standard deviation as the spread
estimator.

Double
Exponential
Distribution
The second histogram is a sample from a double exponential
distribution. The standard deviation is 1.417, the median
absolute deviation is 0.706, and the range is 17.556.
Comparing the double exponential and the normal histograms
shows that the double exponential has a stronger peak at the
center, decays more rapidly near the center, and has much
longer tails. Due to the longer tails, the standard deviation
tends to be inflated compared to the normal. On the other
hand, the median absolute deviation is only slightly larger
than it is for the normal data. The longer tails are clearly
reflected in the value of the range, which shows that the
extremes fall about 6 standard deviations from the mean
compared to about 4 for the normal data.
Cauchy
Distribution
The standard deviation is 998.389, the median absolute
deviation is 1.16, and the range is 118,953.6.
The Cauchy distribution has the interesting property that
collecting more data does not provide a more accurate
estimate for the mean or standard deviation. That is, the
sampling distribution of the means and standard deviation are
equivalent to the sampling distribution of the original data.
That means that for the Cauchy distribution the standard
deviation is useless as a measure of the spread. From the
histogram, it is clear that just about all the data are between
about -5 and 5. However, a few very extreme values cause
both the standard deviation and range to be extremely large.
However, the median absolute deviation is only slightly larger
than it is for the normal distribution. In this case, the median
absolute deviation is clearly the better measure of spread.
Although the Cauchy distribution is an extreme case, it does
illustrate the importance of heavy tails in measuring the
spread. Extreme values in the tails can distort the standard
deviation. However, these extreme values do not distort the
median absolute deviation since the median absolute deviation
is based on ranks. In general, for data with extreme values in
the tails, the median absolute deviation or interquartile range
can provide a more stable estimate of spread than the standard
deviation.
Tukey-
Lambda
Distribution
The fourth histogram is a sample from a Tukey lambda
distribution with shape parameter λ = 1.2. The standard
deviation is 0.49, the median absolute deviation is 0.427, and
the range is 1.666.
The Tukey lambda distribution has a range limited to (-1/λ,1/
λ). That is, it has truncated tails. In this case the standard
deviation and median absolute deviation have closer values
than for the other three examples which have significant tails.
Robustness Tukey and Mosteller defined two types of robustness where
robustness is a lack of susceptibility to the effects of
nonnormality.
1. Robustness of validity means that the confidence
intervals for a measure of the population spread (e.g.,
the standard deviation) have a 95 % chance of covering
the true value (i.e., the population value) of that
measure of spread regardless of the underlying
distribution.
2. Robustness of efficiency refers to high effectiveness in
the face of non-normal tails. That is, confidence
intervals for the measure of spread tend to be almost as
narrow as the best that could be done if we knew the
true shape of the distribution.

The standard deviation is an example of an estimator that is
the best we can do if the underlying distribution is normal.
However, it lacks robustness of validity. That is, confidence
intervals based on the standard deviation tend to lack
precision if the underlying distribution is in fact not normal.
The median absolute deviation and the interquartile range are
estimates of scale that have robustness of validity. However,
they are not particularly strong for robustness of efficiency.
If histograms and probability plots indicate that your data are
in fact reasonably approximated by a normal distribution, then
it makes sense to use the standard deviation as the estimate of
scale. However, if your data are not normal, and in particular
if there are long tails, then using an alternative measure such
as the median absolute deviation, average absolute deviation,
or interquartile range makes sense. The range is used in some
applications, such as quality control, for its simplicity. In
addition, comparing the range to the standard deviation gives
an indication of the spread of the data in the tails.
Since the range is determined by the two most extreme points
in the data set, we should be cautious about its use for large
values of N.
Tukey and Mosteller give a scale estimator that has both
robustness of validity and robustness of efficiency. However,
it is more complicated and we do not give the formula here.
generate at least some of the measures of scale discusssed
above.
Bartlett's Test
Purpose:
Test for
Homogeneity
of Variances
Bartlett's test (Snedecor and Cochran, 1983) is used to test if
k samples have equal variances. Equal variances across
samples is called homogeneity of variances. Some statistical
tests, for example the analysis of variance, assume that
variances are equal across groups or samples. The Bartlett
test can be used to verify that assumption.
Bartlett's test is sensitive to departures from normality. That
is, if your samples come from non-normal distributions, then
Bartlett's test may simply be testing for non-normality. The
Levene test is an alternative to the Bartlett test that is less
sensitive to departures from normality.
Definition The Bartlett test is defined as:
H0: σ1
2 = σ2
2 = ... = σk
2
Ha: σi
2 ≠ σj
2 for at least one pair (i,j).
Test
Statistic:
The Bartlett test statistic is designed to test for
equality of variances across groups against the
alternative that variances are unequal for at
least two groups.
[ T = frac{(N-k) ln{s^{2}_{p}} -
sum_{i=1}^{k}(N_{i} - 1)ln{s^{2}_{i}}}{1
+ (1/(3(k-1)))((sum_{i=1}^{k}{1/(N_{i} -
1))} - 1/(N-k))} ]
In the above, si
2 is the variance of the ith
group, N is the total sample size, Ni is the
sample size of the ith group, k is the number
of groups, and sp
2 is the pooled variance. The

pooled variance is a weighted average of the
group variances and is defined as:
[ s^{2}_{p} = sum_{i=1}^{k}(N_{i} -
1)s^{2}_{i}/(N-k) ]
Significance
Level:
α
Critical
Region:
The variances are judged to be unequal if,
( T > chi^2_{1-alpha, , k-1} )
where ( chi^2_{1-alpha, , k-1} ) is the
critical value of the chi-square distribution
with k - 1 degrees of freedom and a
significance level of α.
An alternate definition (Dixon and Massey, 1969) is based on
an approximation to the F distribution. This definition is
given in the Product and Process Comparisons chapter
(chapter 7).
Example Bartlett's test was performed for the GEAR.DAT data set. The
data set contains 10 measurements of gear diameter for ten
different batches for a total of 100 measurements.
H0: σ1
2 = σ2
2 = ... = σ10
2
Ha: At least one σi
2 is not equal to the others.
Degrees of freedom: k - 1 = 9
Critical value: Χ 2
1-α,k-1 = 16.919
Critical region: Reject H0 if T > 16.919
We are testing the null hypothesis that the batch variances are
all equal. Because the test statistic is larger than the critical
value, we reject the null hypotheses at the 0.05 significance
level and conclude that at least one batch variance is different
from the others.
Question Bartlett's test can be used to answer the following question:
Is the assumption of equal variances valid?
Importance Bartlett's test is useful whenever the assumption of equal
variances is made. In particular, this assumption is made for
the frequently used one-way analysis of variance. In this case,
Bartlett's or Levene's test should be applied to verify the
assumption.
Related
Techniques
Box Plot
Levene Test
Chi-Square Test
Case Study Heat flow meter data
Software The Bartlett test is available in many general purpose statistical
software programs. Both Dataplot code and R code can be
used to generate the analyses in this section. These scripts use
the GEAR.DAT data file.

Chi-Square Test for the Variance
Purpose:
Test if the
variance is
equal to a
specified
value
A chi-square test ( Snedecor and Cochran, 1983) can be used to
test if the variance of a population is equal to a specified value.
This test can be either a two-sided test or a one-sided test. The
two-sided version tests against the alternative that the true
variance is either less than or greater than the specified value. The
one-sided version only tests in one direction. The choice of a two-
sided or one-sided test is determined by the problem. For example,
if we are testing a new process, we may only be concerned if its
variability is greater than the variability of the current process.
Definition The chi-square hypothesis test is defined as:
H0: ( sigma^2 = sigma_0^2 )
Ha: ( sigma^2 < sigma_0^2
)
for a lower one-tailed
test
( sigma^2 > sigma_0^2
)
for an upper one-
tailed test
( sigma^2 ne
sigma_0^2 )
for a two-tailed test
Test
Statistic:
( T = (N-1)(s/sigma_0)^2 )
where N is the sample size and s is the sample
standard deviation. The key element of this formula
is the ratio s/σ0 which compares the ratio of the
sample standard deviation to the target standard
deviation. The more this ratio deviates from 1, the
more likely we are to reject the null hypothesis.
Significance
Level:
α.
Critical
Region:
Reject the null hypothesis that the variance is a
specified value, σ0
2, if
( T > chi^2_{1-alpha, , N-1}
)
for an upper
one-tailed
alternative
( T < chi^2_{alpha, , N-1} ) for a lower one-
tailed
alternative
( T < chi^2_{alpha/2, , N-1}
) or ( T > chi^2_{1-
alpha/2, , N-1} )
for a two-tailed
alternative
where ( chi^2_{., , N-1} ) is the critical value of
the chi-square distribution with N - 1 degrees of
freedom.
The formula for the hypothesis test can easily be converted to
form an interval estimate for the variance:
[ sqrt{frac{(N-1)s^2}{chi^2_{1-alpha/2, , N-1}}} le
sigma le sqrt{frac{(N-1)s^2}{chi^2_{alpha/2, , N-1}}}
]
A confidence interval for the standard deviation is computed by
taking the square root of the upper and lower limits of the
confidence interval for the variance.
Chi-
Square
Test
Example
A chi-square test was performed for the GEAR.DAT data set. The
observed variance for the 100 measurements of gear diameter is
0.00003969 (the standard deviation is 0.0063). We will test the
null hypothesis that the true variance is equal to 0.01.
H0: σ2 = 0.01
Ha: σ2 ≠ 0.01

Degrees of freedom: N - 1 = 99
Critical values: Χ 2
α/2,N-1 = 73.361
Χ 2
1-α/2,N-1 = 128.422
Critical region: Reject H0 if T < 73.361 or T > 128.422
The test statistic value of 0.3903 is much smaller than the lower
critical value, so we reject the null hypothesis and conclude that
the variance is not equal to 0.01.
Questions The chi-square test can be used to answer the following questions:
1. Is the variance equal to some pre-determined threshold
value?
2. Is the variance greater than some pre-determined threshold
value?
3. Is the variance less than some pre-determined threshold
value?
Related
Techniques
F Test
Bartlett Test
Levene Test
Software The chi-square test for the variance is available in many general
purpose statistical software programs. Both Dataplot code and R
code can be used to generate the analyses in this section. These
scripts use the GEAR.DAT data file.
Data Used for Chi-Square Test for the Variance
Data Used
for Chi-
Square
Test for the
Variance
Example
The following are the data used for the chi-square test for the
variance example. The first column is gear diameter and the
second column is batch number. Only the first column is used
for this example.
1.006 1.000
0.996 1.000
0.998 1.000
1.000 1.000
0.992 1.000
0.993 1.000
1.002 1.000
0.999 1.000
0.994 1.000
1.000 1.000
0.998 2.000
1.006 2.000
1.000 2.000
1.002 2.000
0.997 2.000
0.998 2.000
0.996 2.000
1.000 2.000
1.006 2.000
0.988 2.000
0.991 3.000
0.987 3.000
0.997 3.000
0.999 3.000
0.995 3.000
0.994 3.000
1.000 3.000
0.999 3.000
0.996 3.000
0.996 3.000
1.005 4.000
1.002 4.000
0.994 4.000
1.000 4.000
0.995 4.000
0.994 4.000
0.998 4.000

0.996 4.000
1.002 4.000
0.996 4.000
0.998 5.000
0.998 5.000
0.982 5.000
0.990 5.000
1.002 5.000
0.984 5.000
0.996 5.000
0.993 5.000
0.980 5.000
0.996 5.000
1.009 6.000
1.013 6.000
1.009 6.000
0.997 6.000
0.988 6.000
1.002 6.000
0.995 6.000
0.998 6.000
0.981 6.000
0.996 6.000
0.990 7.000
1.004 7.000
0.996 7.000
1.001 7.000
0.998 7.000
1.000 7.000
1.018 7.000
1.010 7.000
0.996 7.000
1.002 7.000
0.998 8.000
1.000 8.000
1.006 8.000
1.000 8.000
1.002 8.000
0.996 8.000
0.998 8.000
0.996 8.000
1.002 8.000
1.006 8.000
1.002 9.000
0.998 9.000
0.996 9.000
0.995 9.000
0.996 9.000
1.004 9.000
1.004 9.000
0.998 9.000
0.999 9.000
0.991 9.000
0.991 10.000
0.995 10.000
0.984 10.000
0.994 10.000
0.997 10.000
0.997 10.000
0.991 10.000
0.998 10.000
1.004 10.000
0.997 10.000
F-Test for Equality of Two Variances
Purpose:
Test if
variances
from two
populations
are equal
An F-test (Snedecor and Cochran, 1983) is used to test if the
variances of two populations are equal. This test can be a two-tailed
test or a one-tailed test. The two-tailed version tests against the
alternative that the variances are not equal. The one-tailed version
only tests in one direction, that is the variance from the first
population is either greater than or less than (but not both) the
second population variance. The choice is determined by the
problem. For example, if we are testing a new process, we may only
be interested in knowing if the new process is less variable than the
old process.

Definition The F hypothesis test is defined as:
H0: ( sigma_{1}^{2} ) = ( sigma_{2}^{2} )
Ha: ( sigma_{1}^{2} <
sigma_{2}^{2} )
for a lower one-
tailed test
( sigma_{1}^{2} >
sigma_{2}^{2} )
for an upper one-
tailed test
( sigma_{1}^{2} ne
sigma_{2}^{2} )
for a two-tailed
test
Test
Statistic:
F = ( s^{2}_{1}/s^{2}_{2} )
where ({s^{2}_{1}}) and ({s^{2}_{2}}) and are
the sample variances. The more this ratio deviates
from 1, the stronger the evidence for unequal
population variances.
Significance
Level:
α
Critical
Region:
The hypothesis that the two variances are equal is
rejected if
( F > F_{alpha,N_1 - 1,N_2 -
1} )
for an upper one-
tailed test
( F < F_{1 - alpha,N_1 -
1,N_2 - 1} )
for a lower one-
tailed test
( F < F_{1 - alpha/2,N_1 -
1,N_2 - 1} )
or
( F > F_{alpha/2,N_1 - 1,N_2
- 1} )
for a two-tailed
test
where Fα, N1-1, N2-1 is the critical value of the F
distribution with N1-1 and N2-1 degrees of freedom
and a significance level of α.
In the above formulas for the critical regions, the
Handbook follows the convention that Fα is the upper
critical value from the F distribution and F1-α is the
lower critical value from the F distribution. Note that
this is the opposite of the designation used by some
texts and software programs.
F Test
Example
The following F-test was generated for the JAHANMI2.DAT data
set. The data set contains 480 ceramic strength measurements for
two batches of material. The summary statistics for each batch are
shown below.
BATCH 1:
MEAN = 688.9987
BATCH 2:
MEAN = 611.1559
We are testing the null hypothesis that the variances for the two
batches are equal.
H0: σ1
2 = σ2
2
Ha: σ1
2 ≠ σ2
2
Test statistic: F = 1.123037
Numerator degrees of freedom: N1 - 1 = 239
Denominator degrees of freedom: N2 - 1 = 239
Critical values: F(1-α/2,N1-1,N2-1) = 0.7756
F(α/2,N1-1,N2-1) = 1.2894
Rejection region: Reject H0 if F < 0.7756 or F > 1.2894

The F test indicates that there is not enough evidence to reject the
null hypothesis that the two batch variancess are equal at the 0.05
significance level.
Questions The F-test can be used to answer the following questions:
1. Do two samples come from populations with equal
variancess?
2. Does a new process, treatment, or test reduce the variability
of the current process?
Related
Techniques
Bihistogram
Chi-Square Test
Bartlett's Test
Levene Test
Case Study Ceramic strength data.
Software The F-test for equality of two variances is available in many
general purpose statistical software programs. Both Dataplot code
and R code can be used to generate the analyses in this section.
These scripts use the AUTO83B.DAT data file.
Levene Test for Equality of Variances
Purpose:
Test for
Homogeneity
of Variances
Levene's test ( Levene 1960) is used to test if k samples have
equal variances. Equal variances across samples is called
homogeneity of variance. Some statistical tests, for example
the analysis of variance, assume that variances are equal
across groups or samples. The Levene test can be used to
verify that assumption.
Levene's test is an alternative to the Bartlett test. The Levene
test is less sensitive than the Bartlett test to departures from
normality. If you have strong evidence that your data do in
fact come from a normal, or nearly normal, distribution, then
Bartlett's test has better performance.
Definition The Levene test is defined as:
H0: ( sigma_{1}^{2} = sigma_{2}^{2} = ldots
= sigma_{k}^{2} )
Ha: ( sigma_{i}^{2} ne sigma_{j}^{2} ) for
at least one pair (i,j).
Test
Statistic:
Given a variable Y with sample of size N
divided into k subgroups, where Ni is the
sample size of the ith subgroup, the Levene test
statistic is defined as:
[ W = frac{(N-k)} {(k-1)}
frac{sum_{i=1}^{k}N_{i}
(bar{Z}_{i.}-bar{Z}_{..})^{2} }
{sum_{i=1}^{k}sum_{j=1}^{N_i}
(Z_{ij}-bar{Z}_{i.})^{2} } ]
where Zij can have one of the following three
definitions:
1. (Z_{ij} = |Y_{ij} - bar{Y}_{i.}|)
where (bar{Y}_{i.}) is the mean of the
i-th subgroup.
2. (Z_{ij} = |Y_{ij} - tilde{Y}_{i.}|)

where (tilde{Y}_{i.}) is the median of
the i-th subgroup.
3. (Z_{ij} = |Y_{ij} - bar{Y}_{i.}'|)
where (bar{Y}_{i.}') is the 10%
trimmed mean of the i-th subgroup.
(bar{Z}_{i.}) are the group means of the Zij
and (bar{Z}_{..}) is the overall mean of the
Zij.
The three choices for defining Zij determine
the robustness and power of Levene's test. By
robustness, we mean the ability of the test to
not falsely detect unequal variances when the
underlying data are not normally distributed
and the variables are in fact equal. By power,
we mean the ability of the test to detect
unequal variances when the variances are in
fact unequal.
Levene's original paper only proposed using
the mean. Brown and Forsythe (1974))
extended Levene's test to use either the median
or the trimmed mean in addition to the mean.
They performed Monte Carlo studies that
indicated that using the trimmed mean
performed best when the underlying data
followed a Cauchy distribution (i.e., heavy-
tailed) and the median performed best when
the underlying data followed a
(chi^{2}_{4}) (i.e., skewed) distribution.
Using the mean provided the best power for
symmetric, moderate-tailed, distributions.
Although the optimal choice depends on the
underlying distribution, the definition based on
the median is recommended as the choice that
provides good robustness against many types
of non-normal data while retaining good
power. If you have knowledge of the
underlying distribution of the data, this may
indicate using one of the other choices.
Significance
Level:
α
Critical
Region:
The Levene test rejects the hypothesis that the
variances are equal if
W > Fα, k-1, N-k
where Fα, k-1, N-k is the upper critical value of
the F distribution with k-1 and N-k degrees of
freedom at a significance level of α.
In the above formulas for the critical regions,
the Handbook follows the convention that Fα
is the upper critical value from the F
distribution and F1-α is the lower critical
value. Note that this is the opposite of some
texts and software programs.
Levene's Test
Example
Levene's test, based on the median, was performed for the
GEAR.DAT data set. The data set includes ten measurements
of gear diameter for each of ten batches for a total of 100
measurements.
H0: σ1
2 = ... = σ10
2
Ha: σ1
2 ≠ ... ≠ σ10
2

Test statistic: W = 1.705910
Degrees of freedom: k-1 = 10-1 = 9
N-k = 100-10 = 90
Critical value (upper tail): Fα,k-1,N-k = 1.9855
Critical region: Reject H0 if F > 1.9855
We are testing the hypothesis that the group variances are
equal. We fail to reject the null hypothesis at the 0.05
significance level since the value of the Levene test statistic
is less than the critical value. We conclude that there is
insufficient evidence to claim that the variances are not
equal.
Question Levene's test can be used to answer the following question:
Is the assumption of equal variances valid?
Related
Techniques
Box Plot
Bartlett Test
Chi-Square Test
Software The Levene test is available in some general purpose
statistical software programs. Both Dataplot code and R code
can be used to generate the analyses in this section. These
scripts use the GEAR.DAT data file.
Measures of Skewness and Kurtosis
Skewness
and
Kurtosis
A fundamental task in many statistical analyses is to
characterize the location and variability of a data set. A further
characterization of the data includes skewness and kurtosis.
Skewness is a measure of symmetry, or more precisely, the
lack of symmetry. A distribution, or data set, is symmetric if it
looks the same to the left and right of the center point.
Kurtosis is a measure of whether the data are heavy-tailed or
light-tailed relative to a normal distribution. That is, data sets
with high kurtosis tend to have heavy tails, or outliers. Data
sets with low kurtosis tend to have light tails, or lack of
outliers. A uniform distribution would be the extreme case.
The histogram is an effective graphical technique for showing
both the skewness and kurtosis of data set.
Definition
of Skewness
For univariate data Y1, Y2, ..., YN, the formula for skewness is:
[ g_{1} = frac{sum_{i=1}^{N}(Y_{i} -
bar{Y})^{3}/N} {s^{3}} ]
where (bar{Y}) is the mean, s is the standard deviation, and
N is the number of data points. Note that in computing the
skewness, the s is computed with N in the denominator rather
than N - 1.
The above formula for skewness is referred to as the Fisher-
Pearson coefficient of skewness. Many software programs
actually compute the adjusted Fisher-Pearson coefficient of
skewness
[ G_{1} = frac{sqrt{N(N-1)}}{N-2}
frac{sum_{i=1}^{N}(Y_{i} - bar{Y})^{3}/N}
{s^{3}} ]

This is an adjustment for sample size. The adjustment
approaches 1 as N gets large. For reference, the adjustment
factor is 1.49 for N = 5, 1.19 for N = 10, 1.08 for N = 20, 1.05
for N = 30, and 1.02 for N = 100.
The skewness for a normal distribution is zero, and any
symmetric data should have a skewness near zero. Negative
values for the skewness indicate data that are skewed left and
positive values for the skewness indicate data that are skewed
right. By skewed left, we mean that the left tail is long relative
to the right tail. Similarly, skewed right means that the right
tail is long relative to the left tail. If the data are multi-modal,
then this may affect the sign of the skewness.
Some measurements have a lower bound and are skewed
right. For example, in reliability studies, failure times cannot
be negative.
It should be noted that there are alternative definitions of
skewness in the literature. For example, the Galton skewness
(also known as Bowley's skewness) is defined as
[ mbox{Galton skewness} = frac{Q_{1} + Q_{3} -2
Q_{2}}{Q_{3} - Q_{1}} ]
where Q1 is the lower quartile, Q3 is the upper quartile, and
Q2 is the median.
The Pearson 2 skewness coefficient is defined as
[ S_{k_2} = 3 frac{(bar{Y} - tilde{Y})}{s} ]
where ( tilde{Y} ) is the sample median.
There are many other definitions for skewness that will not be
discussed here.
Definition
of Kurtosis
For univariate data Y1, Y2, ..., YN, the formula for kurtosis is:
[ mbox{kurtosis} = frac{sum_{i=1}^{N}(Y_{i} -
bar{Y})^{4}/N} {s^{4}} ]
where (bar{Y}) is the mean, s is the standard deviation, and
N is the number of data points. Note that in computing the
kurtosis, the standard deviation is computed using N in the
denominator rather than N - 1.
Alternative
Definition
of Kurtosis
The kurtosis for a standard normal distribution is three. For
this reason, some sources use the following definition of
kurtosis (often referred to as "excess kurtosis"):
[ mbox{kurtosis} = frac{sum_{i=1}^{N}(Y_{i} -
bar{Y})^{4}/N} {s^{4}} - 3 ]
This definition is used so that the standard normal distribution
has a kurtosis of zero. In addition, with the second definition
positive kurtosis indicates a "heavy-tailed" distribution and
negative kurtosis indicates a "light tailed" distribution.
Which definition of kurtosis is used is a matter of convention
(this handbook uses the original definition). When using
software to compute the sample kurtosis, you need to be
aware of which convention is being followed. Many sources
use the term kurtosis when they are actually computing
"excess kurtosis", so it may not always be clear.
Examples The following example shows histograms for 10,000 random
numbers generated from a normal, a double exponential, a
Cauchy, and a Weibull distribution.

Normal
Distribution
behaved tails. This is indicated by the skewness of 0.03. The
kurtosis of 2.96 is near the expected value of 3. The histogram
verifies the symmetry.
Double
Exponential
Distribution
The second histogram is a sample from a double exponential
distribution. The double exponential is a symmetric
distribution. Compared to the normal, it has a stronger peak,
more rapid decay, and heavier tails. That is, we would expect
a skewness near zero and a kurtosis higher than 3. The
skewness is 0.06 and the kurtosis is 5.9.
Cauchy
Distribution
For better visual comparison with the other data sets, we
restricted the histogram of the Cauchy distribution to values
between -10 and 10. The full data set for the Cauchy data in
fact has a minimum of approximately -29,000 and a maximum
of approximately 89,000.
Since it is symmetric, we would expect a skewness near zero.
Due to the heavier tails, we might expect the kurtosis to be
larger than for a normal distribution. In fact the skewness is
69.99 and the kurtosis is 6,693. These extremely high values
can be explained by the heavy tails. Just as the mean and
standard deviation can be distorted by extreme values in the
tails, so too can the skewness and kurtosis measures.
Weibull
Distribution
The fourth histogram is a sample from a Weibull distribution
with shape parameter 1.5. The Weibull distribution is a
skewed distribution with the amount of skewness depending
on the value of the shape parameter. The degree of decay as
we move away from the center also depends on the value of
the shape parameter. For this data set, the skewness is 1.08
and the kurtosis is 4.46, which indicates moderate skewness
and kurtosis.
Dealing
with
Skewness
and
Kurtosis
Many classical statistical tests and intervals depend on
normality assumptions. Significant skewness and kurtosis
clearly indicate that data are not normal. If a data set exhibits
significant skewness or kurtosis (as indicated by a histogram
or the numerical measures), what can we do about it?
One approach is to apply some type of transformation to try to
make the data normal, or more nearly normal. The Box-Cox
transformation is a useful technique for trying to normalize a
data set. In particular, taking the log or square root of a data
set is often useful for data that exhibit moderate right
skewness.
Another approach is to use techniques based on distributions
other than the normal. For example, in reliability studies, the

exponential, Weibull, and lognormal distributions are typically
used as a basis for modeling rather than using the normal
distribution. The probability plot correlation coefficient plot
and the probability plot are useful tools for determining a
good distributional model for the data.
Software The skewness and kurtosis coefficients are available in most
general purpose statistical software programs.
Autocorrelation
Purpose:
Detect Non-
Randomness,
Time Series
Modeling
The autocorrelation ( Box and Jenkins, 1976) function can
be used for the following two purposes:
1. To detect non-randomness in data.
2. To identify an appropriate time series model if the
data are not random.
Definition Given measurements, Y1, Y2, ..., YN at time X1, X2, ..., XN,
the lag k autocorrelation function is defined as
[ r_{k} = frac{sum_{i=1}^{N-k}(Y_{i} -
bar{Y})(Y_{i+k} - bar{Y})} {sum_{i=1}^{N}
(Y_{i} - bar{Y})^{2} } ]
Although the time variable, X, is not used in the formula
for autocorrelation, the assumption is that the observations
are equi-spaced.
Autocorrelation is a correlation coefficient. However,
instead of correlation between two different variables, the
correlation is between two values of the same variable at
times Xi and Xi+k.
When the autocorrelation is used to detect non-
randomness, it is usually only the first (lag 1)
autocorrelation that is of interest. When the autocorrelation
is used to identify an appropriate time series model, the
autocorrelations are usually plotted for many lags.
Autocorrelation
Example
Lag-one autocorrelations were computed for the the
LEW.DAT data set.
lag autocorrelation
0. 1.00
1. -0.31
2. -0.74
3. 0.77
4. 0.21
5. -0.90
6. 0.38
7. 0.63
8. -0.77
9. -0.12
10. 0.82
11. -0.40
12. -0.55
13. 0.73
14. 0.07
15. -0.76
16. 0.40
17. 0.48
18. -0.70
19. -0.03
20. 0.70
21. -0.41
22. -0.43
23. 0.67
24. 0.00
25. -0.66
26. 0.42
27. 0.39
28. -0.65
29. 0.03
30. 0.63
31. -0.42
32. -0.36
33. 0.64
34. -0.05
35. -0.60
36. 0.43
37. 0.32

38. -0.64
39. 0.08
40. 0.58
41. -0.45
42. -0.28
43. 0.62
44. -0.10
45. -0.55
46. 0.45
47. 0.25
48. -0.61
49. 0.14
Questions The autocorrelation function can be used to answer the
1. Was this sample data set generated from a random
process?
2. Would a non-linear or time series model be a more
appropriate model for these data than a simple
constant plus error model?
Importance Randomness is one of the key assumptions in determining
if a univariate statistical process is in control. If the
assumptions of constant location and scale, randomness,
and fixed distribution are reasonable, then the univariate
process can be modeled as:
[ Y_{i} = A_0 + E_{i} ]
where Ei is an error term.
If the randomness assumption is not valid, then a different
model needs to be used. This will typically be either a time
series model or a non-linear model (with time as the
independent variable).
Related
Techniques
Run Sequence Plot
Lag Plot
Runs Test
Case Study The heat flow meter data demonstrate the use of
autocorrelation in determining if the data are from a
random process.
Software The autocorrelation capability is available in most general
purpose statistical software programs. Both Dataplot code
and R code can be used to generate the analyses in this
section. These scripts use the LEW.DAT data file.
Runs Test for Detecting Non-randomness
Purpose:
Detect Non-
Randomness
The runs test (Bradley, 1968) can be used to decide if a data set is
from a random process.
A run is defined as a series of increasing values or a series of
decreasing values. The number of increasing, or decreasing, values
is the length of the run. In a random data set, the probability that
the (I+1)th value is larger or smaller than the Ith value follows a
binomial distribution, which forms the basis of the runs test.
Typical
Analysis
and Test
Statistics
The first step in the runs test is to count the number of runs in the
data sequence. There are several ways to define runs in the
literature, however, in all cases the formulation must produce a
dichotomous sequence of values. For example, a series of 20 coin
tosses might produce the following sequence of heads (H) and
tails (T).

H H T T H T H H H H T H H T T T T T H H
The number of runs for this series is nine. There are 11 heads and
9 tails in the sequence.
Definition We will code values above the median as positive and values
below the median as negative. A run is defined as a series of
consecutive positive (or negative) values. The runs test is defined
as:
H0: the sequence was produced in a random manner
Ha: the sequence was not produced in a random manner
Test
Statistic:
The test statistic is
[ Z = frac{R - bar{R}}{s_R} ]
where R is the observed number of runs,
(bar{R}), is the expected number of runs, and
(s_{R}) is the standard deviation of the number of
runs. The values of (bar{R}) and (s_{R}) are
computed as follows:
[ bar{R} = frac{2 n_1 n_2}{n_1 + n_2} +
1 ]
[ s_{R}^2 = frac{2 n_1 n_2(2 n_1 n_2 - n_1
- n_2)} {(n_1 + n_2)^2 (n_1 + n_2 - 1)} ]
with n1 and n2 denoting the number of positive and
negative values in the series.
Significance
Level:
α
Critical
Region:
The runs test rejects the null hypothesis if
|Z| > Z1-α/2
For a large-sample runs test (where n1 > 10 and n2
> 10), the test statistic is compared to a standard
normal table. That is, at the 5 % significance level,
a test statistic with an absolute value greater than
1.96 indicates non-randomness. For a small-sample
runs test, there are tables to determine critical
values that depend on values of n1 and n2
(Mendenhall, 1982).
Runs Test
Example
A runs test was performed for 200 measurements of beam
deflection contained in the LEW.DAT data set.
Test statistic: Z = 2.6938
Critical value (upper tail): Z1-α/2 = 1.96
Critical region: Reject H0 if |Z| > 1.96
Since the test statistic is greater than the critical value, we
conclude that the data are not random at the 0.05 significance
level.
Question The runs test can be used to answer the following question:
Were these sample data generated from a random process?
Importance Randomness is one of the key assumptions in determining if a
univariate statistical process is in control. If the assumptions of
constant location and scale, randomness, and fixed distribution are
reasonable, then the univariate process can be modeled as:

[ Y_{i} = A_0 + E_{i} ]
where Ei is an error term.
If the randomness assumption is not valid, then a different model
needs to be used. This will typically be either a times series model
or a non-linear model (with time as the independent variable).
Related
Techniques
Autocorrelation
Run Sequence Plot
Lag Plot
Case Study Heat flow meter data
Software Most general purpose statistical software programs support a runs
test. Both Dataplot code and R code can be used to generate the
analyses in this section. These scripts use the LEW.DAT data file.
Anderson-Darling Test
Purpose:
Test for
Distributional
Adequacy
The Anderson-Darling test (Stephens, 1974) is used to test if a sample of data
came from a population with a specific distribution. It is a modification of the
Kolmogorov-Smirnov (K-S) test and gives more weight to the tails than does
the K-S test. The K-S test is distribution free in the sense that the critical
values do not depend on the specific distribution being tested (note that this is
true only for a fully specified distribution, i.e. the parameters are known). The
Anderson-Darling test makes use of the specific distribution in calculating
critical values. This has the advantage of allowing a more sensitive test and
the disadvantage that critical values must be calculated for each distribution.
Currently, tables of critical values are available for the normal, uniform,
lognormal, exponential, Weibull, extreme value type I, generalized Pareto,
and logistic distributions. We do not provide the tables of critical values in
this Handbook (see Stephens 1974, 1976, 1977, and 1979) since this test is
usually applied with a statistical software program that will print the relevant
critical values.
The Anderson-Darling test is an alternative to the chi-square and
Kolmogorov-Smirnov goodness-of-fit tests.
Definition The Anderson-Darling test is defined as:
H0: The data follow a specified distribution.
Ha: The data do not follow the specified distribution
Test
Statistic:
The Anderson-Darling test statistic is defined as
[ A^{2} = -N - S ]
where
[ S = sum_{i=1}^{N}frac{(2i - 1)}{N}[ln{F(Y_{i})} +
ln{(1 - F(Y_{N+1-i}))}] ]
F is the cumulative distribution function of the specified
distribution. Note that the Yi are the ordered data.
Significance
Level:
α
Critical
Region:
The critical values for the Anderson-Darling test are dependent
on the specific distribution that is being tested. Tabulated
values and formulas have been published (Stephens, 1974,
1976, 1977, 1979) for a few specific distributions (normal,
lognormal, exponential, Weibull, logistic, extreme value type 1
and others). The test is a one-sided test and the hypothesis that
the distribution is of a specific form is rejected if the test
statistic, ( A^{2} ), is greater than the critical value.

Note that for a given distribution, the Anderson-Darling
statistic may be multiplied by a constant (which usually
depends on the sample size, n). These constants are given in
the various papers by Stephens. In the sample output below, the
test statistic values are adjusted. Also, be aware that different
constants (and therefore critical values) have been published.
You just need to be aware of what constant was used for a
given set of critical values (the needed constant is typically
given with the critical values).
Sample
Output
We generated 1,000 random numbers for normal, double exponential,
Cauchy, and lognormal distributions. In all four cases, the Anderson-Darling
test was applied to test for a normal distribution.
The normal random numbers were stored in the variable Y1, the double
exponential random numbers were stored in the variable Y2, the Cauchy
random numbers were stored in the variable Y3, and the lognormal random
numbers were stored in the variable Y4.
Distribution Mean Standard Deviation
------------ -------- ------------------
Normal (Y1) 0.004360 1.001816
Double Exponential (Y2) 0.020349 1.321627
Cauchy (Y3) 1.503854 35.130590
Lognormal (Y4) 1.518372 1.719969
H0: the data are normally distributed
Ha: the data are not normally distributed
Y1 adjusted test statistic: A2 = 0.2576
Critical value: 0.752
Critical region: Reject H0 if A2 > 0.752
When the data were generated using a normal distribution, the test statistic
was small and the hypothesis of normality was not rejected. When the data
were generated using the double exponential, Cauchy, and lognormal
distributions, the test statistics were large, and the hypothesis of an underlying
normal distribution was rejected at the 0.05 significance level.
Questions The Anderson-Darling test can be used to answer the following questions:
Are the data from a normal distribution?
Are the data from a log-normal distribution?
Are the data from a Weibull distribution?
Are the data from an exponential distribution?
Are the data from a logistic distribution?
Importance Many statistical tests and procedures are based on specific distributional
assumptions. The assumption of normality is particularly common in classical
statistical tests. Much reliability modeling is based on the assumption that the
data follow a Weibull distribution.
There are many non-parametric and robust techniques that do not make strong
distributional assumptions. However, techniques based on specific
distributional assumptions are in general more powerful than non-parametric
and robust techniques. Therefore, if the distributional assumptions can be
validated, they are generally preferred.
Related
Techniques
Chi-Square goodness-of-fit Test
Kolmogorov-Smirnov Test
Shapiro-Wilk Normality Test
Probability Plot
Case Study Josephson junction cryothermometry case study.

Software The Anderson-Darling goodness-of-fit test is available in some general
purpose statistical software programs. Both Dataplot code and R code can be
used to generate the analyses in this section.
Chi-Square Goodness-of-Fit Test
Purpose:
Test for
distributional
adequacy
The chi-square test (Snedecor and Cochran, 1989) is used to
test if a sample of data came from a population with a
specific distribution.
An attractive feature of the chi-square goodness-of-fit test is
that it can be applied to any univariate distribution for which
you can calculate the cumulative distribution function. The
chi-square goodness-of-fit test is applied to binned data (i.e.,
data put into classes). This is actually not a restriction since
for non-binned data you can simply calculate a histogram or
frequency table before generating the chi-square test.
However, the value of the chi-square test statistic are
dependent on how the data is binned. Another disadvantage
of the chi-square test is that it requires a sufficient sample
size in order for the chi-square approximation to be valid.
The chi-square test is an alternative to the Anderson-Darling
and Kolmogorov-Smirnov goodness-of-fit tests. The chi-
square goodness-of-fit test can be applied to discrete
distributions such as the binomial and the Poisson. The
Kolmogorov-Smirnov and Anderson-Darling tests are
restricted to continuous distributions.
Additional discussion of the chi-square goodness-of-fit test
is contained in the product and process comparisons chapter
(chapter 7).
Definition The chi-square test is defined for the hypothesis:
H0: The data follow a specified distribution.
Ha: The data do not follow the specified
distribution.
Test
Statistic:
To compute the test statistic used in the chi-
square goodness-of-fit test, one first partitions
the range of possible observed values of the
quantity of interest, ( Y ), into ( k ) bins
determined by ( y_{1} < y_{2} < dots <
y_{k+1} ), such that bin ( i ) has lower
endpoint ( y_{i} ) and upper endpoint (
y_{i+1} ) for ( i=1,dots,k ). Note that (
y_{1} ) can be ( -infty ) and ( y_{k+1} )
can be ( +infty ).
The test statistic is defined as (
begin{equation*} chi^{2} = sum_{i=1}^{k}
(O_{i}-E_{i})^{2}/E_{i} end{equation*} )
where ( O_{i} ) denotes the number of
observations that fall in bin ( i ) (under some
convention for how to count observations that
fall on the boundary between two consecutive
bins: for example, that an observation equal to
( y_{i} ) is counted as being in bin ( i+1 )),
and ( E_{i} ) denotes the number of
observations expected to fall in bin ( i ) based
on the probability model under test. If this
model is specified in terms of its cumulative
distribution function, ( F_{theta} ), then the
expected counts are computed as (
begin{equation*} E_{i} = N (F_{theta}
(y_{i+1})-F_{theta}(y_{i}). end{equation*}

) where ( N ) denotes the sample size, and
under the convention that ( F_{theta}(-infty)
= 0 ) and ( F_{theta}(+infty) = 1 ).
Note that, in general, this cumulative
distribution function depends on a possibly
vectorial parameter, ( theta ), hence the
notation ( F_{theta} ) in the foregoing
equation for ( E_{i} ). For example, if the
probability model is Gaussian, then ( theta =
(mu, sigma) ), where ( mu ) and ( sigma
) denote the mean and standard deviation of
the Gaussian distribution. However, if the
probability model is Poisson, then ( theta ) is
a scalar, ( theta = lambda ), the
corresponding mean. In any case, to be able to
compute ( E_{i} ) one needs first to estimate
( theta ).
The chi-square test assumes that the estimate
of ( theta ) is the maximum likelihood
estimate derived from the observed bin counts,
not from the individual observations. Such
estimate, which we denote (
widetilde{theta} ), can be computed by
numerical maximization of the log-likelihood
function, ( ell ), with respect to ( theta ): (
begin{equation*} ell(theta) =
sum_{i=1}^{k} O_{i} log (F_{theta}
(y_{i+1})-F_{theta}(y_{i})).
end{equation*} ) The expected bin counts
are then computed as ( E_{i} = N
(F_{widetilde{theta}}(y_{i+1})-
F_{widetilde{theta}}(y_{i}) ) for (
i=1,dots,k ).
Significance
Level:
α
Critical
Region:
The test statistic follows, approximately, a chi-
square distribution with (k - c) degrees of
freedom where k is the number of non-empty
cells and c = the number of estimated
parameters (including location and scale
parameters and shape parameters) for the
distribution + 1. For example, for a 3-
parameter Weibull distribution, c = 4.
Therefore, the hypothesis that the data are
from a population with the specified
distribution is rejected if
[ chi^2 > chi^2_{1-alpha, , k-c} ]
where (chi^2_{1-alpha, , k-c}) is the chi-
square critical value with k - c degrees of
freedom and significance level α.
Chi-Square
Test Example
We generated 1,000 random numbers for normal, double
exponential, t with 3 degrees of freedom, and lognormal
distributions. In all cases, a chi-square test with k = 32 bins
was applied to test for normally distributed data. Because the
normal distribution has two parameters, c = 2 + 1 = 3
The normal random numbers were stored in the variable Y1,
the double exponential random numbers were stored in the
variable Y2, the t random numbers were stored in the
variable Y3, and the lognormal random numbers were stored
in the variable Y4.

Y1 Test statistic: Χ 2 = 32.256
Degrees of freedom: k - c = 32 - 3 = 29
1-α, k-c = 42.557
Critical region: Reject H0 if Χ 2 > 42.557
As we would hope, the chi-square test fails to reject the null
hypothesis for the normally distributed data set and rejects
the null hypothesis for the three non-normal data sets.
Application
Example
This example uses the data set described in Fatigue Life of
Aluminum Alloy Specimens, which comprises 101 measured
values of the fatigue life (thousands of cycles until rupture)
of rectangular strips of aluminum sheeting that were
subjected to periodic loading until failure. To test whether
these data are consistent with a 3-parameter Weibull
probability model, one can employ the chi-square goodness-
of-fit test, after binning the measured values into ( k=10 )
bins chosen so that each is expected to include about 10
observations. The maximum likelihood estimates of the
Weibull parameters based on the bin counts are slightly
different from their counterparts based on the individual
observations. The ( p-value ) corresponding to the test
statistic is 0.15, thus not questioning the adequacy of the 3-
parameter Weibull distribution as probability model for these
data. The R code mentioned below includes an
implementation of the test, as just described.
Questions The chi-square test can be used to answer the following
types of questions:
Are the data from a binomial distribution?
Importance Many statistical tests and procedures are based on specific
distributional assumptions. The assumption of normality is
particularly common in classical statistical tests. Much
reliability modeling is based on the assumption that the
distribution of the data follows a Weibull distribution.
There are many non-parametric and robust techniques that
are not based on strong distributional assumptions. By non-
parametric, we mean a technique, such as the sign test, that is
not based on a specific distributional assumption. By robust,
we mean a statistical technique that performs well under a
wide range of distributional assumptions. However,
techniques based on specific distributional assumptions are
in general more powerful than these non-parametric and
robust techniques. By power, we mean the ability to detect a
difference when that difference actually exists. Therefore, if
the distributional assumption can be confirmed, the
parametric techniques are generally preferred.
If you are using a technique that makes a normality (or some
other type of distributional) assumption, it is important to
confirm that this assumption is in fact justified. If it is, the
more powerful parametric techniques can be used. If the
distributional assumption is not justified, a non-parametric or
robust technique may be required.
Related
Techniques
Anderson-Darling Goodness-of-Fit Test
Kolmogorov-Smirnov Test

Probability Plots
Software Some general purpose statistical software programs provide
a chi-square goodness-of-fit test for at least some of the
common distributions. Both Dataplot code and R code can
be used to generate the analyses in this section.
Kolmogorov-Smirnov Goodness-of-Fit Test
Purpose:
Test for
Distributional
Adequacy
The Kolmogorov-Smirnov test (Chakravart, Laha, and
Roy, 1967) is used to decide if a sample comes from a
population with a specific distribution.
The Kolmogorov-Smirnov (K-S) test is based on the
empirical distribution function (ECDF). Given N ordered
data points Y1, Y2, ..., YN, the ECDF is defined as
[ E_{N} = n(i)/N ]
where n(i) is the number of points less than Yi and the Yi
are ordered from smallest to largest value. This is a step
function that increases by 1/N at the value of each ordered
data point.
The graph below is a plot of the empirical distribution
function with a normal cumulative distribution function for
100 normal random numbers. The K-S test is based on the
maximum distance between these two curves.
Characteristics
and
Limitations of
the K-S Test
An attractive feature of this test is that the distribution of
the K-S test statistic itself does not depend on the
underlying cumulative distribution function being tested.
Another advantage is that it is an exact test (the chi-square
goodness-of-fit test depends on an adequate sample size for
the approximations to be valid). Despite these advantages,
the K-S test has several important limitations:
1. It only applies to continuous distributions.
2. It tends to be more sensitive near the center of the
distribution than at the tails.
3. Perhaps the most serious limitation is that the
distribution must be fully specified. That is, if
location, scale, and shape parameters are estimated
from the data, the critical region of the K-S test is no
longer valid. It typically must be determined by
simulation.

Several goodness-of-fit tests, such as the Anderson-Darling
test and the Cramer Von-Mises test, are refinements of the
K-S test. As these refined tests are generally considered to
be more powerful than the original K-S test, many analysts
prefer them. Also, the advantage for the K-S test of having
the critical values be indpendendent of the underlying
distribution is not as much of an advantage as first appears.
This is due to limitation 3 above (i.e., the distribution
parameters are typically not known and have to be
estimated from the data). So in practice, the critical values
for the K-S test have to be determined by simulation just as
for the Anderson-Darling and Cramer Von-Mises (and
related) tests.
Note that although the K-S test is typically developed in
the context of continuous distributions for uncensored and
ungrouped data, the test has in fact been extended to
discrete distributions and to censored and grouped data. We
do not discuss those cases here.
Definition The Kolmogorov-Smirnov test is defined by:
H0: The data follow a specified distribution
Ha: The data do not follow the specified
distribution
Test
Statistic:
The Kolmogorov-Smirnov test statistic is
defined as
[ D = max_{1 le i le N} left(
F(Y_{i}) - frac{i-1} {N}, frac{i}
{N} - F(Y_{i}) right) ]
where F is the theoretical cumulative
distribution of the distribution being tested
which must be a continuous distribution
(i.e., no discrete distributions such as the
binomial or Poisson), and it must be fully
specified (i.e., the location, scale, and shape
parameters cannot be estimated from the
data).
Significance
Level:
α
Critical
Values:
The hypothesis regarding the distributional
form is rejected if the test statistic, D, is
greater than the critical value obtained from
a table. There are several variations of these
tables in the literature that use somewhat
different scalings for the K-S test statistic
and critical regions. These alternative
formulations should be equivalent, but it is
necessary to ensure that the test statistic is
calculated in a way that is consistent with
how the critical values were tabulated.
We do not provide the K-S tables in the
Handbook since software programs that
perform a K-S test will provide the relevant
critical values.
Technical Note Previous editions of e-Handbook gave the following
formula for the computation of the Kolmogorov-Smirnov
goodness of fit statistic:
[ D = max_{1 le i le N} left| F(Y_{i}) - frac{i}
{N} right| ]
This formula is in fact not correct. Note that this formula
can be rewritten as:
[ D = max_{1 le i le N} (F(Y_{i}) - frac{i} {N},
frac{i}{N} - F(Y_{i})) ]

This form makes it clear that an upper bound on the
difference between these two formulas is i/N. For actual
data, the difference is likely to be less than the upper
bound.
For example, for N = 20, the upper bound on the difference
between these two formulas is 0.05 (for comparison, the
5% critical value is 0.294). For N = 100, the upper bound is
0.001. In practice, if you have moderate to large sample
sizes (say N ≥ 50), these formulas are essentially
equivalent.
Kolmogorov-
Smirnov Test
Example
We generated 1,000 random numbers for normal, double
exponential, t with 3 degrees of freedom, and lognormal
distributions. In all cases, the Kolmogorov-Smirnov test
was applied to test for a normal distribution.
The normal random numbers were stored in the variable
Y1, the double exponential random numbers were stored in
the variable Y2, the t random numbers were stored in the
variable Y3, and the lognormal random numbers were
stored in the variable Y4.
Y1 test statistic: D = 0.0241492
Critical region: Reject H0 if D > 0.04301
As expected, the null hypothesis is not rejected for the
normally distributed data, but is rejected for the remaining
three data sets that are not normally distributed.
Questions The Kolmogorov-Smirnov test can be used to answer the
following types of questions:
Importance Many statistical tests and procedures are based on specific
distributional assumptions. The assumption of normality is
particularly common in classical statistical tests. Much
reliability modeling is based on the assumption that the
data follow a Weibull distribution.
There are many non-parametric and robust techniques that
are not based on strong distributional assumptions. By non-
parametric, we mean a technique, such as the sign test, that
is not based on a specific distributional assumption. By
robust, we mean a statistical technique that performs well
under a wide range of distributional assumptions.
However, techniques based on specific distributional
assumptions are in general more powerful than these non-
parametric and robust techniques. By power, we mean the
ability to detect a difference when that difference actually
exists. Therefore, if the distributional assumptions can be
confirmed, the parametric techniques are generally
preferred.
If you are using a technique that makes a normality (or
some other type of distributional) assumption, it is
important to confirm that this assumption is in fact

justified. If it is, the more powerful parametric techniques
can be used. If the distributional assumption is not
justified, using a non-parametric or robust technique may
be required.
Related
Techniques
Anderson-Darling goodness-of-fit Test
Chi-Square goodness-of-fit Test
Probability Plots
Software Some general purpose statistical software programs
support the Kolmogorov-Smirnov goodness-of-fit test, at
least for the more common distributions. Both Dataplot
code and R code can be used to generate the analyses in
this section.
Detection of Outliers
Introduction An outlier is an observation that appears to deviate
markedly from other observations in the sample.
Identification of potential outliers is important for the
following reasons.
1. An outlier may indicate bad data. For example, the
data may have been coded incorrectly or an
experiment may not have been run correctly. If it can
be determined that an outlying point is in fact
erroneous, then the outlying value should be deleted
from the analysis (or corrected if possible).
2. In some cases, it may not be possible to determine if
an outlying point is bad data. Outliers may be due to
random variation or may indicate something
scientifically interesting. In any event, we typically
do not want to simply delete the outlying
observation. However, if the data contains significant
outliers, we may need to consider the use of robust
statistical techniques.
Labeling,
Accomodation,
Identification
Iglewicz and Hoaglin distinguish the three following issues
with regards to outliers.
1. outlier labeling - flag potential outliers for further
investigation (i.e., are the potential outliers erroneous
data, indicative of an inappropriate distributional
model, and so on).
2. outlier accomodation - use robust statistical
techniques that will not be unduly affected by
outliers. That is, if we cannot determine that potential
outliers are erroneous observations, do we need
modify our statistical analysis to more appropriately
account for these observations?
3. outlier identification - formally test whether
observations are outliers.
This section focuses on the labeling and identification
issues.
Normality
Assumption
Identifying an observation as an outlier depends on the
underlying distribution of the data. In this section, we limit
the discussion to univariate data sets that are assumed to
follow an approximately normal distribution. If the
normality assumption for the data being tested is not valid,
then a determination that there is an outlier may in fact be

due to the non-normality of the data rather than the
prescence of an outlier.
For this reason, it is recommended that you generate a
normal probability plot of the data before applying an
outlier test. Although you can also perform formal tests for
normality, the prescence of one or more outliers may cause
the tests to reject normality when it is in fact a reasonable
assumption for applying the outlier test.
In addition to checking the normality assumption, the lower
and upper tails of the normal probability plot can be a
useful graphical technique for identifying potential outliers.
In particular, the plot can help determine whether we need
to check for a single outlier or whether we need to check
for multiple outliers.
The box plot and the histogram can also be useful graphical
tools in checking the normality assumption and in
identifying potential outliers.
Single Versus
Multiple
Outliers
Some outlier tests are designed to detect the prescence of a
single outlier while other tests are designed to detect the
prescence of multiple outliers. It is not appropriate to apply
a test for a single outlier sequentially in order to detect
multiple outliers.
In addition, some tests that detect multiple outliers may
require that you specify the number of suspected outliers
exactly.
Masking and
Swamping
Masking can occur when we specify too few outliers in the
test. For example, if we are testing for a single outlier when
there are in fact two (or more) outliers, these additional
outliers may influence the value of the test statistic enough
so that no points are declared as outliers.
On the other hand, swamping can occur when we specify
too many outliers in the test. For example, if we are testing
for two or more outliers when there is in fact only a single
outlier, both points may be declared outliers (many tests
will declare either all or none of the tested points as
outliers).
Due to the possibility of masking and swamping, it is
useful to complement formal outlier tests with graphical
methods. Graphics can often help identify cases where
masking or swamping may be an issue. Swamping and
masking are also the reason that many tests require that the
exact number of outliers being tested must be specified.
Also, masking is one reason that trying to apply a single
outlier test sequentially can fail. For example, if there are
multiple outliers, masking may cause the outlier test for the
first outlier to return a conclusion of no outliers (and so the
testing for any additional outliers is not performed).
Z-Scores and
Modified Z-
Scores
The Z-score of an observation is defined as
[ Z_i = frac{Y_i - bar{Y}} {s} ]
with (bar{Y}) and s denoting the sample mean and
sample standard deviation, respectively. In other words,
data is given in units of how many standard deviations it is
from the mean.
Although it is common practice to use Z-scores to identify
possible outliers, this can be misleading (particularly for
small sample sizes) due to the fact that the maximum Z-
score is at most ((n-1)/sqrt{n})

Iglewicz and Hoaglin recommend using the modified Z-
score
[ M_{i} = frac{0.6745(x_{i} - tilde{x})}
{mbox{MAD}} ]
with MAD denoting the median absolute deviation and
(tilde{x}) denoting the median.
These authors recommend that modified Z-scores with an
absolute value of greater than 3.5 be labeled as potential
outliers.
Formal
Outlier Tests
A number of formal outlier tests have proposed in the
literature. These can be grouped by the following
characteristics:
What is the distributional model for the data? We
restrict our discussion to tests that assume the data
follow an approximately normal distribution.
Is the test designed for a single outlier or is it
designed for multiple outliers?
If the test is designed for multiple outliers, does the
number of outliers need to be specified exactly or can
we specify an upper bound for the number of
outliers?
The following are a few of the more commonly used outlier
tests for normally distributed data. This list is not
exhaustive (a large number of outlier tests have been
proposed in the literature). The tests given here are
essentially based on the criterion of "distance from the
mean". This is not the only criterion that could be used. For
example, the Dixon test, which is not discussed here, is
based a value being too large (or small) compared to its
nearest neighbor.
1. Grubbs' Test - this is the recommended test when
testing for a single outlier.
2. Tietjen-Moore Test - this is a generalization of the
Grubbs' test to the case of more than one outlier. It
has the limitation that the number of outliers must be
specified exactly.
3. Generalized Extreme Studentized Deviate (ESD)
Test - this test requires only an upper bound on the
suspected number of outliers and is the
recommended test when the exact number of outliers
is not known.
Lognormal
Distribution
The tests discussed here are specifically based on the
assumption that the data follow an approximately normal
disribution. If your data follow an approximately lognormal
distribution, you can transform the data to normality by
taking the logarithms of the data and then applying the
outlier tests discussed here.
Further
Information
Iglewicz and Hoaglin provide an extensive discussion of
the outlier tests given above (as well as some not given
above) and also give a good tutorial on the subject of
outliers. Barnett and Lewis provide a book length treatment
of the subject.
In addition to discussing additional tests for data that
follow an approximately normal distribution, these sources
also discuss the case where the data are not normally
distributed.

Grubbs' Test for Outliers
Purpose:
Detection
of Outliers
Grubbs' test (Grubbs 1969 and Stefansky 1972) is used to detect a single outlier in a univariate data
set that follows an approximately normal distribution.
If you suspect more than one outlier may be present, it is recommended that you use either the
Tietjen-Moore test or the generalized extreme studentized deviate test instead of the Grubbs' test.
Grubbs' test is also known as the maximum normed residual test.
Definition Grubbs' test is defined for the hypothesis:
H0: There are no outliers in the data set
Ha: There is exactly one outlier in the data set
Test Statistic: The Grubbs' test statistic is defined as:
( G = frac{max{|Y_{i} - bar{Y}|}} {s} )
with (bar{Y}) and s denoting the sample mean and standard deviation,
respectively. The Grubbs' test statistic is the largest absolute deviation from the
sample mean in units of the sample standard deviation.
This is the two-sided version of the test. The Grubbs' test can also be defined as one
of the following one-sided tests:
1. test whether the minimum value is an outlier
( G = frac{bar{Y} - Y_{min}} {s} )
with Ymin denoting the minimum value.
2. test whether the maximum value is an outlier
( G = frac{Y_{max} - bar{Y}} {s} )
with Ymax denoting the maximum value.
Significance
Level:
α
Critical
Region:
For the two-sided test, the hypothesis of no outliers is rejected if
( G > frac{(N-1)} {sqrt{N}} sqrt{frac{(t_{alpha/(2N),N-2})^2} {N-2+
(t_{alpha/(2N),N-2})^2}} )
with (t_{alpha/(2N),N-2}) denoting the critical value of the t distribution with
(N-2) degrees of freedom and a significance level of α/(2N).
For one-sided tests, we use a significance level of level of α/N.
Grubbs'
Test
Example
The Tietjen and Moore paper gives the following set of 8 mass spectrometer measurements on a
uranium isotope:
199.31 199.53 200.19 200.82 201.92 201.95 202.18 245.57
As a first step, a normal probability plot was generated

This plot indicates that the normality assumption is reasonable with the exception of the maximum
value. We therefore compute Grubbs' test for the case that the maximum value, 245.57, is an outlier.
H0: there are no outliers in the data
Ha: the maximum value is an outlier
Test statistic: G = 2.4687
Critical value for an upper one-tailed test: 2.032
Critical region: Reject H0 if G > 2.032
For this data set, we reject the null hypothesis and conclude that the maximum value is in fact an
outlier at the 0.05 significance level.
Questions Grubbs' test can be used to answer the following questions:
1. Is the maximum value an outlier?
2. Is the minimum value an outlier?
Importance Many statistical techniques are sensitive to the presence of outliers. For example, simple
calculations of the mean and standard deviation may be distorted by a single grossly inaccurate data
point.
Checking for outliers should be a routine part of any data analysis. Potential outliers should be
examined to see if they are possibly erroneous. If the data point is in error, it should be corrected if
possible and deleted if it is not possible. If there is no reason to believe that the outlying point is in
error, it should not be deleted without careful consideration. However, the use of more robust
techniques may be warranted. Robust techniques will often downweight the effect of outlying
points without deleting them.
Related
Techniques
Several graphical techniques can, and should, be used to help detect outliers. A simple normal
probability plot, run sequence plot, a box plot, or a histogram should show any obviously outlying
points. In addition to showing potential outliers, several of these graphics also help assess whether
the data follow an approximately normal distribution.
Run Sequence Plot
Histogram
Box Plot
Lag Plot
Case Study Heat flow meter data.

Software Some general purpose statistical software programs support the Grubbs' test. Both Dataplot code
and R code can be used to generate the analyses in this section.
Tietjen-Moore Test for Outliers
Purpose:
Detection
of Outliers
The Tietjen-Moore test (Tietjen-Moore 1972) is used to detect multiple outliers in a univariate data
set that follows an approximately normal distribution.
The Tietjen-Moore test is a generalization of the Grubbs' test to the case of multiple outliers. If
testing for a single outlier, the Tietjen-Moore test is equivalent to the Grubbs' test.
It is important to note that the Tietjen-Moore test requires that the suspected number of outliers be
specified exactly. If this is not known, it is recommended that the generalized extreme studentized
deviate test be used instead (this test only requires an upper bound on the number of suspected
outliers).
Definition The Tietjen-Moore test is defined for the hypothesis:
Ha: There are exactly k outliers in the data set
Test Statistic: Sort the n data points from smallest to the largest so that yi denotes the ith largest
data value.
The test statistic for the k largest points is
( L_{k} = frac{sum_{i=1}^{n-k}{(y_{i} - bar{y}_{k})^{2}}}
{sum_{i=1}^{n}{(y_{i} - bar{y})^{2}}} )
with (bar{y}) denoting the sample mean for the full sample and (bar{y}_{k})
denoting the sample mean with the largest k points removed.
The test statistic for the k smallest points is
( L_{k} = frac{sum_{i=k+1}^{n}{(y_{i} - bar{y}_{k})^{2}}}
{sum_{i=1}^{n}{(y_{i} - bar{y})^{2}}} )
with (bar{y}) denoting the sample mean for the full sample and (bar{y}_{k})
denoting the sample mean with the smallest k points removed.
To test for outliers in both tails, compute the absolute residuals
( r_{i} = |y_{i} - bar{y}| )
and then let zi denote the yi values sorted by their absolute residuals in ascending
order. The test statistic for this case is
( E_{k} = frac{sum_{i=1}^{n-k}{(z_{i} - bar{z}_{k})^{2}}}
{sum_{i=1}^{n}{(z_{i} - bar{z})^{2}}} )
with (bar{z}) denoting the sample mean for the full data set and (bar{z}_{k})
denoting the sample mean with the largest k points removed.
Significance
Level:
α
Critical
Region:
The critical region for the Tietjen-Moore test is determined by simulation. The
simulation is performed by generating a standard normal random sample of size n
and computing the Tietjen-Moore test statistic. Typically, 10,000 random samples
are used. The value of the Tietjen-Moore statistic obtained from the data is
compared to this reference distribution.
The value of the test statistic is between zero and one. If there are no outliers in the
data, the test statistic is close to 1. If there are outliers in the data, the test statistic
will be closer to zero. Thus, the test is always a lower, one-tailed test regardless of
which test statisic is used, Lk or Ek.
Sample
Output
The Tietjen-Moore paper gives the following 15 observations of vertical semi-diameters of the
planet Venus (this example originally appeared in Grubbs' 1950 paper):

-1.40 -0.44 -0.30 -0.24 -0.22 -0.13 -0.05 0.06 0.10 0.18 0.20 0.39 0.48 0.63 1.01
As a first step, a normal probability plot was generated.
This plot indicates that the normality assumption is reasonable. The minimum value appears to be
an outlier. To a lesser extent, the maximum value may also be an outlier. The Tietjen-Moore test of
the two most extreme points (-1.40 and 1.01) is shown below.
Ha: the two most extreme points are outliers
Test statistic: Ek = 0.292
Critical value for lower tail: 0.317
Critical region: Reject H0 if Ek < 0.317
The Tietjen-Moore test is a lower, one-tailed test, so we reject the null hypothesis that there are no
outliers when the value of the test statistic is less than the critical value. For our example, the null
hypothesis is rejected at the 0.05 level of significance and we conclude that the two most extreme
points are outliers.
Questions The Tietjen-Moore test can be used to answer the following question:
1. Does the data set contain k outliers?
calculations of the mean and standard deviation may be distorted by a single grossly inaccurate data
point.
Checking for outliers should be a routine part of any data analysis. Potential outliers should be
examined to see if they are possibly erroneous. If the data point is in error, it should be corrected if
possible and deleted if it is not possible. If there is no reason to believe that the outlying point is in
error, it should not be deleted without careful consideration. However, the use of more robust
techniques may be warranted. Robust techniques will often downweight the effect of outlying
points without deleting them.
Related
Techniques
Several graphical techniques can, and should, be used to help detect outliers. A simple normal
probability plot, run sequence plot, a box plot, or a histogram should show any obviously outlying
points. In addition to showing potential outliers, several of these graphics also help assess whether
the data follow an approximately normal distribution.
Run Sequence Plot

Histogram
Box Plot
Lag Plot
Software Some general purpose statistical software programs support the Tietjen-Moore test. Both Dataplot
code and R code can be used to generate the analyses in this section. These scripts use the
TIETMOO1.DAT data file.
Generalized ESD Test for Outliers
Purpose:
Detection
of Outliers
The generalized (extreme Studentized deviate) ESD test (Rosner 1983) is used to detect one
or more outliers in a univariate data set that follows an approximately normal distribution.
The primary limitation of the Grubbs test and the Tietjen-Moore test is that the suspected
number of outliers, k, must be specified exactly. If k is not specified correctly, this can distort
the conclusions of these tests. On the other hand, the generalized ESD test (Rosner 1983)
only requires that an upper bound for the suspected number of outliers be specified.
Definition Given the upper bound, r, the generalized ESD test essentially performs r separate tests: a
test for one outlier, a test for two outliers, and so on up to r outliers.
The generalized ESD test is defined for the hypothesis:
Ha: There are up to r outliers in the data set
Test Statistic: Compute
( R_i = frac{mbox{max}_i |x_i - bar{x}|}{s} )
with (bar{x}) and s denoting the sample mean and sample standard
deviation, respectively.
Remove the observation that maximizes (|x_{i} - bar{x}|) and then
recompute the above statistic with n - 1 observations. Repeat this process
until r observations have been removed. This results in the r test statistics R1,
R2, ..., Rr.
Significance
Level:
α
Critical
Region:
Corresponding to the r test statistics, compute the following r critical values
( lambda_{i} = frac{(n-i) , t_{p, , n-i-1}}{sqrt{(n-i-1+t_{p, , n-i-
1}^{2}) (n-i+1)}} hspace{.2in} i = 1, 2, ldots, r )
where tp,ν is the 100p percentage point from the t distribution with ν degrees
of freedom and
( p = 1-frac{alpha}{2(n-i+1)} )
The number of outliers is determined by finding the largest i such that Ri >
λi.
Simulation studies by Rosner indicate that this critical value approximation is
very accurate for n ≥ 25 and reasonably accurate for n ≥ 15.
Note that although the generalized ESD is essentially Grubbs test applied sequentially, there
are a few important distinctions:
The generalized ESD test makes approriate adjustments for the critical values based on
the number of outliers being tested for that the sequential application of Grubbs test
does not.
If there is significant masking, applying Grubbs test sequentially may stop too soon.
The example below identifies three outliers at the 5 % level when using the generalized

ESD test. However, trying to use Grubbs test sequentially would stop at the first
iteration and declare no outliers.
Generalized
ESD Test
Example
The Rosner paper gives an example with the following data.
-0.25 0.68 0.94 1.15 1.20 1.26 1.26
1.34 1.38 1.43 1.49 1.49 1.55 1.56
1.58 1.65 1.69 1.70 1.76 1.77 1.81
1.91 1.94 1.96 1.99 2.06 2.09 2.10
2.14 2.15 2.23 2.24 2.26 2.35 2.37
2.40 2.47 2.54 2.62 2.64 2.90 2.92
2.92 2.93 3.21 3.26 3.30 3.59 3.68
4.30 4.64 5.34 5.42 6.01
As a first step, a normal probability plot was generated
This plot indicates that the normality assumption is questionable.
Following the Rosner paper, we test for up to 10 outliers:
Ha: there are up to 10 outliers in the data
Critical region: Reject H0 if Ri > critical value
Summary Table for Two-Tailed Test
---------------------------------------
Exact Test Critical
Number of Statistic Value, λi
Outliers, i Value, Ri 5 %
---------------------------------------
1 3.118 3.158
2 2.942 3.151
3 3.179 3.143 *
4 2.810 3.136
5 2.815 3.128
6 2.848 3.120
7 2.279 3.111
8 2.310 3.103
9 2.101 3.094
10 2.067 3.085
For the generalized ESD test above, there are essentially 10 separate tests being performed.
For this example, the largest number of outliers for which the test statistic is greater than the

critical value (at the 5 % level) is three. We therefore conclude that there are three outliers in
this data set.
Questions The generalized ESD test can be used to answer the following question:
1. How many outliers does the data set contain?
calculations of the mean and standard deviation may be distorted by a single grossly
inaccurate data point.
Checking for outliers should be a routine part of any data analysis. Potential outliers should
be examined to see if they are possibly erroneous. If the data point is in error, it should be
corrected if possible and deleted if it is not possible. If there is no reason to believe that the
outlying point is in error, it should not be deleted without careful consideration. However, the
use of more robust techniques may be warranted. Robust techniques will often downweight
the effect of outlying points without deleting them.
Related
Techniques
Several graphical techniques can, and should, be used to help detect outliers. A simple
normal probability plot, run sequence plot, a box plot, or a histogram should show any
obviously outlying points. In addition to showing potential outliers, several of these graphics
also help assess whether the data follow an approximately normal distribution.
Run Sequence Plot
Histogram
Box Plot
Lag Plot
Software Some general purpose statistical software programs support the generalized ESD test. Both
Dataplot code and R code can be used to generate the analyses in this section. These scripts
use the ROSNER.DAT data file.
Yates Algorithm
Purpose:
Estimate
Factor Effects
in a 2-Level
Factorial
Design
Full factorial and fractional factorial designs are common in
designed experiments for engineering and scientific applications.
In these designs, each factor is assigned two levels. These are
typically called the low and high levels. For computational
purposes, the factors are scaled so that the low level is assigned a
value of -1 and the high level is assigned a value of +1. These are
also commonly referred to as "-" and "+".
A full factorial design contains all possible combinations of
low/high levels for all the factors. A fractional factorial design
contains a carefully chosen subset of these combinations. The
criterion for choosing the subsets is discussed in detail in the
process improvement chapter.
The Yates algorithm exploits the special structure of these designs
to generate least squares estimates for factor effects for all factors
and all relevant interactions.
The mathematical details of the Yates algorithm are given in chapter
10 of Box, Hunter, and Hunter (1978). Natrella (1963) also provides
a procedure for testing the significance of effect estimates.
The effect estimates are typically complemented by a number of
graphical techniques such as the DOE mean plot and the DOE
contour plot ("DOE" represents "design of experiments"). These are
demonstrated in the eddy current case study.
Yates Order Before performing the Yates algorithm, the data should be arranged
in "Yates order". That is, given k factors, the kth column consists of

2k-1 minus signs (i.e., the low level of the factor) followed by 2k-1
plus signs (i.e., the high level of the factor). For example, for a full
factorial design with three factors, the design matrix is
- - -
+ - -
- + -
+ + -
- - +
+ - +
- + +
+ + +
Determining the Yates order for fractional factorial designs requires
knowledge of the confounding structure of the fractional factorial
design.
Yates
Algorithm
The Yates algorithm is demonstrated for the eddy current data set.
The data set contains eight measurements from a two-level, full
factorial design with three factors. The purpose of the experiment is
to identify factors that have the most effect on eddy current
measurements.
In the "Effect" column, we list the main effects and interactions
from our factorial experiment in standard order. In the "Response"
column, we list the measurement results from our experiment in
Yates order.
Effect Response Col 1 Col 2 Col 3 Estimate
------ -------- ----- ----- ----- --------
Mean 1.70 6.27 10.21 21.27 2.65875
X1 4.57 3.94 11.06 12.41 1.55125
X2 0.55 6.10 5.71 -3.47 -0.43375
X1*X2 3.39 4.96 6.70 0.51 0.06375
X3 1.51 2.87 -2.33 0.85 0.10625
X1*X3 4.59 2.84 -1.14 0.99 0.12375
X2*X3 0.67 3.08 -0.03 1.19 0.14875
X1*X2*X3 4.29 3.62 0.54 0.57 0.07125
Sum of responses: 21.27
Sum-of-squared responses: 77.7707
Sum-of-squared Col 3: 622.1656
The first four values in Col 1 are obtained by adding adjacent pairs
of responses, for example 4.57 + 1.70 = 6.27, and 3.39 + 0.55 =
3.94. The second four values in Col 1 are obtained by subtracting
the same adjacent pairs of responses, for example, 4.57 - 1.70 =
2.87, and 3.39 - 0.55 = 2.84. The values in Col 2 are calculated in
the same way, except that we are adding and subtracting adjacent
values from Col 1. Col 3 is computed using adjacent values from
Col 2. Finally, we obtain the "Estimate" column by dividing the
values in Col 3 by the total number of responses, 8.
We can check our calculations by making sure that the first value in
Col 3 (21.27) is the sum of all the responses. In addition, the sum-
of-squared responses (77.7707) should equal the sum-of-squared
Col 3 values divided by 8 (622.1656/8 = 77.7707).
Practical
Considerations
The Yates algorithm provides a convenient method for computing
effect estimates; however, the same information is easily obtained
from statistical software using either an analysis of variance or
regression procedure. The methods for analyzing data from a
designed experiment are discussed more fully in the chapter on
Process Improvement.
Graphical
Presentation
The following plots may be useful to complement the quantitative
information from the Yates algorithm.
1. Ordered data plot
2. Ordered absolute effects plot
3. Cumulative residual standard deviation plot
Questions The Yates algorithm can be used to answer the following question.

1. What is the estimated effect of a factor on the response?
Related
Techniques
Multi-factor analysis of variance
DOE mean plot
Block plot
DOE contour plot
Case Study The analysis of a full factorial design is demonstrated in the eddy
current case study.
Software All statistical software packages are capable of estimating effects
using an analysis of variance or least squares regression procedure.
Defining Models and Prediction Equations
For
Orthogonal
Designs,
Parameter
Estimates
Don't
Change as
Additional
Terms Are
Added
In most cases of least-squares fitting, the model coefficients
for previously added terms change depending on what was
successively added. For example, the X1 coefficient might
change depending on whether or not an X2 term was included
in the model. This is not the case when the design is
orthogonal, as is a 23 full factorial design. For orthogonal
designs, the estimates for the previously included terms do not
change as additional terms are added. This means the ranked
list of parameter estimates are the least-squares coefficient
estimates for progressively more complicated models.
Example
Prediction
Equation
We use the parameter estimates derived from a least-squares
analysis for the eddy current data set to create an example
prediction equation.
Parameter Estimate
--------- --------
Mean 2.65875
X1 1.55125
X2 -0.43375
X1*X2 0.06375
X3 0.10625
X1*X3 0.12375
X2*X3 0.14875
X1*X2*X3 0.07125
A prediction equation predicts a value of the reponse variable
for given values of the factors. The equation we select can
include all the factors shown above, or it can include a subset
of the factors. For example, one possible prediction equation
using only two factors, X1 and X2, is:
( hat{Y} = 2.65875 + 1.55125 cdot X_1 - 0.43375 cdot X_2
)
The least-squares parameter estimates in the prediction
equation reflect the change in response for a one-unit change
in the factor value. To obtain "full" effect estimates (as
computed using the Yates algorithm) for the change in factor
levels from -1 to +1, the effect estimates (except for the
intercept) would be multiplied by two.
Remember that the Yates algorithm is just a convenient method
for computing effects, any statistical software package with
least-squares regression capabilities will produce the same
effects as well as many other useful analyses.
Model
Selection
We want to select the most appropriate model for our data
while balancing the following two goals.
1. We want the model to include all important factors.
2. We want the model to be parsimonious. That is, the
model should be as simple as possible.

Note that the residual standard deviation alone is insufficient
for determining the most appropriate model as it will always
be decreased by adding additional factors. The next section
describes a number of approaches for determining which
factors (and interactions) to include in the model.
Important Factors
Identify
Important
Factors
We want to select the most appropriate model to represent our data. This requires balancing the
following two goals.
1. We want the model to include all important factors.
2. We want the model to be parsimonious. That is, the model should be as simple as
possible.
In short, we want our model to include all the important factors and interactions and to omit the
unimportant factors and interactions.
Seven criteria are utilized to define important factors. These seven criteria are not all equally
important, nor will they yield identical subsets, in which case a consensus subset or a weighted
consensus subset must be extracted. In practice, some of these criteria may not apply in all
situations.
These criteria will be examined in the context of the eddy current data set. The parameter
estimates computed using least-squares analysis are shown below.
Parameter Estimate
--------- --------
Mean 2.65875
X1 1.55125
X2 -0.43375
X1*X2 0.06375
X3 0.10625
X1*X3 0.12375
X2*X3 0.14875
X1*X2*X3 0.07125
In practice, not all of these criteria will be used with every analysis (and some analysts may
have additional criteria). These critierion are given as useful guidelines. Most analysts will
focus on those criteria that they find most useful.
Criteria for
Including
Terms in the
Model
The seven criteria that we can use in determining whether to keep a factor in the model can be
summarized as follows.
1. Parameters: Engineering Significance
2. Parameters: Order of Magnitude
3. Parameters: Statistical Significance
4. Parameters: Probability Plots
5. Effects: Youden Plot
6. Residual Standard Deviation: Engineering Significance
7. Residual Standard Deviation: Statistical Significance
The first four criteria focus on parameter estimates with three numeric criteria and one
graphical criteria. The fifth criteria focuses on effects, which are twice the parameter estimates.
The last two criteria focus on the residual standard deviation of the model. We discuss each of
these seven criteria in detail in the sections that following.
Parameters:
Engineering
Significance
The minimum engineering significant difference is defined as
( |hat{beta_{i}}| > Delta )
where (|hat{beta_{i}}|) is the absolute value of the parameter estimate and Δ is the minimum
engineering significant difference.
That is, declare a factor as "important" if the parameter estimate is greater than some a priori
declared engineering difference. This implies that the engineering staff have in fact stated what
a minimum difference will be. Oftentimes this is not the case. In the absence of an a priori
difference, a good rough rule for the minimum engineering significant Δ is to keep only those
factors whose parameter estimate is greater than, say, 10% of the current production average. In
this case, let's say that the average detector has a sensitivity of 2.5 ohms. This would suggest

that we would declare all factors whose parameter is greater than 10 % of 2.5 ohms = 0.25 ohm
to be significant (from an engineering point of view).
Based on this minimum engineering significant difference criterion, we conclude that we
should keep two terms: X1 and X2.
Parameters:
Order of
Magnitude
The order of magnitude criterion is defined as
( |hat{beta_{i}}| < 0.10 cdot max |hat{beta_{i}}| )
That is, exclude any factor that is less than 10 % of the maximum parameter size. We may or
may not keep the other factors. This criterion is neither engineering nor statistical, but it does
offer some additional numerical insight. For the current example, the largest parameter is from
X1 (1.55125 ohms), and so 10 % of that is 0.155 ohms, which suggests keeping all factors
whose parameters exceed 0.155 ohms.
Based on the order-of-magnitude criterion, we thus conclude that we should keep two terms:
X1 and X2. A third term, X2*X3 (0.14875), is just slightly under the cutoff level, so we may
consider keeping it based on the other criterion.
Parameters:
Statistical
Significance
Statistical significance is defined as
( |hat{beta_i}| > 2 cdot mbox{s.e.}(hat{beta_i}) )
That is, declare a factor as important if its parameter is more than 2 standard deviations away
from 0 (0, by definition, meaning "no effect").
The "2" comes from normal theory (more specifically, a value of 1.96 yields a 95 % confidence
interval). More precise values would come from t-distribution theory.
The difficulty with this is that in order to invoke this criterion we need the standard deviation, σ
of an observation. This is problematic because
1. the engineer may not know σ;
2. the experiment might not have replication, and so a model-free estimate of σ is not
obtainable;
3. obtaining an estimate of σ by assuming the sometimes employed assumption of ignoring
3-term interactions and higher may be incorrect from an engineering point of view.
For the eddy current example:
1. the engineer did not know σ;
2. the design (a 23 full factorial) did not have replication;
3. ignoring 3-term interactions and higher interactions leads to an estimate of σ based on
omitting only a single term: the X1*X2*X3 interaction.
For the eddy current example, if one assumes that the 3-term interaction is nil and hence
represents a single drawing from a population centered at zero, then an estimate of the standard
deviation of a parameter is simply the estimate of the 3-factor interaction (0.07125). Two
standard deviations is thus 0.1425. For this example, the rule is thus to keep all (
|hat{beta_{i}}| )
This results in keeping three terms: X1 (1.55125), X2 (-0.43375), and X1*X2 (0.14875).
Parameters:
Probability
Plots
Probability plots can be used in the following manner.
1. Normal Probability Plot: Keep a factor as "important" if it is well off the line through
zero on a normal probability plot of the parameter estimates.
2. Half-Normal Probability Plot: Keep a factor as "important" if it is well off the line near
zero on a half-normal probability plot of the absolute value of parameter estimates.
Both of these methods are based on the fact that the least-squares estimates of parameters for
these two-level orthogonal designs are simply half the difference of averages and so the central
limit theorem, loosely applied, suggests that (if no factor were important) the parameter
estimates should have approximately a normal distribution with mean zero and the absolute
value of the estimates should have a half-normal distribution.
Since the half-normal probability plot is only concerned with parmeter magnitudes as opposed
to signed parameters (which are subject to the vagaries of how the initial factor codings +1 and
-1 were assigned), the half-normal probability plot is preferred by some over the normal
probability plot.

Normal
Probablity
Plot of
Parameters
The following normal probability plot shows the parameter estimates for the eddy current data.
For the example at hand, the probability plot clearly shows two factors (X1 and X2) displaced
off the line. All of the remaining five parameters are behaving like random drawings from a
normal distribution centered at zero, and so are deemed to be statistically non-significant. In
conclusion, this rule keeps two factors: X1 (1.55125) and X2 (-0.43375).
Averages:
Youden Plot
A Youden plot can be used in the following way. Keep a factor as "important" if it is displaced
away from the central-tendancy "bunch" in a Youden plot of high and low averages. By
definition, a factor is important when its average response for the low (-1) setting is
significantly different from its average response for the high (+1) setting. (Note that effects are
twice the parameter estimates.) Conversely, if the low and high averages are about the same,
then what difference does it make which setting to use and so why would such a factor be
considered important? This fact in combination with the intrinsic benefits of the Youden plot
for comparing pairs of items leads to the technique of generating a Youden plot of the low and
high averages.
Youden Plot
of Effect
Estimates
The following is the Youden plot of the effect estimatess for the eddy current data.

For the example at hand, the Youden plot clearly shows a cluster of points near the grand
average (2.65875) with two displaced points above (factor 1) and below (factor 2). Based on
the Youden plot, we conclude to keep two factors: X1 (1.55125) and X2 (-0.43375).
Residual
Standard
Deviation:
Engineering
Significance
This criterion is defined as
Residual Standard Deviation > Cutoff
That is, declare a factor as "important" if the cumulative model that includes the factor (and all
larger factors) has a residual standard deviation smaller than an a priori engineering-specified
minimum residual standard deviation.
This criterion is different from the others in that it is model focused. In practice, this criterion
states that starting with the largest parameter, we cumulatively keep adding terms to the model
and monitor how the residual standard deviation for each progressively more complicated
model becomes smaller. At some point, the cumulative model will become complicated enough
and comprehensive enough that the resulting residual standard deviation will drop below the
pre-specified engineering cutoff for the residual standard deviation. At that point, we stop
adding terms and declare all of the model-included terms to be "important" and everything not
in the model to be "unimportant".
This approach implies that the engineer has considered what a minimum residual standard
deviation should be. In effect, this relates to what the engineer can tolerate for the magnitude of
the typical residual (the difference between the raw data and the predicted value from the
model). In other words, how good does the engineer want the prediction equation to be.
Unfortunately, this engineering specification has not always been formulated and so this
criterion can become moot.
In the absence of a prior specified cutoff, a good rough rule for the minimum engineering
residual standard deviation is to keep adding terms until the residual standard deviation just
dips below, say, 5 % of the current production average. For the eddy current data, let's say that
the average detector has a sensitivity of 2.5 ohms. Then this would suggest that we would keep
adding terms to the model until the residual standard deviation falls below 5 % of 2.5 ohms =
0.125 ohms.
Residual
Model Std. Dev.

----------------------------------------------------- ---------
Mean + X1 0.57272
Mean + X1 + X2 0.30429
Mean + X1 + X2 + X2*X3 0.26737
Mean + X1 + X2 + X2*X3 + X1*X3 0.23341
Mean + X1 + X2 + X2*X3 + X1*X3 + X3 0.19121
Mean + X1 + X2 + X2*X3 + X1*X3 + X3 + X1*X2*X3 0.18031
Mean + X1 + X2 + X2*X3 + X1*X3 + X3 + X1*X2*X3 + X1*X2 NA
Based on the minimum residual standard deviation criteria, and we would include all terms in
order to drive the residual standard deviation below 0.125. Again, the 5 % rule is a rough-and-
ready rule that has no basis in engineering or statistics, but is simply a "numerics". Ideally, the
engineer has a better cutoff for the residual standard deviation that is based on how well he/she
wants the equation to peform in practice. If such a number were available, then for this
criterion and data set we would select something less than the entire collection of terms.
Residual
Standard
Deviation:
Statistical
Significance
This criterion is defined as
Residual Standard Deviation > σ
where σ is the standard deviation of an observation under replicated conditions.
That is, declare a term as "important" until the cumulative model that includes the term has a
residual standard deviation smaller than σ. In essence, we are allowing that we cannot demand
a model fit any better than what we would obtain if we had replicated data; that is, we cannot
demand that the residual standard deviation from any fitted model be any smaller than the
(theoretical or actual) replication standard deviation. We can drive the fitted standard deviation
down (by adding terms) until it achieves a value close to σ, but to attempt to drive it down
further means that we are, in effect, trying to fit noise.
In practice, this criterion may be difficult to apply because
1. the engineer may not know σ;
2. the experiment might not have replication, and so a model-free estimate of σ is not
obtainable.
For the current case study:
1. the engineer did not know σ;
2. the design (a 23 full factorial) did not have replication. The most common way of having
replication in such designs is to have replicated center points at the center of the cube
((X1,X2,X3) = (0,0,0)).
Thus for this current case, this criteria could not be used to yield a subset of "important"
factors.
Conclusions In summary, the seven criteria for specifying "important" factors yielded the following for the
eddy current data:
1. Parameters, Engineering Significance: X1, X2
2. Parameters, Numerically Significant: X1, X2
3. Parameters, Statistically Significant: X1, X2, X2*X3
4. Parameters, Probability Plots: X1, X2
5. Effects, Youden Plot: X1, X2
6. Residual SD, Engineering Significance:all 7 terms
7. Residual SD, Statistical Significance: not applicable
Such conflicting results are common. Arguably, the three most important criteria (listed in
order of most important) are:
4. Parameters, Probability Plots: X1, X2
1. Parameters, Engineering Significance: X1, X2
3. Residual SD, Engineering Significance:all 7 terms

Scanning all of the above, we thus declare the following consensus for the eddy current data:
1. Important Factors: X1 and X2
2. Parsimonious Prediction Equation:
( hat{Y} = 2.65875 + 1.55125 cdot X_1 - 0.43375 cdot X_2 )
(with a residual standard deviation of 0.30429 ohms)
Note that this is the initial model selection. We still need to perform model validation with a
residual analysis.
Probability Distributions
Probability
Distributions
Probability distributions are a fundamental concept in
statistics. They are used both on a theoretical level and a
practical level.
Some practical uses of probability distributions are:
To calculate confidence intervals for parameters and to
calculate critical regions for hypothesis tests.
For univariate data, it is often useful to determine a
reasonable distributional model for the data.
Statistical intervals and hypothesis tests are often based
on specific distributional assumptions. Before
computing an interval or test based on a distributional
assumption, we need to verify that the assumption is
justified for the given data set. In this case, the
distribution does not need to be the best-fitting
distribution for the data, but an adequate enough model
so that the statistical technique yields valid
conclusions.
Simulation studies with random numbers generated
from using a specific probability distribution are often
needed.
Table of
Contents
1. What is a probability distribution?
2. Related probability functions
3. Families of distributions
4. Location and scale parameters
5. Estimating the parameters of a distribution
6. A gallery of common distributions
7. Tables for probability distributions
What is a Probability Distribution
Discrete
Distributions
The mathematical definition of a discrete probability
function, p(x), is a function that satisfies the following
properties.
1. The probability that x can take a specific value is p(x).
That is
[ P[X = x] = p(x) = p_{x} ]
2. p(x) is non-negative for all real x.
3. The sum of p(x) over all possible values of x is 1, that
is
[ sum_{j}p_{j} = 1 ]

where j represents all possible values that x can have
and pj is the probability at xj.
One consequence of properties 2 and 3 is that 0 <=
p(x) <= 1.
What does this actually mean? A discrete probability function
is a function that can take a discrete number of values (not
necessarily finite). This is most often the non-negative
integers or some subset of the non-negative integers. There is
no mathematical restriction that discrete probability functions
only be defined at integers, but in practice this is usually
what makes sense. For example, if you toss a coin 6 times,
you can get 2 heads or 3 heads but not 2 1/2 heads. Each of
the discrete values has a certain probability of occurrence
that is between zero and one. That is, a discrete function that
allows negative values or values greater than one is not a
probability function. The condition that the probabilities sum
to one means that at least one of the values has to occur.
Continuous
Distributions
The mathematical definition of a continuous probability
function, f(x), is a function that satisfies the following
properties.
1. The probability that x is between two points a and b is
[ p[a le x le b] = int_{a}^{b} {f(x)dx} ]
2. It is non-negative for all real x.
3. The integral of the probability function is one, that is
[ int_{-infty}^{infty} {f(x)dx} = 1 ]
What does this actually mean? Since continuous probability
functions are defined for an infinite number of points over a
continuous interval, the probability at a single point is always
zero. Probabilities are measured over intervals, not single
points. That is, the area under the curve between two distinct
points defines the probability for that interval. This means
that the height of the probability function can in fact be
greater than one. The property that the integral must equal
one is equivalent to the property for discrete distributions that
the sum of all the probabilities must equal one.
Probability
Mass
Functions
Versus
Probability
Density
Functions
Discrete probability functions are referred to as probability
mass functions and continuous probability functions are
referred to as probability density functions. The term
probability functions covers both discrete and continuous
distributions.
There are a few occasions in the e-Handbook when we use
the term probability density function in a generic sense where
it may apply to either probability density or probability mass
functions. It should be clear from the context whether we are
referring only to continuous distributions or to either
continuous or discrete distributions.
Related Distributions
Probability distributions are typically defined in terms of the
probability density function. However, there are a number of
probability functions used in applications.
Probability
Density
Function
For a continuous function, the probability density function
(pdf) is the probability that the variate has the value x. Since
for continuous distributions the probability at a single point is

zero, this is often expressed in terms of an integral between
two points.
( int_{a}^{b} {f(x) dx} = Pr[a le X le b] )
For a discrete distribution, the pdf is the probability that the
variate takes the value x.
( f(x) = Pr[X = x] )
The following is the plot of the normal probability density
function.
Cumulative
Distribution
Function
The cumulative distribution function (cdf) is the probability
that the variable takes a value less than or equal to x. That is
( F(x) = Pr[X le x] = alpha )
For a continuous distribution, this can be expressed
mathematically as
( F(x) = int_{-infty}^{x} {f(mu) dmu} )
For a discrete distribution, the cdf can be expressed as
( F(x) = sum_{i=0}^{x} {f(i)} )
The following is the plot of the normal cumulative distribution
function.
The horizontal axis is the allowable domain for the given
probability function. Since the vertical axis is a probability, it
must fall between zero and one. It increases from zero to one
as we go from left to right on the horizontal axis.
Percent
Point
Function
The percent point function (ppf) is the inverse of the
cumulative distribution function. For this reason, the percent
point function is also commonly referred to as the inverse
distribution function. That is, for a distribution function we

calculate the probability that the variable is less than or equal
to x for a given x. For the percent point function, we start with
the probability and compute the corresponding x for the
cumulative distribution. Mathematically, this can be expressed
as
( Pr[X le G(alpha)] = alpha )
or alternatively
( x = G(alpha) = G(F(x)) )
The following is the plot of the normal percent point function.
Since the horizontal axis is a probability, it goes from zero to
one. The vertical axis goes from the smallest to the largest
value of the cumulative distribution function.
Hazard
Function
The hazard function is the ratio of the probability density
function to the survival function, S(x).
( h(x) = frac {f(x)} {S(x)} = frac {f(x)} {1 - F(x)} )
The following is the plot of the normal distribution hazard
function.
Hazard plots are most commonly used in reliability
applications. Note that Johnson, Kotz, and Balakrishnan refer
to this as the conditional failure density function rather than
the hazard function.
Cumulative
Hazard
Function
The cumulative hazard function is the integral of the hazard
function.
( H(x) = int_{-infty}^{x} {h(mu) dmu} )
This can alternatively be expressed as
( H(x) = -ln {(1 - F(x))} )

The following is the plot of the normal cumulative hazard
function.
Cumulative hazard plots are most commonly used in
reliability applications. Note that Johnson, Kotz, and
Balakrishnan refer to this as the hazard function rather than
the cumulative hazard function.
Survival
Function
Survival functions are most often used in reliability and
related fields. The survival function is the probability that the
variate takes a value greater than x.
( S(x) = Pr[X > x] = 1 - F(x) )
The following is the plot of the normal distribution survival
function.
For a survival function, the y value on the graph starts at 1 and
monotonically decreases to zero. The survival function should
be compared to the cumulative distribution function.
Inverse
Survival
Function
Just as the percent point function is the inverse of the
cumulative distribution function, the survival function also has
an inverse function. The inverse survival function can be
defined in terms of the percent point function.
( Z(alpha) = G(1 - alpha) )
The following is the plot of the normal distribution inverse
survival function.

As with the percent point function, the horizontal axis is a
probability. Therefore the horizontal axis goes from 0 to 1
regardless of the particular distribution. The appearance is
similar to the percent point function. However, instead of
going from the smallest to the largest value on the vertical
axis, it goes from the largest to the smallest value.
Families of Distributions
Shape
Parameters
Many probability distributions are not a single distribution,
but are in fact a family of distributions. This is due to the
distribution having one or more shape parameters.
Shape parameters allow a distribution to take on a variety of
shapes, depending on the value of the shape parameter. These
distributions are particularly useful in modeling applications
since they are flexible enough to model a variety of data sets.
Example:
Weibull
Distribution
The Weibull distribution is an example of a distribution that
has a shape parameter. The following graph plots the Weibull
pdf with the following values for the shape parameter: 0.5,
1.0, 2.0, and 5.0.
The shapes above include an exponential distribution, a right-
skewed distribution, and a relatively symmetric distribution.
The Weibull distribution has a relatively simple distributional
form. However, the shape parameter allows the Weibull to
assume a wide variety of shapes. This combination of
simplicity and flexibility in the shape of the Weibull
distribution has made it an effective distributional model in
reliability applications. This ability to model a wide variety of
distributional shapes using a relatively simple distributional
form is possible with many other distributional families as
well.

PPCC Plots The PPCC plot is an effective graphical tool for selecting the
member of a distributional family with a single shape
parameter that best fits a given set of data.
Location and Scale Parameters
Normal
PDF
A probability distribution is characterized by location and
scale parameters. Location and scale parameters are typically
used in modeling applications.
For example, the following graph is the probability density
function for the standard normal distribution, which has the
location parameter equal to zero and scale parameter equal to
one.
Location
Parameter
The next plot shows the probability density function for a
normal distribution with a location parameter of 10 and a
scale parameter of 1.
The effect of the location parameter is to translate the graph,
relative to the standard normal distribution, 10 units to the
right on the horizontal axis. A location parameter of -10
would have shifted the graph 10 units to the left on the
horizontal axis.
That is, a location parameter simply shifts the graph left or
right on the horizontal axis.
Scale
Parameter
The next plot has a scale parameter of 3 (and a location
parameter of zero). The effect of the scale parameter is to
stretch out the graph. The maximum y value is approximately
0.13 as opposed 0.4 in the previous graphs. The y value, i.e.,
the vertical axis value, approaches zero at about (+/-) 9 as
opposed to (+/-) 3 with the first graph.

In contrast, the next graph has a scale parameter of 1/3
(=0.333). The effect of this scale parameter is to squeeze the
pdf. That is, the maximum y value is approximately 1.2 as
opposed to 0.4 and the y value is near zero at (+/-) 1 as
opposed to (+/-) 3.
The effect of a scale parameter greater than one is to stretch
the pdf. The greater the magnitude, the greater the stretching.
The effect of a scale parameter less than one is to compress
the pdf. The compressing approaches a spike as the scale
parameter goes to zero. A scale parameter of 1 leaves the pdf
unchanged (if the scale parameter is 1 to begin with) and non-
positive scale parameters are not allowed.
Location
and Scale
Together
The following graph shows the effect of both a location and a
scale parameter. The plot has been shifted right 10 units and
stretched by a factor of 3.
Standard
Form
The standard form of any distribution is the form that has
location parameter zero and scale parameter one.

It is common in statistical software packages to only compute
the standard form of the distribution. There are formulas for
converting from the standard form to the form with other
location and scale parameters. These formulas are
independent of the particular probability distribution.
Formulas
for Location
and Scale
Based on
the
Standard
Form
The following are the formulas for computing various
probability functions based on the standard form of the
distribution. The parameter a refers to the location parameter
and the parameter b refers to the scale parameter. Shape
parameters are not included.
Cumulative Distribution
Function
F(x;a,b) = F((x-a)/b;0,1)
Probability Density Function f(x;a,b) = (1/b)f((x-a)/b;0,1)
Percent Point Function G(α;a,b) = a + bG(α;0,1)
Hazard Function h(x;a,b) = (1/b)h((x-a)/b;0,1)
Cumulative Hazard Function H(x;a,b) = H((x-a)/b;0,1)
Survival Function S(x;a,b) = S((x-a)/b;0,1)
Inverse Survival Function Z(α;a,b) = a + bZ(α;0,1)
Random Numbers Y(a,b) = a + bY(0,1)
Relationship
to Mean and
Standard
Deviation
For the normal distribution, the location and scale parameters
correspond to the mean and standard deviation, respectively.
However, this is not necessarily true for other distributions. In
fact, it is not true for most distributions.
Estimating the Parameters of a Distribution
Model a
univariate
data set
with a
probability
distribution
One common application of probability distributions is
modeling univariate data with a specific probability
distribution. This involves the following two steps:
1. Determination of the "best-fitting" distribution.
2. Estimation of the parameters (shape, location, and scale
parameters) for that distribution.
Various
Methods
There are various methods, both numerical and graphical, for
estimating the parameters of a probability distribution.
1. Method of moments
2. Maximum likelihood
3. Least squares
4. PPCC and probability plots
Method of Moments
Method of
Moments
The method of moments equates sample moments to parameter
estimates. When moment methods are available, they have the
advantage of simplicity. The disadvantage is that they are often
not available and they do not have the desirable optimality
properties of maximum likelihood and least squares estimators.
The primary use of moment estimates is as starting values for
the more precise maximum likelihood and least squares
estimates.
Software Most general purpose statistical software does not include
explicit method of moments parameter estimation commands.
However, when utilized, the method of moment formulas tend
to be straightforward and can be easily implemented in most

Maximum Likelihood
Maximum
Likelihood
Maximum likelihood estimation begins with the
mathematical expression known as a likelihood function of
the sample data. Loosely speaking, the likelihood of a set of
data is the probability of obtaining that particular set of data
given the chosen probability model. This expression
contains the unknown parameters. Those values of the
parameter that maximize the sample likelihood are known
as the maximum likelihood estimates.
The reliability chapter contains some examples of the
likelihood functions for a few of the commonly used
distributions in reliability analysis.
Advantages The advantages of this method are:
Maximum likelihood provides a consistent approach
to parameter estimation problems. This means that
maximum likelihood estimates can be developed for
a large variety of estimation situations. For example,
they can be applied in reliability analysis to censored
data under various censoring models.
Maximum likelihood methods have desirable
mathematical and optimality properties. Specifically,
1. They become minimum variance unbiased
estimators as the sample size increases. By
unbiased, we mean that if we take (a very large
number of) random samples with replacement
from a population, the average value of the
parameter estimates will be theoretically
exactly equal to the population value. By
minimum variance, we mean that the estimator
has the smallest variance, and thus the
narrowest confidence interval, of all estimators
of that type.
2. They have approximate normal distributions
and approximate sample variances that can be
used to generate confidence bounds and
hypothesis tests for the parameters.
Several popular statistical software packages provide
excellent algorithms for maximum likelihood
estimates for many of the commonly used
distributions. This helps mitigate the computational
complexity of maximum likelihood estimation.
Disadvantages The disadvantages of this method are:
The likelihood equations need to be specifically
worked out for a given distribution and estimation
problem. The mathematics is often non-trivial,
particularly if confidence intervals for the parameters
are desired.
The numerical estimation is usually non-trivial.
Except for a few cases where the maximum
likelihood formulas are in fact simple, it is generally
best to rely on high quality statistical software to
obtain maximum likelihood estimates. Fortunately,
high quality maximum likelihood software is
becoming increasingly common.
Maximum likelihood estimates can be heavily biased
for small samples. The optimality properties may not
apply for small samples.

Maximum likelihood can be sensitive to the choice of
starting values.
Software Most general purpose statistical software programs support
maximum likelihood estimation (MLE) in some form. MLE
estimation can be supported in two ways.
1. A software program may provide a generic function
minimization (or equivalently, maximization)
capability. This is also referred to as function
optimization. Maximum likelihood estimation is
essentially a function optimization problem.
This type of capability is particularly common in
mathematical software programs.
2. A software program may provide MLE computations
for a specific problem. For example, it may generate
ML estimates for the parameters of a Weibull
distribution.
Statistical software programs will often provide ML
estimates for many specific problems even when they
do not support general function optimization.
The advantage of function minimization software is that it
can be applied to many different MLE problems. The
drawback is that you have to specify the maximum
likelihood equations to the software. As the functions can
be non-trivial, there is potential for error in entering the
equations.
The advantage of the specific MLE procedures is that
greater efficiency and better numerical stability can often
be obtained by taking advantage of the properties of the
specific estimation problem. The specific methods often
return explicit confidence intervals. In addition, you do not
have to know or specify the likelihood equations to the
software. The disadvantage is that each MLE problem must
be specifically coded.
Least Squares
Least Squares Non-linear least squares provides an alternative to
maximum likelihood.
Non-linear least squares software may be available in
many statistical software packages that do not
support maximum likelihood estimates.
It can be applied more generally than maximum
likelihood. That is, if your software provides non-
linear fitting and it has the ability to specify the
probability function you are interested in, then you
can generate least squares estimates for that
distribution. This will allow you to obtain reasonable
estimates for distributions even if the software does
not provide maximum likelihood estimates.
It is not readily applicable to censored data.
It is generally considered to have less desirable
optimality properties than maximum likelihood.

It can be quite sensitive to the choice of starting
values.
Software Non-linear least squares fitting is available in many general
purpose statistical software programs.
PPCC and Probability Plots
PPCC and
Probability
Plots
The PPCC plot can be used to estimate the shape parameter
of a distribution with a single shape parameter. After
finding the best value of the shape parameter, the
probability plot can be used to estimate the location and
scale parameters of a probability distribution.
It is based on two well-understood concepts.
1. The linearity (i.e., straightness) of the
probability plot is a good measure of the
adequacy of the distributional fit.
2. The correlation coefficient between the points
on the probability plot is a good measure of the
linearity of the probability plot.
It is an easy technique to implement for a wide
variety of distributions with a single shape parameter.
The basic requirement is to be able to compute the
percent point function, which is needed in the
computation of both the probability plot and the
PPCC plot.
The PPCC plot provides insight into the sensitivity of
the shape parameter. That is, if the PPCC plot is
relatively flat in the neighborhood of the optimal
value of the shape parameter, this is a strong
indication that the fitted model will not be sensitive
to small deviations, or even large deviations in some
cases, in the value of the shape parameter.
The maximum correlation value provides a method
for comparing across distributions as well as
identifying the best value of the shape parameter for
a given distribution. For example, we could use the
PPCC and probability fits for the Weibull, lognormal,
and possibly several other distributions. Comparing
the maximum correlation coefficient achieved for
each distribution can help in selecting which is the
best distribution to use.
It is limited to distributions with a single shape
parameter.
PPCC plots are not widely available in statistical
software packages other than Dataplot (Dataplot
provides PPCC plots for 40+ distributions).
Probability plots are generally available. However,
many statistical software packages only provide them
for a limited number of distributions.
Significance levels for the correlation coefficient
(i.e., if the maximum correlation value is above a
given value, then the distribution provides an
adequate fit for the data with a given confidence

level) have only been worked out for a limited
number of distributions.
Other
Graphical
Methods
For reliability applications, the hazard plot and the Weibull
plot are alternative graphical methods that are commonly
used to estimate parameters.
Gallery of Distributions
Gallery of
Common
Distributions
Detailed information on a few of the most common
distributions is available below. There are a large number of
distributions used in statistical applications. It is beyond the
scope of this Handbook to discuss more than a few of these.
Two excellent sources for additional detailed information on
a large array of distributions are Johnson, Kotz, and
Balakrishnan and Evans, Hastings, and Peacock. Equations
for the probability functions are given for the standard form
of the distribution. Formulas exist for defining the functions
with location and scale parameters in terms of the standard
form of the distribution.
The sections on parameter estimation are restricted to the
method of moments and maximum likelihood. This is
because the least squares and PPCC and probability plot
estimation procedures are generic. The maximum likelihood
equations are not listed if they involve solving simultaneous
equations. This is because these methods require
sophisticated computer software to solve. Except where the
maximum likelihood estimates are trivial, you should depend
on a statistical software program to compute them.
References are given for those who are interested.
Be aware that different sources may give formulas that are
different from those shown here. In some cases, these are
simply mathematically equivalent formulations. In other
cases, a different parameterization may be used.
Continuous
Distributions
Normal
Distribution
Uniform
Distribution
Cauchy
Distribution
t Distribution F Distribution Chi-Square
Distribution
Exponential
Distribution
Weibull
Distribution
Lognormal
Distribution
Birnbaum-
Saunders (Fatigue
Gamma
Distribution
Double
Exponential

Life) Distribution Distribution
Power Normal
Distribution
Power Lognormal
Distribution
Tukey-Lambda
Distribution
Extreme Value
Type I
Distribution
Beta Distribution
Discrete
Distributions
Binomial
Distribution
Poisson
Distribution
Normal Distribution
Probability
Density
Function
The general formula for the probability density function of the
normal distribution is
( f(x) = frac{e^{-(x - mu)^{2}/(2sigma^{2}) }}
{sigmasqrt{2pi}} )
where μ is the location parameter and σ is the scale parameter.
The case where μ = 0 and σ = 1 is called the standard normal
distribution. The equation for the standard normal
distribution is
( f(x) = frac{e^{-x^{2}/2}} {sqrt{2pi}} )
Since the general form of probability functions can be
expressed in terms of the standard distribution, all subsequent
formulas in this section are given for the standard form of the
function.
The following is the plot of the standard normal probability
density function.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the
standard normal distribution is

( F(x) = int_{-infty}^{x} frac{e^{-x^{2}/2}} {sqrt{2pi}}
)
Note that this integral does not exist in a simple closed
formula. It is computed numerically.
The following is the plot of the normal cumulative
distribution function.
Percent
Point
Function
The formula for the percent point function of the normal
distribution does not exist in a simple closed formula. It is
computed numerically.
The following is the plot of the normal percent point function.
Hazard
Function
The formula for the hazard function of the normal distribution
is
( h(x) = frac{phi(x)} {Phi(-x)} )
where (Phi) is the cumulative distribution function of the
standard normal distribution and (phi) is the probability
density function of the standard normal distribution.
The following is the plot of the normal hazard function.

Cumulative
Hazard
Function
The normal cumulative hazard function can be computed from
the normal cumulative distribution function.
The following is the plot of the normal cumulative hazard
function.
Survival
Function
The normal survival function can be computed from the
normal cumulative distribution function.
The following is the plot of the normal survival function.
Inverse
Survival
Function
The normal inverse survival function can be computed from
the normal percent point function.
The following is the plot of the normal inverse survival
function.

Common
Statistics
Mean The location parameter μ.
Median The location parameter μ.
Mode The location parameter μ.
Range (-infty) to (infty).
Standard
Deviation
The scale parameter σ.
Coefficient of
Variation
σ/μ
Skewness 0
Kurtosis 3
Parameter
Estimation
The location and scale parameters of the normal distribution
can be estimated with the sample mean and sample standard
Comments For both theoretical and practical reasons, the normal
distribution is probably the most important distribution in
statistics. For example,
Many classical statistical tests are based on the
assumption that the data follow a normal distribution.
This assumption should be tested before applying these
tests.
In modeling applications, such as linear and non-linear
regression, the error term is often assumed to follow a
normal distribution with fixed location and scale.
The normal distribution is used to find significance
levels in many hypothesis tests and confidence
intervals.
Theroretical
Justification
- Central
Limit
Theorem
The normal distribution is widely used. Part of the appeal is
that it is well behaved and mathematically tractable. However,
the central limit theorem provides a theoretical basis for why
it has wide applicability.
The central limit theorem basically states that as the sample
size (N) becomes large, the following occur:
1. The sampling distribution of the mean becomes
approximately normal regardless of the distribution of
the original variable.
2. The sampling distribution of the mean is centered at the
population mean, μ, of the original variable. In addition,
the standard deviation of the sampling distribution of
the mean approaches ( sigma / sqrt{N} ).
Software Most general purpose statistical software programs support at
least some of the probability functions for the normal

distribution.
Uniform Distribution
Probability
Density
Function
uniform distribution is
( f(x) = frac{1} {B - A} ;;;;;;; mbox{for} A le x le B
)
where A is the location parameter and (B - A) is the scale
parameter. The case where A = 0 and B = 1 is called the
standard uniform distribution. The equation for the
standard uniform distribution is
( f(x) = 1 ;;;;;;; mbox{for} 0 le x le 1 )
function.
The following is the plot of the uniform probability density
function.
Cumulative
Distribution
Function
uniform distribution is
( F(x) = x ;;;;;;; mbox{for} 0 le x le 1 )
The following is the plot of the uniform cumulative
Percent
Point
Function
The formula for the percent point function of the uniform
distribution is

( G(p) = p ;;;;;;; mbox{for} 0 le p le 1 )
The following is the plot of the uniform percent point
function.
Hazard
Function
The formula for the hazard function of the uniform
distribution is
( h(x) = frac{1} {1-x} ;;;;;;; mbox{for} 0 le x < 1 )
The following is the plot of the uniform hazard function.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the uniform
distribution is
( H(x) = -ln{(1-x)} ;;;;;;; mbox{for} 0 le x < 1 )
The following is the plot of the uniform cumulative hazard
function.
Survival
Function
The formula for the uniform survival function is

( S(x) = 1 - x ;;;;;;; mbox{for} 0 le x le 1 )
The following is the plot of the uniform survival function.
Inverse
Survival
Function
The formula for the uniform inverse survival function is
( Z(p) = 1 - p ;;;;;;; mbox{for} 0 le p le 1 )
The following is the plot of the uniform inverse survival
function.
Common
Statistics
Mean (A + B)/2
Median (A + B)/2
Range B - A
Standard
Deviation
( sqrt{frac{(B - A)^{2}} {12}} )
Coefficient of
Variation
( frac{(B - A)} {sqrt{3}(B + A)} )
Skewness 0
Kurtosis 9/5
Parameter
Estimation
The method of moments estimators for A and B are
( hat{A} = bar{x} - sqrt{3}s ) ( hat{B} = bar{x} +
sqrt{3}s )
The maximum likelihood estimators are usually given in terms
of the parameters a and h where
A = a - h B = a + h
The maximum likelihood estimators for a and h are
( hat{a} = mbox{midrange}(Y_{1}, Y_{2}, ... ,
Y_{n}) ) ( hat{h} = 0.5[mbox{range}(Y_{1},
Y_{2}, ... , Y_{n})] )

This gives the following maximum likelihood estimators for A
and B
( hat{A} = mbox{midrange}(Y_{1}, Y_{2}, ... ,
Y_{n}) - 0.5[mbox{range}(Y_{1}, Y_{2}, ... ,
Y_{n})] = Y_{1} ) ( hat{B} = mbox{midrange}
(Y_{1}, Y_{2}, ... , Y_{n}) + 0.5[mbox{range}
(Y_{1}, Y_{2}, ... , Y_{n})] = Y_{n} )
Comments The uniform distribution defines equal probability over a
given range for a continuous distribution. For this reason, it is
important as a reference distribution.
One of the most important applications of the uniform
distribution is in the generation of random numbers. That is,
almost all random number generators generate random
numbers on the (0,1) interval. For other distributions, some
transformation is applied to the uniform random numbers.
least some of the probability functions for the uniform
distribution.
Cauchy Distribution
Probability
Density
Function
Cauchy distribution is
( f(x) = frac{1} {spi(1 + ((x - t)/s)^{2})} )
where t is the location parameter and s is the scale parameter.
The case where t = 0 and s = 1 is called the standard Cauchy
distribution. The equation for the standard Cauchy
distribution reduces to
( f(x) = frac{1} {pi(1 + x^{2})} )
function.
The following is the plot of the standard Cauchy probability
density function.
Cumulative
Distribution
Function
The formula for the cumulative distribution function for the
Cauchy distribution is
( F(x) = 0.5 + frac{arctan{(x)}} {pi} )

The following is the plot of the Cauchy cumulative
Percent
Point
Function
The formula for the percent point function of the Cauchy
distribution is
( G(p) = -cot{(pi p)} )
The following is the plot of the Cauchy percent point function.
Hazard
Function
The formula for the Cauchy hazard function is
( h(x) = frac{1} {(1 + x^2)(0.5 pi - arctan{x})} )
The following is the plot of the Cauchy hazard function.
Cumulative
Hazard
Function
The formula for the Cauchy cumulative hazard function is
( H(x) = -ln left( 0.5 - frac{arctan{x}}{pi} right) )
The following is the plot of the Cauchy cumulative hazard
function.

Survival
Function
The formula for the Cauchy survival function is
( S(x) = 0.5 - frac{arctan{(x)}} {pi} )
The following is the plot of the Cauchy survival function.
Inverse
Survival
Function
The formula for the Cauchy inverse survival function is
( Z(p) = -cot{(pi (1 - p))} )
The following is the plot of the Cauchy inverse survival
function.
Common
Statistics
Mean The mean is undefined.
Median The location parameter t.
Mode The location parameter t.
Range ( -infty mbox{ to } infty )
Standard
Deviation
The standard deviation is undefined.
Coefficient of
Variation
The coefficient of variation is undefined.

Skewness The skewness is 0.
Kurtosis The kurtosis is undefined.
Parameter
Estimation
The likelihood functions for the Cauchy maximum likelihood
estimates are given in chapter 16 of Johnson, Kotz, and
Balakrishnan. These equations typically must be solved
numerically on a computer.
Comments The Cauchy distribution is important as an example of a
pathological case. Cauchy distributions look similar to a
normal distribution. However, they have much heavier tails.
When studying hypothesis tests that assume normality, seeing
how the tests perform on data from a Cauchy distribution is a
good indicator of how sensitive the tests are to heavy-tail
departures from normality. Likewise, it is a good check for
robust techniques that are designed to work well under a wide
variety of distributional assumptions.
The mean and standard deviation of the Cauchy distribution
are undefined. The practical meaning of this is that collecting
1,000 data points gives no more accurate an estimate of the
mean and standard deviation than does a single point.
Software Many general purpose statistical software programs support at
least some of the probability functions for the Cauchy
distribution.
t Distribution
Probability
Density
Function
The formula for the probability density function of the t
distribution is
( f(x) = frac{(1 + frac{x^2}{nu})^{frac{-(nu + 1)} {2}}}
{B(0.5,0.5nu)sqrt{nu}} )
where B is the beta function and ν is a positive integer shape
parameter. The formula for the beta function is
( B(alpha,beta) = int_{0}^{1} {t^{alpha-1}(1-t)^{beta-
1}dt} )
In a testing context, the t distribution is treated as a
"standardized distribution" (i.e., no location or scale
parameters). However, in a distributional modeling context (as
with other probability distributions), the t distribution itself
can be transformed with a location parameter, μ, and a scale
parameter, σ.
The following is the plot of the t probability density function
for 4 different values of the shape parameter.

These plots all have a similar shape. The difference is in the
heaviness of the tails. In fact, the t distribution with ν equal to
1 is a Cauchy distribution. The t distribution approaches a
normal distribution as ν becomes large. The approximation is
quite good for values of ν > 30.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the t
distribution is complicated and is not included here. It is given
in the Evans, Hastings, and Peacock book.
The following are the plots of the t cumulative distribution
function with the same values of ν as the pdf plots above.
Percent
Point
Function
The formula for the percent point function of the t distribution
does not exist in a simple closed form. It is computed
numerically.
The following are the plots of the t percent point function with
the same values of ν as the pdf plots above.
Other
Probability
Functions
Since the t distribution is typically used to develop hypothesis
tests and confidence intervals and rarely for modeling
applications, we omit the formulas and plots for the hazard,
cumulative hazard, survival, and inverse survival probability
functions.
Common
Statistics
Mean 0 (It is undefined for ν equal to 1.)
Median 0
Mode 0
Range (-infty mbox{ to } infty)
Standard
Deviation
( sqrt{frac {nu} {(nu - 2)}} )
It is undefined for ν equal to 1 or 2.
Coefficient of
Variation
Undefined
Skewness 0. It is undefined for ν less than or equal to
3. However, the t distribution is symmetric

in all cases.
Kurtosis ( frac {3(nu-2)} {(nu - 4)} )
It is undefined for ν less than or equal to 4.
Parameter
Estimation
Since the t distribution is typically used to develop hypothesis
applications, we omit any discussion of parameter estimation.
Comments The t distribution is used in many cases for the critical regions
for hypothesis tests and in determining confidence intervals.
The most common example is testing if data are consistent
with the assumed process mean.
least some of the probability functions for the t distribution.
F Distribution
Probability
Density
Function
The F distribution is the ratio of two chi-square distributions with
degrees of freedom ν1 and ν2, respectively, where each chi-
square has first been divided by its degrees of freedom. The
formula for the probability density function of the F distribution
is
( f(x) = frac{Gamma(frac{nu_{1} + nu_{2}} {2})
(frac{nu_{1}} {nu_{2}})^{frac{nu_{1}} {2}}
x^{frac{nu_{1}} {2} - 1 }} {Gamma(frac{nu_{1}}
{2}) Gamma(frac{nu_{2}} {2}) (1 + frac{nu_{1}x}
{nu_{2}})^{frac{nu_{1} + nu_{2}} {2}} } )
where ν1 and ν2 are the shape parameters and Γ is the gamma
function. The formula for the gamma function is
( Gamma(a) = int_{0}^{infty} {t^{a-1}e^{-t}dt} )
In a testing context, the F distribution is treated as a
"standardized distribution" (i.e., no location or scale parameters).
However, in a distributional modeling context (as with other
probability distributions), the F distribution itself can be
transformed with a location parameter, μ, and a scale parameter,
σ.
The following is the plot of the F probability density function for
4 different values of the shape parameters.
Cumulative
Distribution
Function
The formula for the Cumulative distribution function of the F
distribution is
( F(x) = 1 - I_{k}(frac{nu_{2}} {2},frac{nu_{1}} {2}
) ) where k = ( nu_2/(nu_2 + nu_1 cdot x) ) and Ik is

the incomplete beta function. The formula for the
incomplete beta function is
( I_{k}(x,alpha,beta) = frac{int_{0}^{x}
{t^{alpha-1}(1-t)^{beta-1}dt}} {B(alpha,beta) }
)
where B is the beta function
( B(alpha,beta) = int_{0}^{1} {t^{alpha-1}(1-
t)^{beta-1}dt} )
The following is the plot of the F cumulative distribution
function with the same values of ν1 and ν2 as the pdf plots
above.
Percent
Point
Function
The formula for the percent point function of the F distribution
does not exist in a simple closed form. It is computed
numerically.
The following is the plot of the F percent point function with the
same values of ν1 and ν2 as the pdf plots above.
Other
Probability
Functions
Since the F distribution is typically used to develop hypothesis
functions.
Common
Statistics
The formulas below are for the case where the location
parameter is zero and the scale parameter is one.
Mean ( frac {nu_{2}} {(nu_{2} - 2)} ;;;;;;;
nu_{2} > 2 )
Mode ( frac {nu_{2}(nu_{1} - 2)} {nu_{1}
(nu_{2} + 2)} ;;;;;;; nu_{1} > 2 )
Range 0 to (infty)

Standard
Deviation
( sqrt{frac {2nu_{2}^{2}(nu_{1} +
nu_{2} - 2)} {nu_{1}(nu_{2} - 2)^{2}
(nu_{2} - 4)}} ;;;;;;; nu_{2} > 4 )
Coefficient of
Variation
( sqrt{frac{2(nu_{1} + nu_{2} - 2)}
{nu_{1}(nu_{2} - 4)}} ;;;;;;; nu_{2}
> 4 )
Skewness ( frac {(2nu_{1} + nu_{2} -
2)sqrt{8(nu_{2} - 4)}} {sqrt{nu_{1}}
(nu_{2} - 6)sqrt{(nu_{1} + nu_{2} -
2)}} ;;;;;;; nu_{2} > 6 )
Parameter
Estimation
Since the F distribution is typically used to develop hypothesis
applications, we omit any discussion of parameter estimation.
Comments The F distribution is used in many cases for the critical regions
for hypothesis tests and in determining confidence intervals. Two
common examples are the analysis of variance and the F test to
determine if the variances of two populations are equal.
least some of the probability functions for the F distribution.
Chi-Square Distribution
Probability
Density
Function
The chi-square distribution results when ν independent
variables with standard normal distributions are squared and
summed. The formula for the probability density function of
the chi-square distribution is
( f(x) = frac{e^{frac{-x} {2}}x^{frac{nu} {2} - 1}}
{2^{frac{nu} {2}}Gamma(frac{nu} {2}) } ;;;;;;;
mbox{for} ; x ge 0 )
where ν is the shape parameter and Γ is the gamma function.
The formula for the gamma function is
In a testing context, the chi-square distribution is treated as a
"standardized distribution" (i.e., no location or scale
parameters). However, in a distributional modeling context (as
with other probability distributions), the chi-square
distribution itself can be transformed with a location
parameter, μ, and a scale parameter, σ.
The following is the plot of the chi-square probability density
function for 4 different values of the shape parameter.
Cumulative
Distribution
chi-square distribution is

Function ( F(x) = frac{gamma(frac{nu} {2},frac{x} {2})}
{Gamma(frac{nu} {2})} ;;;;;;; mbox{for} ; x ge 0 )
where Γ is the gamma function defined above and γ is the
incomplete gamma function. The formula for the incomplete
gamma function is
( Gamma_{x}(a) = int_{0}^{x} {t^{a-1}e^{-t}dt} )
The following is the plot of the chi-square cumulative
distribution function with the same values of ν as the pdf plots
above.
Percent
Point
Function
The formula for the percent point function of the chi-square
distribution does not exist in a simple closed form. It is
The following is the plot of the chi-square percent point
function with the same values of ν as the pdf plots above.
Other
Probability
Functions
Since the chi-square distribution is typically used to develop
hypothesis tests and confidence intervals and rarely for
modeling applications, we omit the formulas and plots for the
hazard, cumulative hazard, survival, and inverse survival
probability functions.
Common
Statistics
Mean ν
Median approximately ν - 2/3 for large ν
Mode ( nu - 2 ;;;;;;; mbox{for} ; nu > 2
)
Range 0 to (infty)
Standard
Deviation
( sqrt{2nu} )
Coefficient of
Variation
( sqrt{frac{2} {nu}} )
Skewness ( frac{2^{1.5} } {sqrt{nu}} )

Kurtosis ( 3 + frac{12} {nu} )
Parameter
Estimation
Since the chi-square distribution is typically used to develop
hypothesis tests and confidence intervals and rarely for
modeling applications, we omit any discussion of parameter
estimation.
Comments The chi-square distribution is used in many cases for the
critical regions for hypothesis tests and in determining
confidence intervals. Two common examples are the chi-
square test for independence in an RxC contingency table and
the chi-square test to determine if the standard deviation of a
population is equal to a pre-specified value.
least some of the probability functions for the chi-square
distribution.
Exponential Distribution
Probability
Density
Function
exponential distribution is
( f(x) = frac{1} {beta} e^{-(x - mu)/beta} hspace{.3in} x
ge mu; beta > 0 )
where μ is the location parameter and β is the scale parameter
(the scale parameter is often referred to as λ which equals 1/β).
The case where μ = 0 and β = 1 is called the standard
exponential distribution. The equation for the standard
( f(x) = e^{-x} ;;;;;;; mbox{for} ; x ge 0 )
The general form of probability functions can be expressed in
terms of the standard distribution. Subsequent formulas in this
section are given for the 1-parameter (i.e., with scale
parameter) form of the function.
The following is the plot of the exponential probability
density function.
Cumulative
Distribution
Function
( F(x) = 1 - e^{-x/beta} hspace{.3in} x ge 0; beta > 0 )
The following is the plot of the exponential cumulative

Percent
Point
Function
The formula for the percent point function of the exponential
distribution is
( G(p) = -betaln(1 - p) hspace{.3in} 0 le p < 1; beta > 0 )
The following is the plot of the exponential percent point
function.
Hazard
Function
The formula for the hazard function of the exponential
distribution is
( h(x) = frac{1} {beta} hspace{.3in} x ge 0; beta > 0 )
The following is the plot of the exponential hazard function.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the
( H(x) = frac{x} {beta} hspace{.3in} x ge 0; beta > 0 )
The following is the plot of the exponential cumulative hazard
function.

Survival
Function
The formula for the survival function of the exponential
distribution is
( S(x) = e^{-x/beta} hspace{.3in} x ge 0; beta > 0 )
The following is the plot of the exponential survival function.
Inverse
Survival
Function
The formula for the inverse survival function of the
( Z(p) = -betaln(p) hspace{.3in} 0 le p < 1; beta > 0 )
The following is the plot of the exponential inverse survival
function.
Common
Statistics
Mean β
Median ( betaln{2} )
Mode μ
Range μ to (infty)
Standard
Deviation
β

Coefficient of
Variation
1
Skewness 2
Kurtosis 9
Parameter
Estimation
For the full sample case, the maximum likelihood estimator of
the scale parameter is the sample mean. Maximum likelihood
estimation for the exponential distribution is discussed in the
chapter on reliability (Chapter 8). It is also discussed in
chapter 19 of Johnson, Kotz, and Balakrishnan.
Comments The exponential distribution is primarily used in reliability
applications. The exponential distribution is used to model
data with a constant failure rate (indicated by the hazard plot
which is simply equal to a constant).
least some of the probability functions for the exponential
distribution.
Weibull Distribution
Probability
Density
Function
The formula for the probability density function of the general
Weibull distribution is
( f(x) = frac{gamma} {alpha} (frac{x-mu}
{alpha})^{(gamma - 1)}exp{(-((x-
mu)/alpha)^{gamma})} hspace{.3in} x ge mu; gamma,
alpha > 0 )
where γ is the shape parameter, μ is the location parameter and
α is the scale parameter. The case where μ = 0 and α = 1 is
called the standard Weibull distribution. The case where μ =
0 is called the 2-parameter Weibull distribution. The equation
for the standard Weibull distribution reduces to
( f(x) = gamma x^{(gamma - 1)}exp(-(x^{gamma}))
hspace{.3in} x ge 0; gamma > 0 )
function.
The following is the plot of the Weibull probability density
function.
Cumulative
Distribution
Function
Weibull distribution is

( F(x) = 1 - e^{-(x^{gamma})} hspace{.3in} x ge 0;
gamma > 0 )
The following is the plot of the Weibull cumulative
distribution function with the same values of γ as the pdf plots
above.
Percent
Point
Function
The formula for the percent point function of the Weibull
distribution is
( G(p) = (-ln(1 - p))^{1/gamma} hspace{.3in} 0 le p < 1;
gamma > 0 )
The following is the plot of the Weibull percent point function
with the same values of γ as the pdf plots above.
Hazard
Function
The formula for the hazard function of the Weibull
distribution is
( h(x) = gamma x^{(gamma - 1)} hspace{.3in} x ge 0;
gamma > 0 )
The following is the plot of the Weibull hazard function with
the same values of γ as the pdf plots above.

Cumulative
Hazard
Function
The formula for the cumulative hazard function of the Weibull
distribution is
( H(x) = x^{gamma} hspace{.3in} x ge 0; gamma > 0 )
The following is the plot of the Weibull cumulative hazard
function with the same values of γ as the pdf plots above.
Survival
Function
The formula for the survival function of the Weibull
distribution is
( S(x) = exp{-(x^{gamma})} hspace{.3in} x ge 0; gamma
> 0 )
The following is the plot of the Weibull survival function with
Inverse
Survival
Function
The formula for the inverse survival function of the Weibull
distribution is
( Z(p) = (-ln(p))^{1/gamma} hspace{.3in} 0 le p < 1;
gamma > 0 )

The following is the plot of the Weibull inverse survival
Common
Statistics
The formulas below are with the location parameter equal to
zero and the scale parameter equal to one.
Mean ( Gamma(frac{gamma + 1} {gamma})
)
where Γ is the gamma function
( Gamma(a) = int_{0}^{infty} {t^{a-
1}e^{-t}dt} )
Median ( ln(2)^{1/gamma} )
Mode ( (1 - frac{1} {gamma})^{1/gamma}
hspace{.2in} gamma > 1 )
( 0 hspace{1.05in} gamma le 1 )
Range 0 to (infty).
Standard
Deviation
( sqrt{Gamma(frac{gamma + 2}
{gamma}) - (Gamma(frac{gamma + 1}
{gamma}))^{2}} )
Coefficient of
Variation
( sqrt{frac{Gamma(frac{gamma + 2}
{gamma})} {(Gamma(frac{gamma +
1} {gamma}))^{2}} - 1} )
Parameter
Estimation
Maximum likelihood estimation for the Weibull distribution is
discussed in the Reliability chapter (Chapter 8). It is also
discussed in Chapter 21 of Johnson, Kotz, and Balakrishnan.
Comments The Weibull distribution is used extensively in reliability
applications to model failure times.
least some of the probability functions for the Weibull
distribution.
Probability
Density
Function
A variable X is lognormally distributed if (Y = ln(X)) is
normally distributed with "LN" denoting the natural
logarithm. The general formula for the probability density
function of the lognormal distribution is
( f(x) = frac{e^{-((ln((x-theta)/m))^{2}/(2sigma^{2}))}}
{(x-theta)sigmasqrt{2pi}} hspace{.2in} x > theta; m,
sigma > 0 )
where σ is the shape parameter (and is the standard deviation
of the log of the distribution), θ is the location parameter and
m is the scale parameter (and is also the median of the
distribution). If x = θ, then f(x) = 0. The case where θ = 0 and
m = 1 is called the standard lognormal distribution. The

case where θ equals zero is called the 2-parameter lognormal
distribution.
The equation for the standard lognormal distribution is
( f(x) = frac{e^{-((ln x)^{2}/2sigma^{2})}}
{xsigmasqrt{2pi}} hspace{.2in} x > 0; sigma > 0 )
function.
Note that the lognormal distribution is commonly
parameterized with
( mu = log(m) )
The μ parameter is the mean of the log of the distribution. If
the μ parameterization is used, the lognormal pdf is
( f(x) = frac{e^{-(ln(x - theta) - mu)^2/(2sigma^2)}} {(x -
theta)sigmasqrt{2pi}} hspace{.2in} x > theta; sigma > 0
)
We prefer to use the m parameterization since m is an explicit
scale parameter.
The following is the plot of the lognormal probability density
function for four values of σ.
There are several common parameterizations of the lognormal
distribution. The form given here is from Evans, Hastings, and
Peacock.
Cumulative
Distribution
Function
lognormal distribution is
( F(x) = Phi(frac{ln(x)} {sigma}) hspace{.2in} x ge 0;
sigma > 0 )
The following is the plot of the lognormal cumulative
distribution function with the same values of σ as the pdf plots
above.

Percent
Point
Function
The formula for the percent point function of the lognormal
distribution is
( G(p) = exp(sigmaPhi^{-1}(p)) hspace{.2in} 0 le p < 1;
sigma > 0 )
where (Phi^{-1}) is the percent point function of the normal
distribution.
The following is the plot of the lognormal percent point
function with the same values of σ as the pdf plots above.
Hazard
Function
The formula for the hazard function of the lognormal
distribution is
( h(x,sigma) = frac{(frac{1} {xsigma})phi(frac{ln x}
{sigma})} {Phi(frac{-ln x} {sigma})} hspace{.2in} x > 0;
sigma > 0 )
where (phi) is the probability density function of the normal
distribution and (Phi) is the cumulative distribution function
of the normal distribution.
The following is the plot of the lognormal hazard function
with the same values of σ as the pdf plots above.

Cumulative
Hazard
Function
The formula for the cumulative hazard function of the
( H(x) = -ln(1 - Phi(frac{ln(x)} {sigma})) hspace{.2in} x
ge 0; sigma > 0 )
The following is the plot of the lognormal cumulative hazard
Survival
Function
The formula for the survival function of the lognormal
distribution is
( S(x) = 1 - Phi(frac{ln(x)} {sigma}) hspace{.2in} x ge
0; sigma > 0 )
The following is the plot of the lognormal survival function
with the same values of σ as the pdf plots above.

Inverse
Survival
Function
The formula for the inverse survival function of the lognormal
distribution is
( Z(p) = exp(sigmaPhi^{-1}(1-p)) hspace{.2in} 0 le p < 1;
sigma > 0 )
where (Phi^{-1}) is the percent point function of the normal
distribution.
The following is the plot of the lognormal inverse survival
Common
Statistics
Mean ( e^{0.5sigma^{2}} )
Median Scale parameter m (= 1 if scale parameter
not specified).
Mode ( frac{1} {e^{sigma^{2}}} )
Range 0 to (infty)
Standard
Deviation
( sqrt{e^{sigma^{2}} (e^{sigma^{2}}
- 1)} )
Skewness ( (e^{sigma^{2}}+2)
sqrt{e^{sigma^{2}} - 1} )
Kurtosis ( (e^{sigma^{2}})^{4} +
2(e^{sigma^{2}})^{3} +
3(e^{sigma^{2}})^{2} - 3 )
Coefficient of
Variation
( sqrt{e^{sigma^{2}} - 1} )
Parameter
Estimation
The maximum likelihood estimates for the scale parameter, m,
and the shape parameter, σ, are
( hat{m} = exp{hat{mu}} )
and

( hat{sigma} = sqrt{frac{sum_{i=1}^{N}
{(ln{(X_{i})} - hat{mu})^{2}}} {N}} )
where
( hat{mu}= frac{sum_{i=1}^{N}{ln{X_i}}} {N}
)
If the location parameter is known, it can be subtracted from
the original data points before computing the maximum
likelihood estimates of the shape and scale parameters.
Comments The lognormal distribution is used extensively in reliability
applications to model failure times. The lognormal and
Weibull distributions are probably the most commonly used
distributions in reliability applications.
least some of the probability functions for the lognormal
distribution.
Birnbaum-Saunders (Fatigue Life) Distribution
Probability
Density
Function
The Birnbaum-Saunders distribution is also commonly known
as the fatigue life distribution. There are several alternative
formulations of the Birnbaum-Saunders distribution in the
literature.
Birnbaum-Saunders distribution is
( f(x) = left (frac{sqrt{frac{x-mu} {beta}} +
sqrt{frac{beta} {x-mu}}} {2gamma (x-mu)} right) phi
left( frac{sqrt{frac{x-mu} {beta}} - sqrt{frac{beta} {x-
mu}}} {gamma} right) hspace{.2in} x > mu; gamma,
beta > 0 )
where γ is the shape parameter, μ is the location parameter, β
is the scale parameter, (phi) is the probability density
function of the standard normal distribution, and (Phi) is the
cumulative distribution function of the standard normal
distribution. The case where μ = 0 and β = 1 is called the
standard Birnbaum-Saunders distribution. The equation
for the standard Birnbaum-Saunders distribution reduces to
( f(x) = left (frac{sqrt{x} + sqrt{frac{1} {x}}} {2gamma
x} right) phi left (frac{sqrt{x} - sqrt{frac{1} {x}}}
{gamma} right) hspace{.2in} x > 0; gamma > 0 )
function.
The following is the plot of the Birnbaum-Saunders
probability density function.

Cumulative
Distribution
Function
Birnbaum-Saunders distribution is
( F(x) = Phi(frac{sqrt{x} - sqrt{frac{1} {x}}} {gamma})
hspace{.2in} x > 0; gamma > 0 )
standard normal distribution. The following is the plot of the
Birnbaum-Saunders cumulative distribution function with the
same values of γ as the pdf plots above.
Percent
Point
Function
The formula for the percent point function of the Birnbaum-
Saunders distribution is
( G(p) = frac{1} {4} left[gamma Phi^{-1}(p) + sqrt{4 +
(gamma Phi^{-1}(p))^{2}}right]^{2} )
where (Phi^{-1}) is the percent point function of the
standard normal distribution. The following is the plot of the
Birnbaum-Saunders percent point function with the same
values of γ as the pdf plots above.

Hazard
Function
The Birnbaum-Saunders hazard function can be computed
from the Birnbaum-Saunders probability density and
cumulative distribution functions.
The following is the plot of the Birnbaum-Saunders hazard
Cumulative
Hazard
Function
The Birnbaum-Saunders cumulative hazard function can be
computed from the Birnbaum-Saunders cumulative
The following is the plot of the Birnbaum-Saunders
cumulative hazard function with the same values of γ as the
pdf plots above.
Survival
Function
The Birnbaum-Saunders survival function can be computed
from the Birnbaum-Saunders cumulative distribution function.
The following is the plot of the Birnbaum-Saunders survival

Inverse
Survival
Function
The Birnbaum-Saunders inverse survival function can be
computed from the Birnbaum-Saunders percent point
function.
The following is the plot of the gamma inverse survival
Common
Statistics
Mean ( 1 + frac{gamma^{2}} {2} )
Range 0 to (infty).
Standard
Deviation
( gammasqrt{1 + frac{5gamma^{2}}
{4}} )
Coefficient of
Variation
( frac{2 + gamma^{2}} {gammasqrt{1
+ 5gamma^{2}}} )
Parameter
Estimation
Maximum likelihood estimation for the Birnbaum-Saunders
distribution is discussed in the Reliability chapter.
Comments The Birnbaum-Saunders distribution is used extensively in
reliability applications to model failure times.
Software Some general purpose statistical software programs, including
Dataplot, support at least some of the probability functions for
the Birnbaum-Saunders distribution. Support for this
distribution is likely to be available for statistical programs
that emphasize reliability applications.
The "bs" package implements support for the Birnbaum-
Saunders distribution for the R package. See
Leiva, V., Hernandez, H., and Riquelme, M. (2006). A
New Package for the Birnbaum-Saunders Distribution.
Rnews, 6/4, 35-40. (http://www.r-project.org)
Gamma Distribution
Probability
Density
Function
gamma distribution is
( f(x) = frac{(frac{x-mu}{beta})^{gamma - 1}exp{(-
frac{x-mu} {beta}})} {betaGamma(gamma)}
hspace{.2in} x ge mu; gamma, beta > 0 )
where γ is the shape parameter, μ is the location parameter, β
is the scale parameter, and Γ is the gamma function which has
the formula

The case where μ = 0 and β = 1 is called the standard gamma
distribution. The equation for the standard gamma
distribution reduces to
( f(x) = frac{x^{gamma - 1}e^{-x}} {Gamma(gamma)}
function.
The following is the plot of the gamma probability density
function.
Cumulative
Distribution
Function
gamma distribution is
( F(x) = frac{Gamma_{x}(gamma)} {Gamma(gamma)}
where Γ is the gamma function defined above and
(Gamma_{x}(a)) is the incomplete gamma function. The
incomplete gamma function has the formula
( Gamma_{x}(a) = int_{0}^{x} {t^{a-1}e^{-t}dt} )
The following is the plot of the gamma cumulative
distribution function with the same values of γ as the pdf plots
above.
Percent
Point
Function
The formula for the percent point function of the gamma
The following is the plot of the gamma percent point function
with the same values of γ as the pdf plots above.

Hazard
Function
The formula for the hazard function of the gamma distribution
is
( h(x) = frac{x^{gamma - 1}e^{-x}} {Gamma(gamma) -
Gamma_{x}(gamma)} hspace{.2in} x ge 0; gamma > 0 )
The following is the plot of the gamma hazard function with
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the gamma
distribution is
( H(x) = -log{(1 - frac{Gamma_{x}(gamma)}
{Gamma(gamma)})} hspace{.2in} x ge 0; gamma > 0 )
(Gamma_{x}(a)) is the incomplete gamma function defined
above.
The following is the plot of the gamma cumulative hazard

Survival
Function
The formula for the survival function of the gamma
distribution is
( S(x) = 1 - frac{Gamma_{x}(gamma)}
{Gamma(gamma)} hspace{.2in} x ge 0; gamma > 0 )
(Gamma_{x}(a)) is the incomplete gamma function defined
above.
The following is the plot of the gamma survival function with
Inverse
Survival
Function
The gamma inverse survival function does not exist in simple
closed form. It is computed numberically.
The following is the plot of the gamma inverse survival
Common
Statistics
Mean γ
Mode γ- 1 γ ≥ 1
Range 0 to (infty).
Standard
Deviation
( sqrt{gamma} )
Skewness ( frac{2} {sqrt{gamma}} )
Kurtosis ( 3 + frac{6} {gamma} )
Coefficient of
Variation
( frac{1} {sqrt{gamma}} )
Parameter
Estimation
The method of moments estimators of the 2-parameter gamma
distribution are
( hat{gamma} = (frac{bar{x}} {s})^{2} )

( hat{beta} = frac{s^{2}} {bar{x}} )
where (bar{x}) and s are the sample mean and standard
The maximum likelihood estimates for the 2-parameter
gamma distribution are the solutions of the following
simultaneous equations
( hat{beta} - frac{bar{x}}{hat{gamma}} = 0 )
( log{hat{gamma}} - psi(hat{gamma}) - log left(
frac{bar{x}} { left( prod_{i=1}^{n}{x_i} right) ^{1/n} }
right) = 0 )
with ψ denoting the digamma function. These equations need
to be solved numerically; this is typically accomplished by
using statistical software packages.
Software Some general purpose statistical software programs support at
least some of the probability functions for the gamma
distribution.
Double Exponential Distribution
Probability
Density
Function
double exponential distribution is
( f(x) = frac{e^{-left| frac{x-mu}{beta} right| }} {2beta}
)
where μ is the location parameter and β is the scale parameter.
The case where μ = 0 and β = 1 is called the standard double
exponential distribution. The equation for the standard
( f(x) = frac{e^{-|x|}} {2} )
function.
Note that the double exponential distribution is also
commonly referred to as the Laplace distribution.
The following is the plot of the double exponential probability
density function.
Cumulative
Distribution
Function

( F(x) = begin{array}{ll} frac{e^{x}} {2} & mbox{for $x
< 0$} 1 - frac{e^{-x}} {2} & mbox{for $x ge 0$}
end{array} )
The following is the plot of the double exponential cumulative
Percent
Point
Function
The formula for the percent point function of the double
( G(P) = begin{array}{ll} log(2p) & mbox{for $p le 0.5$}
-log(2(1 - p)) & mbox{for $p > 0.5$} end{array} )
The following is the plot of the double exponential percent
point function.
Hazard
Function
The formula for the hazard function of the double exponential
distribution is
( h(x) = begin{array}{ll} frac{e^{x}} {2 - e^{x}} &
mbox{for $x < 0$} 1 & mbox{for $x ge 0$} end{array}
)
The following is the plot of the double exponential hazard
function.

Cumulative
Hazard
Function
The formula for the cumulative hazard function of the double
( H(x) = begin{array}{ll} -log{(1 - frac{e^{x}} {2})} &
mbox{for $x < 0$} x + log{(2)} & mbox{for $x ge 0$}
end{array} )
The following is the plot of the double exponential cumulative
hazard function.
Survival
Function
The formula for the survival function of the double
( S(x) = begin{array}{ll} 1 - frac{e^{x}} {2} & mbox{for
$x < 0$} frac{e^{-x}} {2} & mbox{for $x ge 0$}
end{array} )
The following is the plot of the double exponential survival
function.
Inverse
Survival
The formula for the inverse survival function of the double

Function ( Z(P) = begin{array}{ll} log(2(1-p)) & mbox{for $p le
0.5$} -log(2p) & mbox{for $p > 0.5$} end{array} )
The following is the plot of the double exponential inverse
survival function.
Common
Statistics
Mean μ
Median μ
Mode μ
Standard
Deviation
( sqrt{2}beta )
Skewness 0
Kurtosis 6
Coefficient of
Variation
( sqrt{2}(frac{beta} {mu}) )
Parameter
Estimation
The maximum likelihood estimators of the location and scale
parameters of the double exponential distribution are
( hat{mu} = tilde{X} )
( hat{beta} = frac{sum_{i=1}^{N}|X_{i} - tilde{X}|}
{N} )
where (tilde{X}) is the sample median.
least some of the probability functions for the double
exponential distribution.
Power Normal Distribution
Probability
Density
Function
The formula for the probability density function of the
standard form of the power normal distribution is
( f(x;p) = p phi(x) (Phi(-x))^{p-1} hspace{.3in} x, p > 0 )
where p is the shape parameter (also referred to as the power
parameter), (Phi) is the cumulative distribution function of
the standard normal distribution, and (phi) is the probability
As with other probability distributions, the power normal
distribution can be transformed with a location parameter, μ,
and a scale parameter, σ. We omit the equation for the general
form of the power normal distribution. Since the general form
of probability functions can be expressed in terms of the
standard distribution, all subsequent formulas in this section
are given for the standard form of the function.

The following is the plot of the power normal probability
density function with four values of p.
Cumulative
Distribution
Function
power normal distribution is
( F(x;p) = 1 - (Phi(-x))^{p} hspace{.3in} x, p > 0 )
standard normal distribution.
The following is the plot of the power normal cumulative
distribution function with the same values of p as the pdf plots
above.
Percent
Point
Function
The formula for the percent point function of the power
( G(f) = Phi^{-1}(1 - (1 - f)^{1/p}) hspace{.3in} 0 < f < 1; p
> 0 )
where (phi^{-1}) is the percent point function of the
The following is the plot of the power normal percent point
function with the same values of p as the pdf plots above.

Hazard
Function
The formula for the hazard function of the power normal
distribution is
( h(x;p) = frac{pphi(x)} {Phi(-x)} hspace{.3in} x, p > 0 )
The following is the plot of the power normal hazard function
with the same values of p as the pdf plots above.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the power
( H(x,p) = -log{((Phi(-x))^{p})} hspace{.3in} x, p > 0 )
The following is the plot of the power normal cumulative
hazard function with the same values of p as the pdf plots
above.
Survival
Function
The formula for the survival function of the power normal
distribution is
( S(x;p) = (Phi(-x))^{p} hspace{.3in} x, p > 0 )

The following is the plot of the power normal survival
Inverse
Survival
Function
The formula for the inverse survival function of the power
( Z(f) = Phi^{-1}(1 - f^{1/p}) hspace{.3in} 0 < f < 1; p > 0
)
The following is the plot of the power normal inverse survival
Common
Statistics
The statistics for the power normal distribution are
complicated and require tables. Nelson discusses the mean,
median, mode, and standard deviation of the power normal
distribution and provides references to the appropriate tables.
support the probability functions for the power normal
distribution.
Power Lognormal Distribution
Probability
Density
Function
The formula for the probability density function of the
standard form of the power lognormal distribution is
( f(x;p,sigma) = (frac{p} {xsigma})phi(frac{log x}
{sigma}) (Phi(frac{-log x} {sigma}))^{p-1} hspace{.2in}
x, p, sigma > 0 )
where p (also referred to as the power parameter) and σ are the
shape parameters, (Phi) is the cumulative distribution
function of the standard normal distribution, and (phi) is the
probability density function of the standard normal
distribution.

As with other probability distributions, the power lognormal
and a scale parameter, B. We omit the equation for the general
form of the power lognormal distribution. Since the general
form of probability functions can be expressed in terms of the
standard distribution, all subsequent formulas in this section
are given for the standard form of the function.
The following is the plot of the power lognormal probability
density function with four values of p and σ set to 1.
Cumulative
Distribution
Function
power lognormal distribution is
( F(x;p,sigma) = 1 - (Phi(frac{-log x} {sigma}))^{p}
hspace{.2in} x, p, sigma > 0 )
The following is the plot of the power lognormal cumulative
above.
Percent
Point
Function
The formula for the percent point function of the power
( G(f;p,sigma) = exp{(Phi^{-1}(1 - (1 - f)^{1/p})sigma)}
hspace{.2in} 0 < p < 1; p, sigma > 0 )
where (phi^{-1}) is the percent point function of the
The following is the plot of the power lognormal percent point

Hazard
Function
The formula for the hazard function of the power lognormal
distribution is
( h(x,p,sigma) = frac{p(frac{1} {xsigma})phi(frac{log
x} {sigma})} {Phi(frac{-log x} {sigma})} hspace{.2in}
x, p, sigma > 0 )
standard normal distribution, and (phi) is the probability
Note that this is simply a multiple (p) of the lognormal hazard
function.
The following is the plot of the power lognormal hazard
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the power
( H(x;p,sigma) = -log{((Phi(frac{-log x}
{sigma}))^{p})} hspace{.2in} x, p, sigma > 0 )
The following is the plot of the power lognormal cumulative
hazard function with the same values of p as the pdf plots
above.

Survival
Function
The formula for the survival function of the power lognormal
distribution is
( S(x;p,sigma) = (Phi(frac{-log x} {sigma}))^{p}
hspace{.2in} x, p, sigma > 0 )
The following is the plot of the power lognormal survival
Inverse
Survival
Function
The formula for the inverse survival function of the power
( Z(f;p,sigma) = exp{(Phi^{-1}(1 - f^{1/p})sigma)}
hspace{.2in} 0 < p < 1; p, sigma > 0 )
The following is the plot of the power lognormal inverse
survival function with the same values of p as the pdf plots
above.
Common
Statistics
The statistics for the power lognormal distribution are
complicated and require tables. Nelson discusses the mean,

median, mode, and standard deviation of the power lognormal
distribution and provides references to the appropriate tables.
Parameter
Estimation
Nelson discusses maximum likelihood estimation for the
power lognormal distribution. These estimates need to be
performed with computer software. Software for maximum
likelihood estimation of the parameters of the power
lognormal distribution is not as readily available as for other
reliability distributions such as the exponential, Weibull, and
lognormal.
support the probability functions for the power lognormal
distribution.
Tukey-Lambda Distribution
Probability
Density
Function
The Tukey-Lambda density function does not have a simple,
closed form. It is computed numerically.
The Tukey-Lambda distribution has the shape parameter λ. As
with other probability distributions, the Tukey-Lambda
and a scale parameter, σ. Since the general form of probability
functions can be expressed in terms of the standard
distribution, all subsequent formulas in this section are given
for the standard form of the function.
The following is the plot of the Tukey-Lambda probability
density function for four values of λ.
Cumulative
Distribution
Function
The Tukey-Lambda distribution does not have a simple,
closed form. It is computed numerically.
The following is the plot of the Tukey-Lambda cumulative
distribution function with the same values of λ as the pdf plots
above.

Percent
Point
Function
The formula for the percent point function of the standard
form of the Tukey-Lambda distribution is
( G(p;lambda) = frac{p^{lambda} - (1 - p)^{lambda}}
{lambda} )
The following is the plot of the Tukey-Lambda percent point
function with the same values of λ as the pdf plots above.
Other
Probability
Functions
The Tukey-Lambda distribution is typically used to identify an
appropriate distribution (see the comments below) and not
used in statistical models directly. For this reason, we omit the
formulas, and plots for the hazard, cumulative hazard,
survival, and inverse survival functions. We also omit the
common statistics and parameter estimation sections.
Comments The Tukey-Lambda distribution is actually a family of
distributions that can approximate a number of common
distributions. For example,
λ = -1 approximately Cauchy
λ = 0 exactly logistic
λ = 0.14 approximately normal
λ = 0.5 U-shaped
λ = 1 exactly uniform (from -1 to +1)
The most common use of this distribution is to generate a
Tukey-Lambda PPCC plot of a data set. Based on the ppcc
plot, an appropriate model for the data is suggested. For
example, if the maximum correlation occurs for a value of λ at
or near 0.14, then the data can be modeled with a normal
distribution. Values of λ less than this imply a heavy-tailed
distribution (with -1 approximating a Cauchy). That is, as the
optimal value of λ goes from 0.14 to -1, increasingly heavy
tails are implied. Similarly, as the optimal value of λ becomes
greater than 0.14, shorter tails are implied.

As the Tukey-Lambda distribution is a symmetric distribution,
the use of the Tukey-Lambda PPCC plot to determine a
reasonable distribution to model the data only applies to
symmetric distributions. A histogram of the data should
provide evidence as to whether the data can be reasonably
modeled with a symmetric distribution.
support the probability functions for the Tukey-Lambda
distribution.
Extreme Value Type I Distribution
Probability
Density
Function
The extreme value type I distribution has two forms. One is
based on the smallest extreme and the other is based on the
largest extreme. We call these the minimum and maximum
cases, respectively. Formulas and plots for both cases are
given. The extreme value type I distribution is also referred to
as the Gumbel distribution.
Gumbel (minimum) distribution is
( f(x) = frac{1} {beta} e^{frac{x-mu}{beta}}e^{-
e^{frac{x-mu} {beta}}} )
Gumbel distribution. The equation for the standard Gumbel
distribution (minimum) reduces to
( f(x) = e^{x}e^{-e^{x}} )
The following is the plot of the Gumbel probability density
function for the minimum case.
Gumbel (maximum) distribution is
( f(x) = frac{1}{beta} e^{-frac{x-mu}{beta}}e^{-e^{-
frac{x-mu} {beta}}} )
Gumbel distribution. The equation for the standard Gumbel
distribution (maximum) reduces to
( f(x) = e^{-x}e^{-e^{-x}} )
The following is the plot of the Gumbel probability density
function for the maximum case.

function.
Cumulative
Distribution
Function
Gumbel distribution (minimum) is
( F(x) = 1 - e^{-e^{x}} )
The following is the plot of the Gumbel cumulative
distribution function for the minimum case.
Gumbel distribution (maximum) is
( F(x) = e^{-e^{-x}} )
The following is the plot of the Gumbel cumulative
distribution function for the maximum case.

Percent
Point
Function
The formula for the percent point function of the Gumbel
distribution (minimum) is
( G(p) = ln(ln(frac{1} {1 - p})) )
The following is the plot of the Gumbel percent point function
for the minimum case.
The formula for the percent point function of the Gumbel
distribution (maximum) is
( G(p) = -ln(ln(frac{1} {p})) )
The following is the plot of the Gumbel percent point function
for the maximum case.
Hazard
Function
The formula for the hazard function of the Gumbel
( h(x) = e^{x} )
The following is the plot of the Gumbel hazard function for
the minimum case.

The formula for the hazard function of the Gumbel
( h(x) = frac{e^{-x}} {e^{e^{-x}} - 1} )
The following is the plot of the Gumbel hazard function for
the maximum case.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the Gumbel
( H(x) = e^{x} )
The following is the plot of the Gumbel cumulative hazard
The formula for the cumulative hazard function of the Gumbel
( H(x) = -ln(1 - e^{-e^{-x}}) )
The following is the plot of the Gumbel cumulative hazard

Survival
Function
The formula for the survival function of the Gumbel
( S(x) = e^{-e^{x}} )
The following is the plot of the Gumbel survival function for
the minimum case.
The formula for the survival function of the Gumbel
( S(x) = 1 - e^{-e^{-x}} )
The following is the plot of the Gumbel survival function for
the maximum case.
Inverse
Survival
Function
The formula for the inverse survival function of the Gumbel
( Z(p) = ln(ln(frac{1} {p})) )

The following is the plot of the Gumbel inverse survival
The formula for the inverse survival function of the Gumbel
( Z(p) = -ln(ln(frac{1} {1-p})) )
The following is the plot of the Gumbel inverse survival
Common
Statistics
The formulas below are for the maximum order statistic case.
Mean ( mu + 0.5772beta )
The constant 0.5772 is Euler's number.
Median ( mu - betaln(ln(2)) )
Mode μ
Standard
Deviation
( frac{betapi} {sqrt{6}} )
Skewness 1.13955
Kurtosis 5.4
Coefficient of
Variation
( frac {betapi} {sqrt{6}(mu +
0.5772beta)} )
Parameter
Estimation
The method of moments estimators of the Gumbel
(maximum) distribution are
( tilde{beta} = frac{ssqrt{6}} {pi} )
( tilde{mu} = bar{X} - 0.5772 tilde{beta} = bar{X} -
0.45006 s )
where ( bar{X} ) and s are the sample mean and standard

The method of moments estimators of the Gumbel (minimum)
distribution are
( tilde{beta} = frac{ssqrt{6}} {pi} )
( tilde{mu} = bar{X} + 0.5772 tilde{beta} = bar{X} +
0.45006 s )
where ( bar{X} ) and s are the sample mean and standard
The maximum likelihood estimates for the maximum case are
the solution to the following simultaneous equations
( bar{x} - frac{sum_{i=1}^{n}{x_i exp(-
x_i/hat{beta})}} {sum_{i=1}^{n}{exp(-x_i/hat{beta})}}
- hat{beta} = 0 )
( -hat{beta} log left( frac{1}{n} sum_{i=1}^{n}{exp(-
x_i/hat{beta})} right) - hat{mu} = 0 )
For the minimum case, replace (-x_i) with (x_i) in the
above equations.
These equations need to be solved numerically and this is
typically accomplished by using statistical software packages.
least some of the probability functions for the extreme value
type I distribution.
Beta Distribution
Probability
Density
Function
beta distribution is
( f(x) = frac{(x-a)^{p-1}(b-x)^{q-1}}{B(p,q) (b-a)^{p+q-
1}} hspace{.3in} a le x le b; p, q > 0 )
where p and q are the shape parameters, a and b are the lower
and upper bounds, respectively, of the distribution, and B(p,q)
is the beta function. The beta function has the formula
( B(alpha,beta) = int_{0}^{1} {t^{alpha-1}(1-t)^{beta-
1}dt} )
The case where a = 0 and b = 1 is called the standard beta
distribution. The equation for the standard beta distribution is
( f(x) = frac{x^{p-1}(1-x)^{q-1}}{B(p,q)} hspace{.3in} 0
le x le 1; p, q > 0 )
Typically we define the general form of a distribution in terms
of location and scale parameters. The beta is different in that
we define the general distribution in terms of the lower and
upper bounds. However, the location and scale parameters can
be defined in terms of the lower and upper limits as follows:
location = a
scale = b - a
function.
The following is the plot of the beta probability density
function for four different values of the shape parameters.

Cumulative
Distribution
Function
beta distribution is also called the incomplete beta function
ratio (commonly denoted by Ix) and is defined as
( F(x) = I_{x}(p,q) = frac{int_{0}^{x}{t^{p-1}(1-
t)^{q-1}dt}}{B(p,q)} hspace{.2in} 0 le x le 1; p, q >
0 )
where B is the beta function defined above.
The following is the plot of the beta cumulative distribution
function with the same values of the shape parameters as the
pdf plots above.
Percent
Point
Function
The formula for the percent point function of the beta
The following is the plot of the beta percent point function
with the same values of the shape parameters as the pdf plots
above.

Other
Probability
Functions
Since the beta distribution is not typically used for reliability
functions.
Common
Statistics
The formulas below are for the case where the lower limit is
zero and the upper limit is one.
Mean ( frac {p}{p + q} )
Mode ( frac {p-1}{p+q-2} hspace{.3in} p, q >
1 )
Range 0 to 1
Standard
Deviation
( sqrt{frac{pq}{(p+q)^{2}(p+q+1)}} )
Coefficient of
Variation
( sqrt{frac{q}{p(p+q+1)}} )
Skewness ( frac {2(q-p)sqrt{p+q+1}}
{(p+q+2)sqrt{pq}} )
Parameter
Estimation
First consider the case where a and b are assumed to be
known. For this case, the method of moments estimates are
( p = bar{x}(frac{bar{x}(1 - bar{x})}{s^2} - 1) )
( q = (1 - bar{x})(frac{bar{x}(1 - bar{x})}{s^2} - 1)
)
where (bar{x}) is the sample mean and s2 is the sample
variance. If a and b are not 0 and 1, respectively, then replace
(bar{x}) with (frac{bar{x} - a}{b-a}) and s2 with
(frac{s^2}{(b-a)^2}) in the above equations.
For the case when a and b are known, the maximum
likelihood estimates can be obtained by solving the following
set of equations
( psi(hat{p}) - psi(hat{p} + hat{q}) = frac{1}{n}
sum_{i=1}^{n}{log(frac{Y_i - a}{b - a})} )
( psi(hat{q}) - psi(hat{p} + hat{q}) = frac{1}{n}
sum_{i=1}^{n}{log(frac{b - Y_i}{b - a})} )
Maximum likelihood estimation for the case when a and b are
not known can sometimes be problematic. Chapter 14 of Bury
discusses both moment and maximum likelihood estimation
for this case.
least some of the probability functions for the beta
distribution.
Binomial Distribution
Probability
Mass
Function
The binomial distribution is used when there are exactly two
mutually exclusive outcomes of a trial. These outcomes are
appropriately labeled "success" and "failure". The binomial
distribution is used to obtain the probability of observing x
successes in N trials, with the probability of success on a
single trial denoted by p. The binomial distribution assumes
that p is fixed for all trials.
The formula for the binomial probability mass function is
( P(x;p,n) = left( begin{array}{c} n x end{array} right)
(p)^{x}(1 - p)^{(n-x)} ;;;;;; mbox{for $x = 0, 1, 2, cdots
, n$} )
where

( left( begin{array}{c} n x end{array} right) = frac{n!}
{x!(n-x)! } )
The following is the plot of the binomial probability density
function for four values of p and n = 100.
Cumulative
Distribution
Function
The formula for the binomial cumulative probability function
is
( F(x;p,n) = sum_{i=0}^{x}{left( begin{array}{c} n i
end{array} right) (p)^{i}(1 - p)^{(n-i)}} )
The following is the plot of the binomial cumulative
above.
Percent
Point
Function
The binomial percent point function does not exist in simple
closed form. It is computed numerically. Note that because
this is a discrete distribution that is only defined for integer
values of x, the percent point function is not smooth in the
way the percent point function typically is for a continuous
distribution.
The following is the plot of the binomial percent point

Common
Statistics
Mean np
Mode p(n + 1) - 1 ≤ x ≤ p(n + 1)
Range 0 to n
Standard
Deviation
( sqrt{np(1 - p)} )
Coefficient of
Variation
( sqrt{frac{(1-p)} {np}} )
Skewness ( frac{(1-2p)} {sqrt{np(1 - p)}} )
Kurtosis ( 3 - frac{6} {n} + frac{1} {np(1 - p)} )
Comments The binomial distribution is probably the most commonly
used discrete distribution.
Parameter
Estimation
The maximum likelihood estimator of p (for fixed n) is
( tilde{p} = frac{x} {n} )
least some of the probability functions for the binomial
distribution.
Poisson Distribution
Probability
Mass
Function
The Poisson distribution is used to model the number of
events occurring within a given time interval.
The formula for the Poisson probability mass function is
( p(x;lambda) = frac{e^{-lambda}lambda^{x}} {x!}
mbox{ for } x = 0, 1, 2, cdots )
λ is the shape parameter which indicates the average number
of events in the given time interval.
The following is the plot of the Poisson probability density
function for four values of λ.

Cumulative
Distribution
Function
The formula for the Poisson cumulative probability function is
( F(x;lambda) = sum_{i=0}^{x}{frac{e^{-
lambda}lambda^{i}} {i!}} )
The following is the plot of the Poisson cumulative
distribution function with the same values of λ as the pdf plots
above.
Percent
Point
Function
The Poisson percent point function does not exist in simple
closed form. It is computed numerically. Note that because
this is a discrete distribution that is only defined for integer
values of x, the percent point function is not smooth in the
way the percent point function typically is for a continuous
distribution.
The following is the plot of the Poisson percent point function
with the same values of λ as the pdf plots above.
Common
Statistics
Mean λ
Mode For non-integer λ, it is the largest integer
less than λ. For integer λ, x = λ and x = λ -

1 are both the mode.
Range 0 to (infty)
Standard
Deviation
( sqrt{lambda} )
Coefficient of
Variation
( frac{1} {sqrt{lambda}} )
Skewness ( frac{1} {sqrt{lambda}} )
Kurtosis ( 3 + frac{1} {lambda} )
Parameter
Estimation
The maximum likelihood estimator of λ is
(tilde{lambda} = bar{X})
where (bar{X}) is the sample mean.
least some of the probability functions for the Poisson
distribution.
Tables for Probability Distributions
Tables Several commonly used tables for probability distributions can
be referenced below.
The values from these tables can also be obtained from most
general purpose statistical software programs. Most
introductory statistics textbooks (e.g., Snedecor and Cochran)
contain more extensive tables than are included here. These
tables are included for convenience.
1. Cumulative distribution function for the standard normal
distribution
2. Upper critical values of Student's t-distribution with
degrees of freedom
3. Upper critical values of the F-distribution with ν1 and ν2
degrees of freedom
4. Upper critical values of the chi-square distribution with ν
degrees of freedom
5. Critical values of t* distribution for testing the output of
a linear calibration line at 3 points
6. Upper critical values of the normal PPCC distribution
Cumulative Distribution Function of the
Standard Normal Distribution
How to
Use This
Table
The table below contains the area under the standard normal
curve from 0 to z. This can be used to compute the cumulative
distribution function values for the standard normal
distribution.
The table utilizes the symmetry of the normal distribution, so
what in fact is given is
( P[0 le x le |a|] )
where a is the value of interest. This is demonstrated in the
graph below for a = 0.5. The shaded area of the curve
represents the probability that x is between 0 and a.

This can be clarified by a few simple examples.
1. What is the probability that x is less than or equal to
1.53? Look for 1.5 in the X column, go right to the 0.03
column to find the value 0.43699. Now add 0.5 (for the
probability less than zero) to obtain the final result of
0.93699.
2. What is the probability that x is less than or equal to
-1.53? For negative values, use the relationship
( P[x le a] = 1 - P[x le |a|] ;;;;; mbox{for $x < 0$}
)
From the first example, this gives 1 - 0.93699 = 0.06301.
3. What is the probability that x is between -1 and 0.5?
Look up the values for 0.5 (0.5 + 0.19146 = 0.69146)
and -1 (1 - (0.5 + 0.34134) = 0.15866). Then subtract the
results (0.69146 - 0.15866) to obtain the result 0.5328.
To use this table with a non-standard normal distribution
(either the location parameter is not 0 or the scale parameter is
not 1), standardize your value by subtracting the mean and
dividing the result by the standard deviation. Then look up the
value for this standardized value.
A few particularly important numbers derived from the table
below, specifically numbers that are commonly used in
significance tests, are summarized in the following table:
p 0.001 0.005 0.010 0.025 0.050 0.100
Zp -3.090 -2.576 -2.326 -1.960 -1.645 -1.282
p 0.999 0.995 0.990 0.975 0.950 0.900
Zp +3.090 +2.576 +2.326 +1.960 +1.645 +1.282
These are critical values for the normal distribution.
Area under the Normal Curve from 0 to X
X 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.00000 0.00399 0.00798 0.01197 0.01595 0.01994 0.02392 0.02790 0.03188 0.03586
0.1 0.03983 0.04380 0.04776 0.05172 0.05567 0.05962 0.06356 0.06749 0.07142 0.07535
0.2 0.07926 0.08317 0.08706 0.09095 0.09483 0.09871 0.10257 0.10642 0.11026 0.11409
0.3 0.11791 0.12172 0.12552 0.12930 0.13307 0.13683 0.14058 0.14431 0.14803 0.15173
0.4 0.15542 0.15910 0.16276 0.16640 0.17003 0.17364 0.17724 0.18082 0.18439 0.18793
0.5 0.19146 0.19497 0.19847 0.20194 0.20540 0.20884 0.21226 0.21566 0.21904 0.22240
0.6 0.22575 0.22907 0.23237 0.23565 0.23891 0.24215 0.24537 0.24857 0.25175 0.25490
0.7 0.25804 0.26115 0.26424 0.26730 0.27035 0.27337 0.27637 0.27935 0.28230 0.28524
0.8 0.28814 0.29103 0.29389 0.29673 0.29955 0.30234 0.30511 0.30785 0.31057 0.31327
0.9 0.31594 0.31859 0.32121 0.32381 0.32639 0.32894 0.33147 0.33398 0.33646 0.33891

1.0 0.34134 0.34375 0.34614 0.34849 0.35083 0.35314 0.35543 0.35769 0.35993 0.36214
1.1 0.36433 0.36650 0.36864 0.37076 0.37286 0.37493 0.37698 0.37900 0.38100 0.38298
1.2 0.38493 0.38686 0.38877 0.39065 0.39251 0.39435 0.39617 0.39796 0.39973 0.40147
1.3 0.40320 0.40490 0.40658 0.40824 0.40988 0.41149 0.41308 0.41466 0.41621 0.41774
1.4 0.41924 0.42073 0.42220 0.42364 0.42507 0.42647 0.42785 0.42922 0.43056 0.43189
1.5 0.43319 0.43448 0.43574 0.43699 0.43822 0.43943 0.44062 0.44179 0.44295 0.44408
1.6 0.44520 0.44630 0.44738 0.44845 0.44950 0.45053 0.45154 0.45254 0.45352 0.45449
1.7 0.45543 0.45637 0.45728 0.45818 0.45907 0.45994 0.46080 0.46164 0.46246 0.46327
1.8 0.46407 0.46485 0.46562 0.46638 0.46712 0.46784 0.46856 0.46926 0.46995 0.47062
1.9 0.47128 0.47193 0.47257 0.47320 0.47381 0.47441 0.47500 0.47558 0.47615 0.47670
2.0 0.47725 0.47778 0.47831 0.47882 0.47932 0.47982 0.48030 0.48077 0.48124 0.48169
2.1 0.48214 0.48257 0.48300 0.48341 0.48382 0.48422 0.48461 0.48500 0.48537 0.48574
2.2 0.48610 0.48645 0.48679 0.48713 0.48745 0.48778 0.48809 0.48840 0.48870 0.48899
2.3 0.48928 0.48956 0.48983 0.49010 0.49036 0.49061 0.49086 0.49111 0.49134 0.49158
2.4 0.49180 0.49202 0.49224 0.49245 0.49266 0.49286 0.49305 0.49324 0.49343 0.49361
2.5 0.49379 0.49396 0.49413 0.49430 0.49446 0.49461 0.49477 0.49492 0.49506 0.49520
2.6 0.49534 0.49547 0.49560 0.49573 0.49585 0.49598 0.49609 0.49621 0.49632 0.49643
2.7 0.49653 0.49664 0.49674 0.49683 0.49693 0.49702 0.49711 0.49720 0.49728 0.49736
2.8 0.49744 0.49752 0.49760 0.49767 0.49774 0.49781 0.49788 0.49795 0.49801 0.49807
2.9 0.49813 0.49819 0.49825 0.49831 0.49836 0.49841 0.49846 0.49851 0.49856 0.49861
3.0 0.49865 0.49869 0.49874 0.49878 0.49882 0.49886 0.49889 0.49893 0.49896 0.49900
3.1 0.49903 0.49906 0.49910 0.49913 0.49916 0.49918 0.49921 0.49924 0.49926 0.49929
3.2 0.49931 0.49934 0.49936 0.49938 0.49940 0.49942 0.49944 0.49946 0.49948 0.49950
3.3 0.49952 0.49953 0.49955 0.49957 0.49958 0.49960 0.49961 0.49962 0.49964 0.49965
3.4 0.49966 0.49968 0.49969 0.49970 0.49971 0.49972 0.49973 0.49974 0.49975 0.49976
3.5 0.49977 0.49978 0.49978 0.49979 0.49980 0.49981 0.49981 0.49982 0.49983 0.49983
3.6 0.49984 0.49985 0.49985 0.49986 0.49986 0.49987 0.49987 0.49988 0.49988 0.49989
3.7 0.49989 0.49990 0.49990 0.49990 0.49991 0.49991 0.49992 0.49992 0.49992 0.49992
3.8 0.49993 0.49993 0.49993 0.49994 0.49994 0.49994 0.49994 0.49995 0.49995 0.49995
3.9 0.49995 0.49995 0.49996 0.49996 0.49996 0.49996 0.49996 0.49996 0.49997 0.49997
4.0 0.49997 0.49997 0.49997 0.49997 0.49997 0.49997 0.49998 0.49998 0.49998 0.49998
Critical Values of the Student's t Distribution
How to Use
This Table
This table contains critical values of the Student's t distribution
computed using the cumulative distribution function. The t
distribution is symmetric so that
t1-α,ν = -tα,ν.
The t table can be used for both one-sided (lower and upper) and
two-sided tests using the appropriate value of α.
The significance level, α, is demonstrated in the graph below, which
displays a t distribution with 10 degrees of freedom. The most
commonly used significance level is α = 0.05. For a two-sided test,
we compute 1 - α/2, or 1 - 0.05/2 = 0.975 when α = 0.05. If the
absolute value of the test statistic is greater than the critical value
(0.975), then we reject the null hypothesis. Due to the symmetry of
the t distribution, we only tabulate the positive critical values in the
table below.

Given a specified value for α :
1. For a two-sided test, find the column corresponding to 1-α/2
and reject the null hypothesis if the absolute value of the test
statistic is greater than the value of t1-α/2,ν in the table below.
2. For an upper, one-sided test, find the column corresponding to
1-α and reject the null hypothesis if the test statistic is greater
than the table value.
3. For a lower, one-sided test, find the column corresponding to
1-α and reject the null hypothesis if the test statistic is less
than the negative of the table value.
Critical values of Student's t distribution with ν degrees of freedom
Probability less than the critical value (t1-α,ν)
ν 0.90 0.95 0.975 0.99 0.995 0.999
1. 3.078 6.314 12.706 31.821 63.657 318.309
2. 1.886 2.920 4.303 6.965 9.925 22.327
3. 1.638 2.353 3.182 4.541 5.841 10.215
4. 1.533 2.132 2.776 3.747 4.604 7.173
5. 1.476 2.015 2.571 3.365 4.032 5.893
6. 1.440 1.943 2.447 3.143 3.707 5.208
7. 1.415 1.895 2.365 2.998 3.499 4.785
8. 1.397 1.860 2.306 2.896 3.355 4.501
9. 1.383 1.833 2.262 2.821 3.250 4.297
10. 1.372 1.812 2.228 2.764 3.169 4.144
11. 1.363 1.796 2.201 2.718 3.106 4.025
12. 1.356 1.782 2.179 2.681 3.055 3.930
13. 1.350 1.771 2.160 2.650 3.012 3.852
14. 1.345 1.761 2.145 2.624 2.977 3.787
15. 1.341 1.753 2.131 2.602 2.947 3.733
16. 1.337 1.746 2.120 2.583 2.921 3.686
17. 1.333 1.740 2.110 2.567 2.898 3.646
18. 1.330 1.734 2.101 2.552 2.878 3.610
19. 1.328 1.729 2.093 2.539 2.861 3.579
20. 1.325 1.725 2.086 2.528 2.845 3.552
21. 1.323 1.721 2.080 2.518 2.831 3.527
22. 1.321 1.717 2.074 2.508 2.819 3.505
23. 1.319 1.714 2.069 2.500 2.807 3.485
24. 1.318 1.711 2.064 2.492 2.797 3.467
25. 1.316 1.708 2.060 2.485 2.787 3.450
26. 1.315 1.706 2.056 2.479 2.779 3.435
27. 1.314 1.703 2.052 2.473 2.771 3.421
28. 1.313 1.701 2.048 2.467 2.763 3.408
29. 1.311 1.699 2.045 2.462 2.756 3.396
30. 1.310 1.697 2.042 2.457 2.750 3.385
31. 1.309 1.696 2.040 2.453 2.744 3.375
32. 1.309 1.694 2.037 2.449 2.738 3.365
33. 1.308 1.692 2.035 2.445 2.733 3.356
34. 1.307 1.691 2.032 2.441 2.728 3.348
35. 1.306 1.690 2.030 2.438 2.724 3.340
36. 1.306 1.688 2.028 2.434 2.719 3.333
37. 1.305 1.687 2.026 2.431 2.715 3.326
38. 1.304 1.686 2.024 2.429 2.712 3.319
39. 1.304 1.685 2.023 2.426 2.708 3.313
40. 1.303 1.684 2.021 2.423 2.704 3.307
41. 1.303 1.683 2.020 2.421 2.701 3.301
42. 1.302 1.682 2.018 2.418 2.698 3.296
43. 1.302 1.681 2.017 2.416 2.695 3.291
44. 1.301 1.680 2.015 2.414 2.692 3.286
45. 1.301 1.679 2.014 2.412 2.690 3.281
46. 1.300 1.679 2.013 2.410 2.687 3.277

47. 1.300 1.678 2.012 2.408 2.685 3.273
48. 1.299 1.677 2.011 2.407 2.682 3.269
49. 1.299 1.677 2.010 2.405 2.680 3.265
50. 1.299 1.676 2.009 2.403 2.678 3.261
51. 1.298 1.675 2.008 2.402 2.676 3.258
52. 1.298 1.675 2.007 2.400 2.674 3.255
53. 1.298 1.674 2.006 2.399 2.672 3.251
54. 1.297 1.674 2.005 2.397 2.670 3.248
55. 1.297 1.673 2.004 2.396 2.668 3.245
56. 1.297 1.673 2.003 2.395 2.667 3.242
57. 1.297 1.672 2.002 2.394 2.665 3.239
58. 1.296 1.672 2.002 2.392 2.663 3.237
59. 1.296 1.671 2.001 2.391 2.662 3.234
60. 1.296 1.671 2.000 2.390 2.660 3.232
61. 1.296 1.670 2.000 2.389 2.659 3.229
62. 1.295 1.670 1.999 2.388 2.657 3.227
63. 1.295 1.669 1.998 2.387 2.656 3.225
64. 1.295 1.669 1.998 2.386 2.655 3.223
65. 1.295 1.669 1.997 2.385 2.654 3.220
66. 1.295 1.668 1.997 2.384 2.652 3.218
67. 1.294 1.668 1.996 2.383 2.651 3.216
68. 1.294 1.668 1.995 2.382 2.650 3.214
69. 1.294 1.667 1.995 2.382 2.649 3.213
70. 1.294 1.667 1.994 2.381 2.648 3.211
71. 1.294 1.667 1.994 2.380 2.647 3.209
72. 1.293 1.666 1.993 2.379 2.646 3.207
73. 1.293 1.666 1.993 2.379 2.645 3.206
74. 1.293 1.666 1.993 2.378 2.644 3.204
75. 1.293 1.665 1.992 2.377 2.643 3.202
76. 1.293 1.665 1.992 2.376 2.642 3.201
77. 1.293 1.665 1.991 2.376 2.641 3.199
78. 1.292 1.665 1.991 2.375 2.640 3.198
79. 1.292 1.664 1.990 2.374 2.640 3.197
80. 1.292 1.664 1.990 2.374 2.639 3.195
81. 1.292 1.664 1.990 2.373 2.638 3.194
82. 1.292 1.664 1.989 2.373 2.637 3.193
83. 1.292 1.663 1.989 2.372 2.636 3.191
84. 1.292 1.663 1.989 2.372 2.636 3.190
85. 1.292 1.663 1.988 2.371 2.635 3.189
86. 1.291 1.663 1.988 2.370 2.634 3.188
87. 1.291 1.663 1.988 2.370 2.634 3.187
88. 1.291 1.662 1.987 2.369 2.633 3.185
89. 1.291 1.662 1.987 2.369 2.632 3.184
90. 1.291 1.662 1.987 2.368 2.632 3.183
91. 1.291 1.662 1.986 2.368 2.631 3.182
92. 1.291 1.662 1.986 2.368 2.630 3.181
93. 1.291 1.661 1.986 2.367 2.630 3.180
94. 1.291 1.661 1.986 2.367 2.629 3.179
95. 1.291 1.661 1.985 2.366 2.629 3.178
96. 1.290 1.661 1.985 2.366 2.628 3.177
97. 1.290 1.661 1.985 2.365 2.627 3.176
98. 1.290 1.661 1.984 2.365 2.627 3.175
99. 1.290 1.660 1.984 2.365 2.626 3.175
100. 1.290 1.660 1.984 2.364 2.626 3.174
Normal Values
100. 1.282 1.645 1.960 2.326 2.576 3.090
1.282 1.645 1.960 2.326 2.576 3.090
Upper Critical Values of the F Distribution
How to
Use This
This table contains the upper critical values of the F
distribution. This table is used for one-sided F tests at the α =

Table 0.05, 0.10, and 0.01 levels.
More specifically, a test statistic is computed with ν1 and ν2
degrees of freedom, and the result is compared to this table.
For a one-sided test, the null hypothesis is rejected when the
test statistic is greater than the tabled value. This is
demonstrated with the graph of an F distribution with ν1 = 10
and ν2 = 10. The shaded area of the graph indicates the
rejection region at the α significance level. Since this is a one-
sided test, we have α probability in the upper tail of exceeding
the critical value and zero in the lower tail. Because the F
distribution is asymmetric, a two-sided test requires a set of of
tables (not included here) that contain the rejection regions for
both the lower and upper tails.
Contents The following tables for ν2 from 1 to 100 are included:
1. ν1 = 1 - 10
2. One sided, 5% significance level, ν1 = 11 - 20
Upper critical values of the F distribution for ν1 numerator degrees of freedom and ν2 denominator degrees of
freedom
5% significance level
( F_{.05}(nu_{1},nu_{2}) )
ν1 1 2 3 4 5 6 7 8 9 10
ν2
1 161.448 199.500 215.707 224.583 230.162 233.986 236.768 238.882 240.543 241.882
2 18.513 19.000 19.164 19.247 19.296 19.330 19.353 19.371 19.385 19.396
3 10.128 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786
4 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964
5 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772 4.735
6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.060
7 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637
8 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347
9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137
10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978
11 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.896 2.854
12 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753
13 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671
14 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602

15 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544
16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.538 2.494
17 4.451 3.592 3.197 2.965 2.810 2.699 2.614 2.548 2.494 2.450
18 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.456 2.412
19 4.381 3.522 3.127 2.895 2.740 2.628 2.544 2.477 2.423 2.378
20 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348
21 4.325 3.467 3.072 2.840 2.685 2.573 2.488 2.420 2.366 2.321
22 4.301 3.443 3.049 2.817 2.661 2.549 2.464 2.397 2.342 2.297
23 4.279 3.422 3.028 2.796 2.640 2.528 2.442 2.375 2.320 2.275
24 4.260 3.403 3.009 2.776 2.621 2.508 2.423 2.355 2.300 2.255
25 4.242 3.385 2.991 2.759 2.603 2.490 2.405 2.337 2.282 2.236
26 4.225 3.369 2.975 2.743 2.587 2.474 2.388 2.321 2.265 2.220
27 4.210 3.354 2.960 2.728 2.572 2.459 2.373 2.305 2.250 2.204
28 4.196 3.340 2.947 2.714 2.558 2.445 2.359 2.291 2.236 2.190
29 4.183 3.328 2.934 2.701 2.545 2.432 2.346 2.278 2.223 2.177
30 4.171 3.316 2.922 2.690 2.534 2.421 2.334 2.266 2.211 2.165
31 4.160 3.305 2.911 2.679 2.523 2.409 2.323 2.255 2.199 2.153
32 4.149 3.295 2.901 2.668 2.512 2.399 2.313 2.244 2.189 2.142
33 4.139 3.285 2.892 2.659 2.503 2.389 2.303 2.235 2.179 2.133
34 4.130 3.276 2.883 2.650 2.494 2.380 2.294 2.225 2.170 2.123
35 4.121 3.267 2.874 2.641 2.485 2.372 2.285 2.217 2.161 2.114
36 4.113 3.259 2.866 2.634 2.477 2.364 2.277 2.209 2.153 2.106
37 4.105 3.252 2.859 2.626 2.470 2.356 2.270 2.201 2.145 2.098
38 4.098 3.245 2.852 2.619 2.463 2.349 2.262 2.194 2.138 2.091
39 4.091 3.238 2.845 2.612 2.456 2.342 2.255 2.187 2.131 2.084
40 4.085 3.232 2.839 2.606 2.449 2.336 2.249 2.180 2.124 2.077
41 4.079 3.226 2.833 2.600 2.443 2.330 2.243 2.174 2.118 2.071
42 4.073 3.220 2.827 2.594 2.438 2.324 2.237 2.168 2.112 2.065
43 4.067 3.214 2.822 2.589 2.432 2.318 2.232 2.163 2.106 2.059
44 4.062 3.209 2.816 2.584 2.427 2.313 2.226 2.157 2.101 2.054
45 4.057 3.204 2.812 2.579 2.422 2.308 2.221 2.152 2.096 2.049
46 4.052 3.200 2.807 2.574 2.417 2.304 2.216 2.147 2.091 2.044
47 4.047 3.195 2.802 2.570 2.413 2.299 2.212 2.143 2.086 2.039
48 4.043 3.191 2.798 2.565 2.409 2.295 2.207 2.138 2.082 2.035
49 4.038 3.187 2.794 2.561 2.404 2.290 2.203 2.134 2.077 2.030
50 4.034 3.183 2.790 2.557 2.400 2.286 2.199 2.130 2.073 2.026
51 4.030 3.179 2.786 2.553 2.397 2.283 2.195 2.126 2.069 2.022
52 4.027 3.175 2.783 2.550 2.393 2.279 2.192 2.122 2.066 2.018
53 4.023 3.172 2.779 2.546 2.389 2.275 2.188 2.119 2.062 2.015
54 4.020 3.168 2.776 2.543 2.386 2.272 2.185 2.115 2.059 2.011
55 4.016 3.165 2.773 2.540 2.383 2.269 2.181 2.112 2.055 2.008
56 4.013 3.162 2.769 2.537 2.380 2.266 2.178 2.109 2.052 2.005
57 4.010 3.159 2.766 2.534 2.377 2.263 2.175 2.106 2.049 2.001
58 4.007 3.156 2.764 2.531 2.374 2.260 2.172 2.103 2.046 1.998
59 4.004 3.153 2.761 2.528 2.371 2.257 2.169 2.100 2.043 1.995
60 4.001 3.150 2.758 2.525 2.368 2.254 2.167 2.097 2.040 1.993
61 3.998 3.148 2.755 2.523 2.366 2.251 2.164 2.094 2.037 1.990
62 3.996 3.145 2.753 2.520 2.363 2.249 2.161 2.092 2.035 1.987
63 3.993 3.143 2.751 2.518 2.361 2.246 2.159 2.089 2.032 1.985
64 3.991 3.140 2.748 2.515 2.358 2.244 2.156 2.087 2.030 1.982
65 3.989 3.138 2.746 2.513 2.356 2.242 2.154 2.084 2.027 1.980
66 3.986 3.136 2.744 2.511 2.354 2.239 2.152 2.082 2.025 1.977
67 3.984 3.134 2.742 2.509 2.352 2.237 2.150 2.080 2.023 1.975
68 3.982 3.132 2.740 2.507 2.350 2.235 2.148 2.078 2.021 1.973
69 3.980 3.130 2.737 2.505 2.348 2.233 2.145 2.076 2.019 1.971
70 3.978 3.128 2.736 2.503 2.346 2.231 2.143 2.074 2.017 1.969
71 3.976 3.126 2.734 2.501 2.344 2.229 2.142 2.072 2.015 1.967
72 3.974 3.124 2.732 2.499 2.342 2.227 2.140 2.070 2.013 1.965
73 3.972 3.122 2.730 2.497 2.340 2.226 2.138 2.068 2.011 1.963
74 3.970 3.120 2.728 2.495 2.338 2.224 2.136 2.066 2.009 1.961
75 3.968 3.119 2.727 2.494 2.337 2.222 2.134 2.064 2.007 1.959
76 3.967 3.117 2.725 2.492 2.335 2.220 2.133 2.063 2.006 1.958
77 3.965 3.115 2.723 2.490 2.333 2.219 2.131 2.061 2.004 1.956
78 3.963 3.114 2.722 2.489 2.332 2.217 2.129 2.059 2.002 1.954
79 3.962 3.112 2.720 2.487 2.330 2.216 2.128 2.058 2.001 1.953
80 3.960 3.111 2.719 2.486 2.329 2.214 2.126 2.056 1.999 1.951
81 3.959 3.109 2.717 2.484 2.327 2.213 2.125 2.055 1.998 1.950
82 3.957 3.108 2.716 2.483 2.326 2.211 2.123 2.053 1.996 1.948

83 3.956 3.107 2.715 2.482 2.324 2.210 2.122 2.052 1.995 1.947
84 3.955 3.105 2.713 2.480 2.323 2.209 2.121 2.051 1.993 1.945
85 3.953 3.104 2.712 2.479 2.322 2.207 2.119 2.049 1.992 1.944
86 3.952 3.103 2.711 2.478 2.321 2.206 2.118 2.048 1.991 1.943
87 3.951 3.101 2.709 2.476 2.319 2.205 2.117 2.047 1.989 1.941
88 3.949 3.100 2.708 2.475 2.318 2.203 2.115 2.045 1.988 1.940
89 3.948 3.099 2.707 2.474 2.317 2.202 2.114 2.044 1.987 1.939
90 3.947 3.098 2.706 2.473 2.316 2.201 2.113 2.043 1.986 1.938
91 3.946 3.097 2.705 2.472 2.315 2.200 2.112 2.042 1.984 1.936
92 3.945 3.095 2.704 2.471 2.313 2.199 2.111 2.041 1.983 1.935
93 3.943 3.094 2.703 2.470 2.312 2.198 2.110 2.040 1.982 1.934
94 3.942 3.093 2.701 2.469 2.311 2.197 2.109 2.038 1.981 1.933
95 3.941 3.092 2.700 2.467 2.310 2.196 2.108 2.037 1.980 1.932
96 3.940 3.091 2.699 2.466 2.309 2.195 2.106 2.036 1.979 1.931
97 3.939 3.090 2.698 2.465 2.308 2.194 2.105 2.035 1.978 1.930
98 3.938 3.089 2.697 2.465 2.307 2.193 2.104 2.034 1.977 1.929
99 3.937 3.088 2.696 2.464 2.306 2.192 2.103 2.033 1.976 1.928
100 3.936 3.087 2.696 2.463 2.305 2.191 2.103 2.032 1.975 1.927
ν1 11 12 13 14 15 16 17 18 19 20
ν2
1 242.983 243.906 244.690 245.364 245.950 246.464 246.918 247.323 247.686 248.013
2 19.405 19.413 19.419 19.424 19.429 19.433 19.437 19.440 19.443 19.446
3 8.763 8.745 8.729 8.715 8.703 8.692 8.683 8.675 8.667 8.660
4 5.936 5.912 5.891 5.873 5.858 5.844 5.832 5.821 5.811 5.803
5 4.704 4.678 4.655 4.636 4.619 4.604 4.590 4.579 4.568 4.558
6 4.027 4.000 3.976 3.956 3.938 3.922 3.908 3.896 3.884 3.874
7 3.603 3.575 3.550 3.529 3.511 3.494 3.480 3.467 3.455 3.445
8 3.313 3.284 3.259 3.237 3.218 3.202 3.187 3.173 3.161 3.150
9 3.102 3.073 3.048 3.025 3.006 2.989 2.974 2.960 2.948 2.936
10 2.943 2.913 2.887 2.865 2.845 2.828 2.812 2.798 2.785 2.774
11 2.818 2.788 2.761 2.739 2.719 2.701 2.685 2.671 2.658 2.646
12 2.717 2.687 2.660 2.637 2.617 2.599 2.583 2.568 2.555 2.544
13 2.635 2.604 2.577 2.554 2.533 2.515 2.499 2.484 2.471 2.459
14 2.565 2.534 2.507 2.484 2.463 2.445 2.428 2.413 2.400 2.388
15 2.507 2.475 2.448 2.424 2.403 2.385 2.368 2.353 2.340 2.328
16 2.456 2.425 2.397 2.373 2.352 2.333 2.317 2.302 2.288 2.276
17 2.413 2.381 2.353 2.329 2.308 2.289 2.272 2.257 2.243 2.230
18 2.374 2.342 2.314 2.290 2.269 2.250 2.233 2.217 2.203 2.191
19 2.340 2.308 2.280 2.256 2.234 2.215 2.198 2.182 2.168 2.155
20 2.310 2.278 2.250 2.225 2.203 2.184 2.167 2.151 2.137 2.124
21 2.283 2.250 2.222 2.197 2.176 2.156 2.139 2.123 2.109 2.096
22 2.259 2.226 2.198 2.173 2.151 2.131 2.114 2.098 2.084 2.071
23 2.236 2.204 2.175 2.150 2.128 2.109 2.091 2.075 2.061 2.048
24 2.216 2.183 2.155 2.130 2.108 2.088 2.070 2.054 2.040 2.027
25 2.198 2.165 2.136 2.111 2.089 2.069 2.051 2.035 2.021 2.007
26 2.181 2.148 2.119 2.094 2.072 2.052 2.034 2.018 2.003 1.990
27 2.166 2.132 2.103 2.078 2.056 2.036 2.018 2.002 1.987 1.974
28 2.151 2.118 2.089 2.064 2.041 2.021 2.003 1.987 1.972 1.959
29 2.138 2.104 2.075 2.050 2.027 2.007 1.989 1.973 1.958 1.945
30 2.126 2.092 2.063 2.037 2.015 1.995 1.976 1.960 1.945 1.932
31 2.114 2.080 2.051 2.026 2.003 1.983 1.965 1.948 1.933 1.920
32 2.103 2.070 2.040 2.015 1.992 1.972 1.953 1.937 1.922 1.908
33 2.093 2.060 2.030 2.004 1.982 1.961 1.943 1.926 1.911 1.898
34 2.084 2.050 2.021 1.995 1.972 1.952 1.933 1.917 1.902 1.888
35 2.075 2.041 2.012 1.986 1.963 1.942 1.924 1.907 1.892 1.878
36 2.067 2.033 2.003 1.977 1.954 1.934 1.915 1.899 1.883 1.870
37 2.059 2.025 1.995 1.969 1.946 1.926 1.907 1.890 1.875 1.861
38 2.051 2.017 1.988 1.962 1.939 1.918 1.899 1.883 1.867 1.853
39 2.044 2.010 1.981 1.954 1.931 1.911 1.892 1.875 1.860 1.846
40 2.038 2.003 1.974 1.948 1.924 1.904 1.885 1.868 1.853 1.839
41 2.031 1.997 1.967 1.941 1.918 1.897 1.879 1.862 1.846 1.832
42 2.025 1.991 1.961 1.935 1.912 1.891 1.872 1.855 1.840 1.826
43 2.020 1.985 1.955 1.929 1.906 1.885 1.866 1.849 1.834 1.820

44 2.014 1.980 1.950 1.924 1.900 1.879 1.861 1.844 1.828 1.814
45 2.009 1.974 1.945 1.918 1.895 1.874 1.855 1.838 1.823 1.808
46 2.004 1.969 1.940 1.913 1.890 1.869 1.850 1.833 1.817 1.803
47 1.999 1.965 1.935 1.908 1.885 1.864 1.845 1.828 1.812 1.798
48 1.995 1.960 1.930 1.904 1.880 1.859 1.840 1.823 1.807 1.793
49 1.990 1.956 1.926 1.899 1.876 1.855 1.836 1.819 1.803 1.789
50 1.986 1.952 1.921 1.895 1.871 1.850 1.831 1.814 1.798 1.784
51 1.982 1.947 1.917 1.891 1.867 1.846 1.827 1.810 1.794 1.780
52 1.978 1.944 1.913 1.887 1.863 1.842 1.823 1.806 1.790 1.776
53 1.975 1.940 1.910 1.883 1.859 1.838 1.819 1.802 1.786 1.772
54 1.971 1.936 1.906 1.879 1.856 1.835 1.816 1.798 1.782 1.768
55 1.968 1.933 1.903 1.876 1.852 1.831 1.812 1.795 1.779 1.764
56 1.964 1.930 1.899 1.873 1.849 1.828 1.809 1.791 1.775 1.761
57 1.961 1.926 1.896 1.869 1.846 1.824 1.805 1.788 1.772 1.757
58 1.958 1.923 1.893 1.866 1.842 1.821 1.802 1.785 1.769 1.754
59 1.955 1.920 1.890 1.863 1.839 1.818 1.799 1.781 1.766 1.751
60 1.952 1.917 1.887 1.860 1.836 1.815 1.796 1.778 1.763 1.748
61 1.949 1.915 1.884 1.857 1.834 1.812 1.793 1.776 1.760 1.745
62 1.947 1.912 1.882 1.855 1.831 1.809 1.790 1.773 1.757 1.742
63 1.944 1.909 1.879 1.852 1.828 1.807 1.787 1.770 1.754 1.739
64 1.942 1.907 1.876 1.849 1.826 1.804 1.785 1.767 1.751 1.737
65 1.939 1.904 1.874 1.847 1.823 1.802 1.782 1.765 1.749 1.734
66 1.937 1.902 1.871 1.845 1.821 1.799 1.780 1.762 1.746 1.732
67 1.935 1.900 1.869 1.842 1.818 1.797 1.777 1.760 1.744 1.729
68 1.932 1.897 1.867 1.840 1.816 1.795 1.775 1.758 1.742 1.727
69 1.930 1.895 1.865 1.838 1.814 1.792 1.773 1.755 1.739 1.725
70 1.928 1.893 1.863 1.836 1.812 1.790 1.771 1.753 1.737 1.722
71 1.926 1.891 1.861 1.834 1.810 1.788 1.769 1.751 1.735 1.720
72 1.924 1.889 1.859 1.832 1.808 1.786 1.767 1.749 1.733 1.718
73 1.922 1.887 1.857 1.830 1.806 1.784 1.765 1.747 1.731 1.716
74 1.921 1.885 1.855 1.828 1.804 1.782 1.763 1.745 1.729 1.714
75 1.919 1.884 1.853 1.826 1.802 1.780 1.761 1.743 1.727 1.712
76 1.917 1.882 1.851 1.824 1.800 1.778 1.759 1.741 1.725 1.710
77 1.915 1.880 1.849 1.822 1.798 1.777 1.757 1.739 1.723 1.708
78 1.914 1.878 1.848 1.821 1.797 1.775 1.755 1.738 1.721 1.707
79 1.912 1.877 1.846 1.819 1.795 1.773 1.754 1.736 1.720 1.705
80 1.910 1.875 1.845 1.817 1.793 1.772 1.752 1.734 1.718 1.703
81 1.909 1.874 1.843 1.816 1.792 1.770 1.750 1.733 1.716 1.702
82 1.907 1.872 1.841 1.814 1.790 1.768 1.749 1.731 1.715 1.700
83 1.906 1.871 1.840 1.813 1.789 1.767 1.747 1.729 1.713 1.698
84 1.905 1.869 1.838 1.811 1.787 1.765 1.746 1.728 1.712 1.697
85 1.903 1.868 1.837 1.810 1.786 1.764 1.744 1.726 1.710 1.695
86 1.902 1.867 1.836 1.808 1.784 1.762 1.743 1.725 1.709 1.694
87 1.900 1.865 1.834 1.807 1.783 1.761 1.741 1.724 1.707 1.692
88 1.899 1.864 1.833 1.806 1.782 1.760 1.740 1.722 1.706 1.691
89 1.898 1.863 1.832 1.804 1.780 1.758 1.739 1.721 1.705 1.690
90 1.897 1.861 1.830 1.803 1.779 1.757 1.737 1.720 1.703 1.688
91 1.895 1.860 1.829 1.802 1.778 1.756 1.736 1.718 1.702 1.687
92 1.894 1.859 1.828 1.801 1.776 1.755 1.735 1.717 1.701 1.686
93 1.893 1.858 1.827 1.800 1.775 1.753 1.734 1.716 1.699 1.684
94 1.892 1.857 1.826 1.798 1.774 1.752 1.733 1.715 1.698 1.683
95 1.891 1.856 1.825 1.797 1.773 1.751 1.731 1.713 1.697 1.682
96 1.890 1.854 1.823 1.796 1.772 1.750 1.730 1.712 1.696 1.681
97 1.889 1.853 1.822 1.795 1.771 1.749 1.729 1.711 1.695 1.680
98 1.888 1.852 1.821 1.794 1.770 1.748 1.728 1.710 1.694 1.679
99 1.887 1.851 1.820 1.793 1.769 1.747 1.727 1.709 1.693 1.678
100 1.886 1.850 1.819 1.792 1.768 1.746 1.726 1.708 1.691 1.676
freedom
( F_{.10}(nu_{1},nu_{2}) )

ν1 1 2 3 4 5 6 7 8 9 10
ν2
1 39.863 49.500 53.593 55.833 57.240 58.204 58.906 59.439 59.858 60.195
2 8.526 9.000 9.162 9.243 9.293 9.326 9.349 9.367 9.381 9.392
3 5.538 5.462 5.391 5.343 5.309 5.285 5.266 5.252 5.240 5.230
4 4.545 4.325 4.191 4.107 4.051 4.010 3.979 3.955 3.936 3.920
5 4.060 3.780 3.619 3.520 3.453 3.405 3.368 3.339 3.316 3.297
6 3.776 3.463 3.289 3.181 3.108 3.055 3.014 2.983 2.958 2.937
7 3.589 3.257 3.074 2.961 2.883 2.827 2.785 2.752 2.725 2.703
8 3.458 3.113 2.924 2.806 2.726 2.668 2.624 2.589 2.561 2.538
9 3.360 3.006 2.813 2.693 2.611 2.551 2.505 2.469 2.440 2.416
10 3.285 2.924 2.728 2.605 2.522 2.461 2.414 2.377 2.347 2.323
11 3.225 2.860 2.660 2.536 2.451 2.389 2.342 2.304 2.274 2.248
12 3.177 2.807 2.606 2.480 2.394 2.331 2.283 2.245 2.214 2.188
13 3.136 2.763 2.560 2.434 2.347 2.283 2.234 2.195 2.164 2.138
14 3.102 2.726 2.522 2.395 2.307 2.243 2.193 2.154 2.122 2.095
15 3.073 2.695 2.490 2.361 2.273 2.208 2.158 2.119 2.086 2.059
16 3.048 2.668 2.462 2.333 2.244 2.178 2.128 2.088 2.055 2.028
17 3.026 2.645 2.437 2.308 2.218 2.152 2.102 2.061 2.028 2.001
18 3.007 2.624 2.416 2.286 2.196 2.130 2.079 2.038 2.005 1.977
19 2.990 2.606 2.397 2.266 2.176 2.109 2.058 2.017 1.984 1.956
20 2.975 2.589 2.380 2.249 2.158 2.091 2.040 1.999 1.965 1.937
21 2.961 2.575 2.365 2.233 2.142 2.075 2.023 1.982 1.948 1.920
22 2.949 2.561 2.351 2.219 2.128 2.060 2.008 1.967 1.933 1.904
23 2.937 2.549 2.339 2.207 2.115 2.047 1.995 1.953 1.919 1.890
24 2.927 2.538 2.327 2.195 2.103 2.035 1.983 1.941 1.906 1.877
25 2.918 2.528 2.317 2.184 2.092 2.024 1.971 1.929 1.895 1.866
26 2.909 2.519 2.307 2.174 2.082 2.014 1.961 1.919 1.884 1.855
27 2.901 2.511 2.299 2.165 2.073 2.005 1.952 1.909 1.874 1.845
28 2.894 2.503 2.291 2.157 2.064 1.996 1.943 1.900 1.865 1.836
29 2.887 2.495 2.283 2.149 2.057 1.988 1.935 1.892 1.857 1.827
30 2.881 2.489 2.276 2.142 2.049 1.980 1.927 1.884 1.849 1.819
31 2.875 2.482 2.270 2.136 2.042 1.973 1.920 1.877 1.842 1.812
32 2.869 2.477 2.263 2.129 2.036 1.967 1.913 1.870 1.835 1.805
33 2.864 2.471 2.258 2.123 2.030 1.961 1.907 1.864 1.828 1.799
34 2.859 2.466 2.252 2.118 2.024 1.955 1.901 1.858 1.822 1.793
35 2.855 2.461 2.247 2.113 2.019 1.950 1.896 1.852 1.817 1.787
36 2.850 2.456 2.243 2.108 2.014 1.945 1.891 1.847 1.811 1.781
37 2.846 2.452 2.238 2.103 2.009 1.940 1.886 1.842 1.806 1.776
38 2.842 2.448 2.234 2.099 2.005 1.935 1.881 1.838 1.802 1.772
39 2.839 2.444 2.230 2.095 2.001 1.931 1.877 1.833 1.797 1.767
40 2.835 2.440 2.226 2.091 1.997 1.927 1.873 1.829 1.793 1.763
41 2.832 2.437 2.222 2.087 1.993 1.923 1.869 1.825 1.789 1.759
42 2.829 2.434 2.219 2.084 1.989 1.919 1.865 1.821 1.785 1.755
43 2.826 2.430 2.216 2.080 1.986 1.916 1.861 1.817 1.781 1.751
44 2.823 2.427 2.213 2.077 1.983 1.913 1.858 1.814 1.778 1.747
45 2.820 2.425 2.210 2.074 1.980 1.909 1.855 1.811 1.774 1.744
46 2.818 2.422 2.207 2.071 1.977 1.906 1.852 1.808 1.771 1.741
47 2.815 2.419 2.204 2.068 1.974 1.903 1.849 1.805 1.768 1.738
48 2.813 2.417 2.202 2.066 1.971 1.901 1.846 1.802 1.765 1.735
49 2.811 2.414 2.199 2.063 1.968 1.898 1.843 1.799 1.763 1.732
50 2.809 2.412 2.197 2.061 1.966 1.895 1.840 1.796 1.760 1.729
51 2.807 2.410 2.194 2.058 1.964 1.893 1.838 1.794 1.757 1.727
52 2.805 2.408 2.192 2.056 1.961 1.891 1.836 1.791 1.755 1.724
53 2.803 2.406 2.190 2.054 1.959 1.888 1.833 1.789 1.752 1.722
54 2.801 2.404 2.188 2.052 1.957 1.886 1.831 1.787 1.750 1.719
55 2.799 2.402 2.186 2.050 1.955 1.884 1.829 1.785 1.748 1.717
56 2.797 2.400 2.184 2.048 1.953 1.882 1.827 1.782 1.746 1.715
57 2.796 2.398 2.182 2.046 1.951 1.880 1.825 1.780 1.744 1.713
58 2.794 2.396 2.181 2.044 1.949 1.878 1.823 1.779 1.742 1.711
59 2.793 2.395 2.179 2.043 1.947 1.876 1.821 1.777 1.740 1.709
60 2.791 2.393 2.177 2.041 1.946 1.875 1.819 1.775 1.738 1.707
61 2.790 2.392 2.176 2.039 1.944 1.873 1.818 1.773 1.736 1.705
62 2.788 2.390 2.174 2.038 1.942 1.871 1.816 1.771 1.735 1.703
63 2.787 2.389 2.173 2.036 1.941 1.870 1.814 1.770 1.733 1.702
64 2.786 2.387 2.171 2.035 1.939 1.868 1.813 1.768 1.731 1.700

65 2.784 2.386 2.170 2.033 1.938 1.867 1.811 1.767 1.730 1.699
66 2.783 2.385 2.169 2.032 1.937 1.865 1.810 1.765 1.728 1.697
67 2.782 2.384 2.167 2.031 1.935 1.864 1.808 1.764 1.727 1.696
68 2.781 2.382 2.166 2.029 1.934 1.863 1.807 1.762 1.725 1.694
69 2.780 2.381 2.165 2.028 1.933 1.861 1.806 1.761 1.724 1.693
70 2.779 2.380 2.164 2.027 1.931 1.860 1.804 1.760 1.723 1.691
71 2.778 2.379 2.163 2.026 1.930 1.859 1.803 1.758 1.721 1.690
72 2.777 2.378 2.161 2.025 1.929 1.858 1.802 1.757 1.720 1.689
73 2.776 2.377 2.160 2.024 1.928 1.856 1.801 1.756 1.719 1.687
74 2.775 2.376 2.159 2.022 1.927 1.855 1.800 1.755 1.718 1.686
75 2.774 2.375 2.158 2.021 1.926 1.854 1.798 1.754 1.716 1.685
76 2.773 2.374 2.157 2.020 1.925 1.853 1.797 1.752 1.715 1.684
77 2.772 2.373 2.156 2.019 1.924 1.852 1.796 1.751 1.714 1.683
78 2.771 2.372 2.155 2.018 1.923 1.851 1.795 1.750 1.713 1.682
79 2.770 2.371 2.154 2.017 1.922 1.850 1.794 1.749 1.712 1.681
80 2.769 2.370 2.154 2.016 1.921 1.849 1.793 1.748 1.711 1.680
81 2.769 2.369 2.153 2.016 1.920 1.848 1.792 1.747 1.710 1.679
82 2.768 2.368 2.152 2.015 1.919 1.847 1.791 1.746 1.709 1.678
83 2.767 2.368 2.151 2.014 1.918 1.846 1.790 1.745 1.708 1.677
84 2.766 2.367 2.150 2.013 1.917 1.845 1.790 1.744 1.707 1.676
85 2.765 2.366 2.149 2.012 1.916 1.845 1.789 1.744 1.706 1.675
86 2.765 2.365 2.149 2.011 1.915 1.844 1.788 1.743 1.705 1.674
87 2.764 2.365 2.148 2.011 1.915 1.843 1.787 1.742 1.705 1.673
88 2.763 2.364 2.147 2.010 1.914 1.842 1.786 1.741 1.704 1.672
89 2.763 2.363 2.146 2.009 1.913 1.841 1.785 1.740 1.703 1.671
90 2.762 2.363 2.146 2.008 1.912 1.841 1.785 1.739 1.702 1.670
91 2.761 2.362 2.145 2.008 1.912 1.840 1.784 1.739 1.701 1.670
92 2.761 2.361 2.144 2.007 1.911 1.839 1.783 1.738 1.701 1.669
93 2.760 2.361 2.144 2.006 1.910 1.838 1.782 1.737 1.700 1.668
94 2.760 2.360 2.143 2.006 1.910 1.838 1.782 1.736 1.699 1.667
95 2.759 2.359 2.142 2.005 1.909 1.837 1.781 1.736 1.698 1.667
96 2.759 2.359 2.142 2.004 1.908 1.836 1.780 1.735 1.698 1.666
97 2.758 2.358 2.141 2.004 1.908 1.836 1.780 1.734 1.697 1.665
98 2.757 2.358 2.141 2.003 1.907 1.835 1.779 1.734 1.696 1.665
99 2.757 2.357 2.140 2.003 1.906 1.835 1.778 1.733 1.696 1.664
100 2.756 2.356 2.139 2.002 1.906 1.834 1.778 1.732 1.695 1.663
ν1 11 12 13 14 15 16 17 18 19 20
ν2
1 60.473 60.705 60.903 61.073 61.220 61.350 61.464 61.566 61.658 61.740
2 9.401 9.408 9.415 9.420 9.425 9.429 9.433 9.436 9.439 9.441
3 5.222 5.216 5.210 5.205 5.200 5.196 5.193 5.190 5.187 5.184
4 3.907 3.896 3.886 3.878 3.870 3.864 3.858 3.853 3.849 3.844
5 3.282 3.268 3.257 3.247 3.238 3.230 3.223 3.217 3.212 3.207
6 2.920 2.905 2.892 2.881 2.871 2.863 2.855 2.848 2.842 2.836
7 2.684 2.668 2.654 2.643 2.632 2.623 2.615 2.607 2.601 2.595
8 2.519 2.502 2.488 2.475 2.464 2.455 2.446 2.438 2.431 2.425
9 2.396 2.379 2.364 2.351 2.340 2.329 2.320 2.312 2.305 2.298
10 2.302 2.284 2.269 2.255 2.244 2.233 2.224 2.215 2.208 2.201
11 2.227 2.209 2.193 2.179 2.167 2.156 2.147 2.138 2.130 2.123
12 2.166 2.147 2.131 2.117 2.105 2.094 2.084 2.075 2.067 2.060
13 2.116 2.097 2.080 2.066 2.053 2.042 2.032 2.023 2.014 2.007
14 2.073 2.054 2.037 2.022 2.010 1.998 1.988 1.978 1.970 1.962
15 2.037 2.017 2.000 1.985 1.972 1.961 1.950 1.941 1.932 1.924
16 2.005 1.985 1.968 1.953 1.940 1.928 1.917 1.908 1.899 1.891
17 1.978 1.958 1.940 1.925 1.912 1.900 1.889 1.879 1.870 1.862
18 1.954 1.933 1.916 1.900 1.887 1.875 1.864 1.854 1.845 1.837
19 1.932 1.912 1.894 1.878 1.865 1.852 1.841 1.831 1.822 1.814
20 1.913 1.892 1.875 1.859 1.845 1.833 1.821 1.811 1.802 1.794
21 1.896 1.875 1.857 1.841 1.827 1.815 1.803 1.793 1.784 1.776
22 1.880 1.859 1.841 1.825 1.811 1.798 1.787 1.777 1.768 1.759
23 1.866 1.845 1.827 1.811 1.796 1.784 1.772 1.762 1.753 1.744
24 1.853 1.832 1.814 1.797 1.783 1.770 1.759 1.748 1.739 1.730
25 1.841 1.820 1.802 1.785 1.771 1.758 1.746 1.736 1.726 1.718

26 1.830 1.809 1.790 1.774 1.760 1.747 1.735 1.724 1.715 1.706
27 1.820 1.799 1.780 1.764 1.749 1.736 1.724 1.714 1.704 1.695
28 1.811 1.790 1.771 1.754 1.740 1.726 1.715 1.704 1.694 1.685
29 1.802 1.781 1.762 1.745 1.731 1.717 1.705 1.695 1.685 1.676
30 1.794 1.773 1.754 1.737 1.722 1.709 1.697 1.686 1.676 1.667
31 1.787 1.765 1.746 1.729 1.714 1.701 1.689 1.678 1.668 1.659
32 1.780 1.758 1.739 1.722 1.707 1.694 1.682 1.671 1.661 1.652
33 1.773 1.751 1.732 1.715 1.700 1.687 1.675 1.664 1.654 1.645
34 1.767 1.745 1.726 1.709 1.694 1.680 1.668 1.657 1.647 1.638
35 1.761 1.739 1.720 1.703 1.688 1.674 1.662 1.651 1.641 1.632
36 1.756 1.734 1.715 1.697 1.682 1.669 1.656 1.645 1.635 1.626
37 1.751 1.729 1.709 1.692 1.677 1.663 1.651 1.640 1.630 1.620
38 1.746 1.724 1.704 1.687 1.672 1.658 1.646 1.635 1.624 1.615
39 1.741 1.719 1.700 1.682 1.667 1.653 1.641 1.630 1.619 1.610
40 1.737 1.715 1.695 1.678 1.662 1.649 1.636 1.625 1.615 1.605
41 1.733 1.710 1.691 1.673 1.658 1.644 1.632 1.620 1.610 1.601
42 1.729 1.706 1.687 1.669 1.654 1.640 1.628 1.616 1.606 1.596
43 1.725 1.703 1.683 1.665 1.650 1.636 1.624 1.612 1.602 1.592
44 1.721 1.699 1.679 1.662 1.646 1.632 1.620 1.608 1.598 1.588
45 1.718 1.695 1.676 1.658 1.643 1.629 1.616 1.605 1.594 1.585
46 1.715 1.692 1.672 1.655 1.639 1.625 1.613 1.601 1.591 1.581
47 1.712 1.689 1.669 1.652 1.636 1.622 1.609 1.598 1.587 1.578
48 1.709 1.686 1.666 1.648 1.633 1.619 1.606 1.594 1.584 1.574
49 1.706 1.683 1.663 1.645 1.630 1.616 1.603 1.591 1.581 1.571
50 1.703 1.680 1.660 1.643 1.627 1.613 1.600 1.588 1.578 1.568
51 1.700 1.677 1.658 1.640 1.624 1.610 1.597 1.586 1.575 1.565
52 1.698 1.675 1.655 1.637 1.621 1.607 1.594 1.583 1.572 1.562
53 1.695 1.672 1.652 1.635 1.619 1.605 1.592 1.580 1.570 1.560
54 1.693 1.670 1.650 1.632 1.616 1.602 1.589 1.578 1.567 1.557
55 1.691 1.668 1.648 1.630 1.614 1.600 1.587 1.575 1.564 1.555
56 1.688 1.666 1.645 1.628 1.612 1.597 1.585 1.573 1.562 1.552
57 1.686 1.663 1.643 1.625 1.610 1.595 1.582 1.571 1.560 1.550
58 1.684 1.661 1.641 1.623 1.607 1.593 1.580 1.568 1.558 1.548
59 1.682 1.659 1.639 1.621 1.605 1.591 1.578 1.566 1.555 1.546
60 1.680 1.657 1.637 1.619 1.603 1.589 1.576 1.564 1.553 1.543
61 1.679 1.656 1.635 1.617 1.601 1.587 1.574 1.562 1.551 1.541
62 1.677 1.654 1.634 1.616 1.600 1.585 1.572 1.560 1.549 1.540
63 1.675 1.652 1.632 1.614 1.598 1.583 1.570 1.558 1.548 1.538
64 1.673 1.650 1.630 1.612 1.596 1.582 1.569 1.557 1.546 1.536
65 1.672 1.649 1.628 1.610 1.594 1.580 1.567 1.555 1.544 1.534
66 1.670 1.647 1.627 1.609 1.593 1.578 1.565 1.553 1.542 1.532
67 1.669 1.646 1.625 1.607 1.591 1.577 1.564 1.552 1.541 1.531
68 1.667 1.644 1.624 1.606 1.590 1.575 1.562 1.550 1.539 1.529
69 1.666 1.643 1.622 1.604 1.588 1.574 1.560 1.548 1.538 1.527
70 1.665 1.641 1.621 1.603 1.587 1.572 1.559 1.547 1.536 1.526
71 1.663 1.640 1.619 1.601 1.585 1.571 1.557 1.545 1.535 1.524
72 1.662 1.639 1.618 1.600 1.584 1.569 1.556 1.544 1.533 1.523
73 1.661 1.637 1.617 1.599 1.583 1.568 1.555 1.543 1.532 1.522
74 1.659 1.636 1.616 1.597 1.581 1.567 1.553 1.541 1.530 1.520
75 1.658 1.635 1.614 1.596 1.580 1.565 1.552 1.540 1.529 1.519
76 1.657 1.634 1.613 1.595 1.579 1.564 1.551 1.539 1.528 1.518
77 1.656 1.632 1.612 1.594 1.578 1.563 1.550 1.538 1.527 1.516
78 1.655 1.631 1.611 1.593 1.576 1.562 1.548 1.536 1.525 1.515
79 1.654 1.630 1.610 1.592 1.575 1.561 1.547 1.535 1.524 1.514
80 1.653 1.629 1.609 1.590 1.574 1.559 1.546 1.534 1.523 1.513
81 1.652 1.628 1.608 1.589 1.573 1.558 1.545 1.533 1.522 1.512
82 1.651 1.627 1.607 1.588 1.572 1.557 1.544 1.532 1.521 1.511
83 1.650 1.626 1.606 1.587 1.571 1.556 1.543 1.531 1.520 1.509
84 1.649 1.625 1.605 1.586 1.570 1.555 1.542 1.530 1.519 1.508
85 1.648 1.624 1.604 1.585 1.569 1.554 1.541 1.529 1.518 1.507
86 1.647 1.623 1.603 1.584 1.568 1.553 1.540 1.528 1.517 1.506
87 1.646 1.622 1.602 1.583 1.567 1.552 1.539 1.527 1.516 1.505
88 1.645 1.622 1.601 1.583 1.566 1.551 1.538 1.526 1.515 1.504
89 1.644 1.621 1.600 1.582 1.565 1.550 1.537 1.525 1.514 1.503
90 1.643 1.620 1.599 1.581 1.564 1.550 1.536 1.524 1.513 1.503
91 1.643 1.619 1.598 1.580 1.564 1.549 1.535 1.523 1.512 1.502
92 1.642 1.618 1.598 1.579 1.563 1.548 1.534 1.522 1.511 1.501
93 1.641 1.617 1.597 1.578 1.562 1.547 1.534 1.521 1.510 1.500

94 1.640 1.617 1.596 1.578 1.561 1.546 1.533 1.521 1.509 1.499
95 1.640 1.616 1.595 1.577 1.560 1.545 1.532 1.520 1.509 1.498
96 1.639 1.615 1.594 1.576 1.560 1.545 1.531 1.519 1.508 1.497
97 1.638 1.614 1.594 1.575 1.559 1.544 1.530 1.518 1.507 1.497
98 1.637 1.614 1.593 1.575 1.558 1.543 1.530 1.517 1.506 1.496
99 1.637 1.613 1.592 1.574 1.557 1.542 1.529 1.517 1.505 1.495
100 1.636 1.612 1.592 1.573 1.557 1.542 1.528 1.516 1.505 1.494
freedom
( F_{.01}(nu_{1},nu_{2}) )
ν1 1 2 3 4 5 6 7 8 9 10
ν2
1 4052.19 4999.52 5403.34 5624.62 5763.65 5858.97 5928.33 5981.10 6022.50 6055.85
2 98.502 99.000 99.166 99.249 99.300 99.333 99.356 99.374 99.388 99.399
3 34.116 30.816 29.457 28.710 28.237 27.911 27.672 27.489 27.345 27.229
4 21.198 18.000 16.694 15.977 15.522 15.207 14.976 14.799 14.659 14.546
5 16.258 13.274 12.060 11.392 10.967 10.672 10.456 10.289 10.158 10.051
6 13.745 10.925 9.780 9.148 8.746 8.466 8.260 8.102 7.976 7.874
7 12.246 9.547 8.451 7.847 7.460 7.191 6.993 6.840 6.719 6.620
8 11.259 8.649 7.591 7.006 6.632 6.371 6.178 6.029 5.911 5.814
9 10.561 8.022 6.992 6.422 6.057 5.802 5.613 5.467 5.351 5.257
10 10.044 7.559 6.552 5.994 5.636 5.386 5.200 5.057 4.942 4.849
11 9.646 7.206 6.217 5.668 5.316 5.069 4.886 4.744 4.632 4.539
12 9.330 6.927 5.953 5.412 5.064 4.821 4.640 4.499 4.388 4.296
13 9.074 6.701 5.739 5.205 4.862 4.620 4.441 4.302 4.191 4.100
14 8.862 6.515 5.564 5.035 4.695 4.456 4.278 4.140 4.030 3.939
15 8.683 6.359 5.417 4.893 4.556 4.318 4.142 4.004 3.895 3.805
16 8.531 6.226 5.292 4.773 4.437 4.202 4.026 3.890 3.780 3.691
17 8.400 6.112 5.185 4.669 4.336 4.102 3.927 3.791 3.682 3.593
18 8.285 6.013 5.092 4.579 4.248 4.015 3.841 3.705 3.597 3.508
19 8.185 5.926 5.010 4.500 4.171 3.939 3.765 3.631 3.523 3.434
20 8.096 5.849 4.938 4.431 4.103 3.871 3.699 3.564 3.457 3.368
21 8.017 5.780 4.874 4.369 4.042 3.812 3.640 3.506 3.398 3.310
22 7.945 5.719 4.817 4.313 3.988 3.758 3.587 3.453 3.346 3.258
23 7.881 5.664 4.765 4.264 3.939 3.710 3.539 3.406 3.299 3.211
24 7.823 5.614 4.718 4.218 3.895 3.667 3.496 3.363 3.256 3.168
25 7.770 5.568 4.675 4.177 3.855 3.627 3.457 3.324 3.217 3.129
26 7.721 5.526 4.637 4.140 3.818 3.591 3.421 3.288 3.182 3.094
27 7.677 5.488 4.601 4.106 3.785 3.558 3.388 3.256 3.149 3.062
28 7.636 5.453 4.568 4.074 3.754 3.528 3.358 3.226 3.120 3.032
29 7.598 5.420 4.538 4.045 3.725 3.499 3.330 3.198 3.092 3.005
30 7.562 5.390 4.510 4.018 3.699 3.473 3.305 3.173 3.067 2.979
31 7.530 5.362 4.484 3.993 3.675 3.449 3.281 3.149 3.043 2.955
32 7.499 5.336 4.459 3.969 3.652 3.427 3.258 3.127 3.021 2.934
33 7.471 5.312 4.437 3.948 3.630 3.406 3.238 3.106 3.000 2.913
34 7.444 5.289 4.416 3.927 3.611 3.386 3.218 3.087 2.981 2.894
35 7.419 5.268 4.396 3.908 3.592 3.368 3.200 3.069 2.963 2.876
36 7.396 5.248 4.377 3.890 3.574 3.351 3.183 3.052 2.946 2.859
37 7.373 5.229 4.360 3.873 3.558 3.334 3.167 3.036 2.930 2.843
38 7.353 5.211 4.343 3.858 3.542 3.319 3.152 3.021 2.915 2.828
39 7.333 5.194 4.327 3.843 3.528 3.305 3.137 3.006 2.901 2.814
40 7.314 5.179 4.313 3.828 3.514 3.291 3.124 2.993 2.888 2.801
41 7.296 5.163 4.299 3.815 3.501 3.278 3.111 2.980 2.875 2.788
42 7.280 5.149 4.285 3.802 3.488 3.266 3.099 2.968 2.863 2.776
43 7.264 5.136 4.273 3.790 3.476 3.254 3.087 2.957 2.851 2.764
44 7.248 5.123 4.261 3.778 3.465 3.243 3.076 2.946 2.840 2.754
45 7.234 5.110 4.249 3.767 3.454 3.232 3.066 2.935 2.830 2.743
46 7.220 5.099 4.238 3.757 3.444 3.222 3.056 2.925 2.820 2.733
47 7.207 5.087 4.228 3.747 3.434 3.213 3.046 2.916 2.811 2.724

48 7.194 5.077 4.218 3.737 3.425 3.204 3.037 2.907 2.802 2.715
49 7.182 5.066 4.208 3.728 3.416 3.195 3.028 2.898 2.793 2.706
50 7.171 5.057 4.199 3.720 3.408 3.186 3.020 2.890 2.785 2.698
51 7.159 5.047 4.191 3.711 3.400 3.178 3.012 2.882 2.777 2.690
52 7.149 5.038 4.182 3.703 3.392 3.171 3.005 2.874 2.769 2.683
53 7.139 5.030 4.174 3.695 3.384 3.163 2.997 2.867 2.762 2.675
54 7.129 5.021 4.167 3.688 3.377 3.156 2.990 2.860 2.755 2.668
55 7.119 5.013 4.159 3.681 3.370 3.149 2.983 2.853 2.748 2.662
56 7.110 5.006 4.152 3.674 3.363 3.143 2.977 2.847 2.742 2.655
57 7.102 4.998 4.145 3.667 3.357 3.136 2.971 2.841 2.736 2.649
58 7.093 4.991 4.138 3.661 3.351 3.130 2.965 2.835 2.730 2.643
59 7.085 4.984 4.132 3.655 3.345 3.124 2.959 2.829 2.724 2.637
60 7.077 4.977 4.126 3.649 3.339 3.119 2.953 2.823 2.718 2.632
61 7.070 4.971 4.120 3.643 3.333 3.113 2.948 2.818 2.713 2.626
62 7.062 4.965 4.114 3.638 3.328 3.108 2.942 2.813 2.708 2.621
63 7.055 4.959 4.109 3.632 3.323 3.103 2.937 2.808 2.703 2.616
64 7.048 4.953 4.103 3.627 3.318 3.098 2.932 2.803 2.698 2.611
65 7.042 4.947 4.098 3.622 3.313 3.093 2.928 2.798 2.693 2.607
66 7.035 4.942 4.093 3.618 3.308 3.088 2.923 2.793 2.689 2.602
67 7.029 4.937 4.088 3.613 3.304 3.084 2.919 2.789 2.684 2.598
68 7.023 4.932 4.083 3.608 3.299 3.080 2.914 2.785 2.680 2.593
69 7.017 4.927 4.079 3.604 3.295 3.075 2.910 2.781 2.676 2.589
70 7.011 4.922 4.074 3.600 3.291 3.071 2.906 2.777 2.672 2.585
71 7.006 4.917 4.070 3.596 3.287 3.067 2.902 2.773 2.668 2.581
72 7.001 4.913 4.066 3.591 3.283 3.063 2.898 2.769 2.664 2.578
73 6.995 4.908 4.062 3.588 3.279 3.060 2.895 2.765 2.660 2.574
74 6.990 4.904 4.058 3.584 3.275 3.056 2.891 2.762 2.657 2.570
75 6.985 4.900 4.054 3.580 3.272 3.052 2.887 2.758 2.653 2.567
76 6.981 4.896 4.050 3.577 3.268 3.049 2.884 2.755 2.650 2.563
77 6.976 4.892 4.047 3.573 3.265 3.046 2.881 2.751 2.647 2.560
78 6.971 4.888 4.043 3.570 3.261 3.042 2.877 2.748 2.644 2.557
79 6.967 4.884 4.040 3.566 3.258 3.039 2.874 2.745 2.640 2.554
80 6.963 4.881 4.036 3.563 3.255 3.036 2.871 2.742 2.637 2.551
81 6.958 4.877 4.033 3.560 3.252 3.033 2.868 2.739 2.634 2.548
82 6.954 4.874 4.030 3.557 3.249 3.030 2.865 2.736 2.632 2.545
83 6.950 4.870 4.027 3.554 3.246 3.027 2.863 2.733 2.629 2.542
84 6.947 4.867 4.024 3.551 3.243 3.025 2.860 2.731 2.626 2.539
85 6.943 4.864 4.021 3.548 3.240 3.022 2.857 2.728 2.623 2.537
86 6.939 4.861 4.018 3.545 3.238 3.019 2.854 2.725 2.621 2.534
87 6.935 4.858 4.015 3.543 3.235 3.017 2.852 2.723 2.618 2.532
88 6.932 4.855 4.012 3.540 3.233 3.014 2.849 2.720 2.616 2.529
89 6.928 4.852 4.010 3.538 3.230 3.012 2.847 2.718 2.613 2.527
90 6.925 4.849 4.007 3.535 3.228 3.009 2.845 2.715 2.611 2.524
91 6.922 4.846 4.004 3.533 3.225 3.007 2.842 2.713 2.609 2.522
92 6.919 4.844 4.002 3.530 3.223 3.004 2.840 2.711 2.606 2.520
93 6.915 4.841 3.999 3.528 3.221 3.002 2.838 2.709 2.604 2.518
94 6.912 4.838 3.997 3.525 3.218 3.000 2.835 2.706 2.602 2.515
95 6.909 4.836 3.995 3.523 3.216 2.998 2.833 2.704 2.600 2.513
96 6.906 4.833 3.992 3.521 3.214 2.996 2.831 2.702 2.598 2.511
97 6.904 4.831 3.990 3.519 3.212 2.994 2.829 2.700 2.596 2.509
98 6.901 4.829 3.988 3.517 3.210 2.992 2.827 2.698 2.594 2.507
99 6.898 4.826 3.986 3.515 3.208 2.990 2.825 2.696 2.592 2.505
100 6.895 4.824 3.984 3.513 3.206 2.988 2.823 2.694 2.590 2.503
ν1 11 12 13 14 15 16 17 18 19 20
ν2
1. 6083.35 6106.35 6125.86 6142.70 6157.28 6170.12 6181.42 6191.52 6200.58 6208.74
2. 99.408 99.416 99.422 99.428 99.432 99.437 99.440 99.444 99.447 99.449
3. 27.133 27.052 26.983 26.924 26.872 26.827 26.787 26.751 26.719 26.690
4. 14.452 14.374 14.307 14.249 14.198 14.154 14.115 14.080 14.048 14.020
5. 9.963 9.888 9.825 9.770 9.722 9.680 9.643 9.610 9.580 9.553
6. 7.790 7.718 7.657 7.605 7.559 7.519 7.483 7.451 7.422 7.396
7. 6.538 6.469 6.410 6.359 6.314 6.275 6.240 6.209 6.181 6.155
8. 5.734 5.667 5.609 5.559 5.515 5.477 5.442 5.412 5.384 5.359

9. 5.178 5.111 5.055 5.005 4.962 4.924 4.890 4.860 4.833 4.808
10. 4.772 4.706 4.650 4.601 4.558 4.520 4.487 4.457 4.430 4.405
11. 4.462 4.397 4.342 4.293 4.251 4.213 4.180 4.150 4.123 4.099
12. 4.220 4.155 4.100 4.052 4.010 3.972 3.939 3.909 3.883 3.858
13. 4.025 3.960 3.905 3.857 3.815 3.778 3.745 3.716 3.689 3.665
14. 3.864 3.800 3.745 3.698 3.656 3.619 3.586 3.556 3.529 3.505
15. 3.730 3.666 3.612 3.564 3.522 3.485 3.452 3.423 3.396 3.372
16. 3.616 3.553 3.498 3.451 3.409 3.372 3.339 3.310 3.283 3.259
17. 3.519 3.455 3.401 3.353 3.312 3.275 3.242 3.212 3.186 3.162
18. 3.434 3.371 3.316 3.269 3.227 3.190 3.158 3.128 3.101 3.077
19. 3.360 3.297 3.242 3.195 3.153 3.116 3.084 3.054 3.027 3.003
20. 3.294 3.231 3.177 3.130 3.088 3.051 3.018 2.989 2.962 2.938
21. 3.236 3.173 3.119 3.072 3.030 2.993 2.960 2.931 2.904 2.880
22. 3.184 3.121 3.067 3.019 2.978 2.941 2.908 2.879 2.852 2.827
23. 3.137 3.074 3.020 2.973 2.931 2.894 2.861 2.832 2.805 2.781
24. 3.094 3.032 2.977 2.930 2.889 2.852 2.819 2.789 2.762 2.738
25. 3.056 2.993 2.939 2.892 2.850 2.813 2.780 2.751 2.724 2.699
26. 3.021 2.958 2.904 2.857 2.815 2.778 2.745 2.715 2.688 2.664
27. 2.988 2.926 2.871 2.824 2.783 2.746 2.713 2.683 2.656 2.632
28. 2.959 2.896 2.842 2.795 2.753 2.716 2.683 2.653 2.626 2.602
29. 2.931 2.868 2.814 2.767 2.726 2.689 2.656 2.626 2.599 2.574
30. 2.906 2.843 2.789 2.742 2.700 2.663 2.630 2.600 2.573 2.549
31. 2.882 2.820 2.765 2.718 2.677 2.640 2.606 2.577 2.550 2.525
32. 2.860 2.798 2.744 2.696 2.655 2.618 2.584 2.555 2.527 2.503
33. 2.840 2.777 2.723 2.676 2.634 2.597 2.564 2.534 2.507 2.482
34. 2.821 2.758 2.704 2.657 2.615 2.578 2.545 2.515 2.488 2.463
35. 2.803 2.740 2.686 2.639 2.597 2.560 2.527 2.497 2.470 2.445
36. 2.786 2.723 2.669 2.622 2.580 2.543 2.510 2.480 2.453 2.428
37. 2.770 2.707 2.653 2.606 2.564 2.527 2.494 2.464 2.437 2.412
38. 2.755 2.692 2.638 2.591 2.549 2.512 2.479 2.449 2.421 2.397
39. 2.741 2.678 2.624 2.577 2.535 2.498 2.465 2.434 2.407 2.382
40. 2.727 2.665 2.611 2.563 2.522 2.484 2.451 2.421 2.394 2.369
41. 2.715 2.652 2.598 2.551 2.509 2.472 2.438 2.408 2.381 2.356
42. 2.703 2.640 2.586 2.539 2.497 2.460 2.426 2.396 2.369 2.344
43. 2.691 2.629 2.575 2.527 2.485 2.448 2.415 2.385 2.357 2.332
44. 2.680 2.618 2.564 2.516 2.475 2.437 2.404 2.374 2.346 2.321
45. 2.670 2.608 2.553 2.506 2.464 2.427 2.393 2.363 2.336 2.311
46. 2.660 2.598 2.544 2.496 2.454 2.417 2.384 2.353 2.326 2.301
47. 2.651 2.588 2.534 2.487 2.445 2.408 2.374 2.344 2.316 2.291
48. 2.642 2.579 2.525 2.478 2.436 2.399 2.365 2.335 2.307 2.282
49. 2.633 2.571 2.517 2.469 2.427 2.390 2.356 2.326 2.299 2.274
50. 2.625 2.562 2.508 2.461 2.419 2.382 2.348 2.318 2.290 2.265
51. 2.617 2.555 2.500 2.453 2.411 2.374 2.340 2.310 2.282 2.257
52. 2.610 2.547 2.493 2.445 2.403 2.366 2.333 2.302 2.275 2.250
53. 2.602 2.540 2.486 2.438 2.396 2.359 2.325 2.295 2.267 2.242
54. 2.595 2.533 2.479 2.431 2.389 2.352 2.318 2.288 2.260 2.235
55. 2.589 2.526 2.472 2.424 2.382 2.345 2.311 2.281 2.253 2.228
56. 2.582 2.520 2.465 2.418 2.376 2.339 2.305 2.275 2.247 2.222
57. 2.576 2.513 2.459 2.412 2.370 2.332 2.299 2.268 2.241 2.215
58. 2.570 2.507 2.453 2.406 2.364 2.326 2.293 2.262 2.235 2.209
59. 2.564 2.502 2.447 2.400 2.358 2.320 2.287 2.256 2.229 2.203
60. 2.559 2.496 2.442 2.394 2.352 2.315 2.281 2.251 2.223 2.198
61. 2.553 2.491 2.436 2.389 2.347 2.309 2.276 2.245 2.218 2.192
62. 2.548 2.486 2.431 2.384 2.342 2.304 2.270 2.240 2.212 2.187
63. 2.543 2.481 2.426 2.379 2.337 2.299 2.265 2.235 2.207 2.182
64. 2.538 2.476 2.421 2.374 2.332 2.294 2.260 2.230 2.202 2.177
65. 2.534 2.471 2.417 2.369 2.327 2.289 2.256 2.225 2.198 2.172
66. 2.529 2.466 2.412 2.365 2.322 2.285 2.251 2.221 2.193 2.168
67. 2.525 2.462 2.408 2.360 2.318 2.280 2.247 2.216 2.188 2.163
68. 2.520 2.458 2.403 2.356 2.314 2.276 2.242 2.212 2.184 2.159
69. 2.516 2.454 2.399 2.352 2.310 2.272 2.238 2.208 2.180 2.155
70. 2.512 2.450 2.395 2.348 2.306 2.268 2.234 2.204 2.176 2.150
71. 2.508 2.446 2.391 2.344 2.302 2.264 2.230 2.200 2.172 2.146
72. 2.504 2.442 2.388 2.340 2.298 2.260 2.226 2.196 2.168 2.143
73. 2.501 2.438 2.384 2.336 2.294 2.256 2.223 2.192 2.164 2.139
74. 2.497 2.435 2.380 2.333 2.290 2.253 2.219 2.188 2.161 2.135
75. 2.494 2.431 2.377 2.329 2.287 2.249 2.215 2.185 2.157 2.132
76. 2.490 2.428 2.373 2.326 2.284 2.246 2.212 2.181 2.154 2.128

77. 2.487 2.424 2.370 2.322 2.280 2.243 2.209 2.178 2.150 2.125
78. 2.484 2.421 2.367 2.319 2.277 2.239 2.206 2.175 2.147 2.122
79. 2.481 2.418 2.364 2.316 2.274 2.236 2.202 2.172 2.144 2.118
80. 2.478 2.415 2.361 2.313 2.271 2.233 2.199 2.169 2.141 2.115
81. 2.475 2.412 2.358 2.310 2.268 2.230 2.196 2.166 2.138 2.112
82. 2.472 2.409 2.355 2.307 2.265 2.227 2.193 2.163 2.135 2.109
83. 2.469 2.406 2.352 2.304 2.262 2.224 2.191 2.160 2.132 2.106
84. 2.466 2.404 2.349 2.302 2.259 2.222 2.188 2.157 2.129 2.104
85. 2.464 2.401 2.347 2.299 2.257 2.219 2.185 2.154 2.126 2.101
86. 2.461 2.398 2.344 2.296 2.254 2.216 2.182 2.152 2.124 2.098
87. 2.459 2.396 2.342 2.294 2.252 2.214 2.180 2.149 2.121 2.096
88. 2.456 2.393 2.339 2.291 2.249 2.211 2.177 2.147 2.119 2.093
89. 2.454 2.391 2.337 2.289 2.247 2.209 2.175 2.144 2.116 2.091
90. 2.451 2.389 2.334 2.286 2.244 2.206 2.172 2.142 2.114 2.088
91. 2.449 2.386 2.332 2.284 2.242 2.204 2.170 2.139 2.111 2.086
92. 2.447 2.384 2.330 2.282 2.240 2.202 2.168 2.137 2.109 2.083
93. 2.444 2.382 2.327 2.280 2.237 2.200 2.166 2.135 2.107 2.081
94. 2.442 2.380 2.325 2.277 2.235 2.197 2.163 2.133 2.105 2.079
95. 2.440 2.378 2.323 2.275 2.233 2.195 2.161 2.130 2.102 2.077
96. 2.438 2.375 2.321 2.273 2.231 2.193 2.159 2.128 2.100 2.075
97. 2.436 2.373 2.319 2.271 2.229 2.191 2.157 2.126 2.098 2.073
98. 2.434 2.371 2.317 2.269 2.227 2.189 2.155 2.124 2.096 2.071
99. 2.432 2.369 2.315 2.267 2.225 2.187 2.153 2.122 2.094 2.069
100. 2.430 2.368 2.313 2.265 2.223 2.185 2.151 2.120 2.092 2.067
Critical Values of the Chi-Square Distribution
How to
Use This
Table
This table contains the critical values of the chi-square
distribution. Because of the lack of symmetry of the chi-square
distribution, separate tables are provided for the upper and
lower tails of the distribution.
A test statistic with ν degrees of freedom is computed from the
data. For upper-tail one-sided tests, the test statistic is
compared with a value from the table of upper-tail critical
values. For two-sided tests, the test statistic is compared with
values from both the table for the upper-tail critical values and
the table for the lower-tail critical values.
The significance level, α, is demonstrated with the graph below
which shows a chi-square distribution with 3 degrees of
freedom for a two-sided test at significance level α = 0.05. If
the test statistic is greater than the upper-tail critical value or
less than the lower-tail critical value, we reject the null
hypothesis. Specific instructions are given below.
Given a specified value of α:
1. For a two-sided test, find the column corresponding to 1-
α/2 in the table for upper-tail critical values and reject the

null hypothesis if the test statistic is greater than the
tabled value. Similarly, find the column corresponding to
α/2 in the table for lower-tail critical values and reject the
null hypothesis if the test statistic is less than the tabled
value.
2. For an upper-tail one-sided test, find the column
corresponding to 1-α in the table containing upper-tail
critical and reject the null hypothesis if the test statistic is
greater than the tabled value.
3. For a lower-tail one-sided test, find the column
corresponding to α in the lower-tail critical values table
and reject the null hypothesis if the computed test
statistic is less than the tabled value.
Upper-tail critical values of chi-square distribution with ν degrees of
freedom
Probability less than the critical value
ν 0.90 0.95 0.975 0.99 0.999
1 2.706 3.841 5.024 6.635 10.828
2 4.605 5.991 7.378 9.210 13.816
3 6.251 7.815 9.348 11.345 16.266
4 7.779 9.488 11.143 13.277 18.467
5 9.236 11.070 12.833 15.086 20.515
6 10.645 12.592 14.449 16.812 22.458
7 12.017 14.067 16.013 18.475 24.322
8 13.362 15.507 17.535 20.090 26.125
9 14.684 16.919 19.023 21.666 27.877
10 15.987 18.307 20.483 23.209 29.588
11 17.275 19.675 21.920 24.725 31.264
12 18.549 21.026 23.337 26.217 32.910
13 19.812 22.362 24.736 27.688 34.528
14 21.064 23.685 26.119 29.141 36.123
15 22.307 24.996 27.488 30.578 37.697
16 23.542 26.296 28.845 32.000 39.252
17 24.769 27.587 30.191 33.409 40.790
18 25.989 28.869 31.526 34.805 42.312
19 27.204 30.144 32.852 36.191 43.820
20 28.412 31.410 34.170 37.566 45.315
21 29.615 32.671 35.479 38.932 46.797
22 30.813 33.924 36.781 40.289 48.268
23 32.007 35.172 38.076 41.638 49.728
24 33.196 36.415 39.364 42.980 51.179
25 34.382 37.652 40.646 44.314 52.620
26 35.563 38.885 41.923 45.642 54.052
27 36.741 40.113 43.195 46.963 55.476
28 37.916 41.337 44.461 48.278 56.892
29 39.087 42.557 45.722 49.588 58.301
30 40.256 43.773 46.979 50.892 59.703
31 41.422 44.985 48.232 52.191 61.098
32 42.585 46.194 49.480 53.486 62.487
33 43.745 47.400 50.725 54.776 63.870
34 44.903 48.602 51.966 56.061 65.247
35 46.059 49.802 53.203 57.342 66.619
36 47.212 50.998 54.437 58.619 67.985
37 48.363 52.192 55.668 59.893 69.347
38 49.513 53.384 56.896 61.162 70.703
39 50.660 54.572 58.120 62.428 72.055
40 51.805 55.758 59.342 63.691 73.402
41 52.949 56.942 60.561 64.950 74.745
42 54.090 58.124 61.777 66.206 76.084
43 55.230 59.304 62.990 67.459 77.419

44 56.369 60.481 64.201 68.710 78.750
45 57.505 61.656 65.410 69.957 80.077
46 58.641 62.830 66.617 71.201 81.400
47 59.774 64.001 67.821 72.443 82.720
48 60.907 65.171 69.023 73.683 84.037
49 62.038 66.339 70.222 74.919 85.351
50 63.167 67.505 71.420 76.154 86.661
51 64.295 68.669 72.616 77.386 87.968
52 65.422 69.832 73.810 78.616 89.272
53 66.548 70.993 75.002 79.843 90.573
54 67.673 72.153 76.192 81.069 91.872
55 68.796 73.311 77.380 82.292 93.168
56 69.919 74.468 78.567 83.513 94.461
57 71.040 75.624 79.752 84.733 95.751
58 72.160 76.778 80.936 85.950 97.039
59 73.279 77.931 82.117 87.166 98.324
60 74.397 79.082 83.298 88.379 99.607
61 75.514 80.232 84.476 89.591 100.888
62 76.630 81.381 85.654 90.802 102.166
63 77.745 82.529 86.830 92.010 103.442
64 78.860 83.675 88.004 93.217 104.716
65 79.973 84.821 89.177 94.422 105.988
66 81.085 85.965 90.349 95.626 107.258
67 82.197 87.108 91.519 96.828 108.526
68 83.308 88.250 92.689 98.028 109.791
69 84.418 89.391 93.856 99.228 111.055
70 85.527 90.531 95.023 100.425 112.317
71 86.635 91.670 96.189 101.621 113.577
72 87.743 92.808 97.353 102.816 114.835
73 88.850 93.945 98.516 104.010 116.092
74 89.956 95.081 99.678 105.202 117.346
75 91.061 96.217 100.839 106.393 118.599
76 92.166 97.351 101.999 107.583 119.850
77 93.270 98.484 103.158 108.771 121.100
78 94.374 99.617 104.316 109.958 122.348
79 95.476 100.749 105.473 111.144 123.594
80 96.578 101.879 106.629 112.329 124.839
81 97.680 103.010 107.783 113.512 126.083
82 98.780 104.139 108.937 114.695 127.324
83 99.880 105.267 110.090 115.876 128.565
84 100.980 106.395 111.242 117.057 129.804
85 102.079 107.522 112.393 118.236 131.041
86 103.177 108.648 113.544 119.414 132.277
87 104.275 109.773 114.693 120.591 133.512
88 105.372 110.898 115.841 121.767 134.746
89 106.469 112.022 116.989 122.942 135.978
90 107.565 113.145 118.136 124.116 137.208
91 108.661 114.268 119.282 125.289 138.438
92 109.756 115.390 120.427 126.462 139.666
93 110.850 116.511 121.571 127.633 140.893
94 111.944 117.632 122.715 128.803 142.119
95 113.038 118.752 123.858 129.973 143.344
96 114.131 119.871 125.000 131.141 144.567
97 115.223 120.990 126.141 132.309 145.789
98 116.315 122.108 127.282 133.476 147.010
99 117.407 123.225 128.422 134.642 148.230
100 118.498 124.342 129.561 135.807 149.449
100 118.498 124.342 129.561 135.807 149.449
Lower-tail critical values of chi-square distribution with ν degrees of
freedom

Probability less than the critical value
ν 0.10 0.05 0.025 0.01 0.001
1. .016 .004 .001 .000 .000
2. .211 .103 .051 .020 .002
3. .584 .352 .216 .115 .024
4. 1.064 .711 .484 .297 .091
5. 1.610 1.145 .831 .554 .210
6. 2.204 1.635 1.237 .872 .381
7. 2.833 2.167 1.690 1.239 .598
8. 3.490 2.733 2.180 1.646 .857
9. 4.168 3.325 2.700 2.088 1.152
10. 4.865 3.940 3.247 2.558 1.479
11. 5.578 4.575 3.816 3.053 1.834
12. 6.304 5.226 4.404 3.571 2.214
13. 7.042 5.892 5.009 4.107 2.617
14. 7.790 6.571 5.629 4.660 3.041
15. 8.547 7.261 6.262 5.229 3.483
16. 9.312 7.962 6.908 5.812 3.942
17. 10.085 8.672 7.564 6.408 4.416
18. 10.865 9.390 8.231 7.015 4.905
19. 11.651 10.117 8.907 7.633 5.407
20. 12.443 10.851 9.591 8.260 5.921
21. 13.240 11.591 10.283 8.897 6.447
22. 14.041 12.338 10.982 9.542 6.983
23. 14.848 13.091 11.689 10.196 7.529
24. 15.659 13.848 12.401 10.856 8.085
25. 16.473 14.611 13.120 11.524 8.649
26. 17.292 15.379 13.844 12.198 9.222
27. 18.114 16.151 14.573 12.879 9.803
28. 18.939 16.928 15.308 13.565 10.391
29. 19.768 17.708 16.047 14.256 10.986
30. 20.599 18.493 16.791 14.953 11.588
31. 21.434 19.281 17.539 15.655 12.196
32. 22.271 20.072 18.291 16.362 12.811
33. 23.110 20.867 19.047 17.074 13.431
34. 23.952 21.664 19.806 17.789 14.057
35. 24.797 22.465 20.569 18.509 14.688
36. 25.643 23.269 21.336 19.233 15.324
37. 26.492 24.075 22.106 19.960 15.965
38. 27.343 24.884 22.878 20.691 16.611
39. 28.196 25.695 23.654 21.426 17.262
40. 29.051 26.509 24.433 22.164 17.916
41. 29.907 27.326 25.215 22.906 18.575
42. 30.765 28.144 25.999 23.650 19.239
43. 31.625 28.965 26.785 24.398 19.906
44. 32.487 29.787 27.575 25.148 20.576
45. 33.350 30.612 28.366 25.901 21.251
46. 34.215 31.439 29.160 26.657 21.929
47. 35.081 32.268 29.956 27.416 22.610
48. 35.949 33.098 30.755 28.177 23.295
49. 36.818 33.930 31.555 28.941 23.983
50. 37.689 34.764 32.357 29.707 24.674
51. 38.560 35.600 33.162 30.475 25.368
52. 39.433 36.437 33.968 31.246 26.065
53. 40.308 37.276 34.776 32.018 26.765
54. 41.183 38.116 35.586 32.793 27.468
55. 42.060 38.958 36.398 33.570 28.173
56. 42.937 39.801 37.212 34.350 28.881
57. 43.816 40.646 38.027 35.131 29.592
58. 44.696 41.492 38.844 35.913 30.305
59. 45.577 42.339 39.662 36.698 31.020
60. 46.459 43.188 40.482 37.485 31.738
61. 47.342 44.038 41.303 38.273 32.459
62. 48.226 44.889 42.126 39.063 33.181

63. 49.111 45.741 42.950 39.855 33.906
64. 49.996 46.595 43.776 40.649 34.633
65. 50.883 47.450 44.603 41.444 35.362
66. 51.770 48.305 45.431 42.240 36.093
67. 52.659 49.162 46.261 43.038 36.826
68. 53.548 50.020 47.092 43.838 37.561
69. 54.438 50.879 47.924 44.639 38.298
70. 55.329 51.739 48.758 45.442 39.036
71. 56.221 52.600 49.592 46.246 39.777
72. 57.113 53.462 50.428 47.051 40.519
73. 58.006 54.325 51.265 47.858 41.264
74. 58.900 55.189 52.103 48.666 42.010
75. 59.795 56.054 52.942 49.475 42.757
76. 60.690 56.920 53.782 50.286 43.507
77. 61.586 57.786 54.623 51.097 44.258
78. 62.483 58.654 55.466 51.910 45.010
79. 63.380 59.522 56.309 52.725 45.764
80. 64.278 60.391 57.153 53.540 46.520
81. 65.176 61.261 57.998 54.357 47.277
82. 66.076 62.132 58.845 55.174 48.036
83. 66.976 63.004 59.692 55.993 48.796
84. 67.876 63.876 60.540 56.813 49.557
85. 68.777 64.749 61.389 57.634 50.320
86. 69.679 65.623 62.239 58.456 51.085
87. 70.581 66.498 63.089 59.279 51.850
88. 71.484 67.373 63.941 60.103 52.617
89. 72.387 68.249 64.793 60.928 53.386
90. 73.291 69.126 65.647 61.754 54.155
91. 74.196 70.003 66.501 62.581 54.926
92. 75.100 70.882 67.356 63.409 55.698
93. 76.006 71.760 68.211 64.238 56.472
94. 76.912 72.640 69.068 65.068 57.246
95. 77.818 73.520 69.925 65.898 58.022
96. 78.725 74.401 70.783 66.730 58.799
97. 79.633 75.282 71.642 67.562 59.577
98. 80.541 76.164 72.501 68.396 60.356
99. 81.449 77.046 73.361 69.230 61.137
100. 82.358 77.929 74.222 70.065 61.918
Critical Values of the t* Distribution
How to
Use This
Table
This table contains upper critical values of the t* distribution
that are appropriate for determining whether or not a
calibration line is in a state of statistical control from
measurements on a check standard at three points in the
calibration interval. A test statistic with ν degrees of freedom is
compared with the critical value. If the absolute value of the
test statistic exceeds the tabled value, the calibration of the
instrument is judged to be out of control.
Upper critical values of t* distribution at significance level 0.05
for testing the output of a linear calibration line at 3 points
ν ν
1 37.544 61 2.455
2 7.582 62 2.454
3 4.826 63 2.453
4 3.941 64 2.452
5 3.518 65 2.451
6 3.274 66 2.450
7 3.115 67 2.449
8 3.004 68 2.448
9 2.923 69 2.447
10 2.860 70 2.446
11 2.811 71 2.445
12 2.770 72 2.445
13 2.737 73 2.444

14 2.709 74 2.443
15 2.685 75 2.442
16 2.665 76 2.441
17 2.647 77 2.441
18 2.631 78 2.440
19 2.617 79 2.439
20 2.605 80 2.439
21 2.594 81 2.438
22 2.584 82 2.437
23 2.574 83 2.437
24 2.566 84 2.436
25 2.558 85 2.436
26 2.551 86 2.435
27 2.545 87 2.435
28 2.539 88 2.434
29 2.534 89 2.434
30 2.528 90 2.433
31 2.524 91 2.432
32 2.519 92 2.432
33 2.515 93 2.431
34 2.511 94 2.431
35 2.507 95 2.431
36 2.504 96 2.430
37 2.501 97 2.430
38 2.498 98 2.429
39 2.495 99 2.429
40 2.492 100 2.428
41 2.489 101 2.428
42 2.487 102 2.428
43 2.484 103 2.427
44 2.482 104 2.427
45 2.480 105 2.426
46 2.478 106 2.426
47 2.476 107 2.426
48 2.474 108 2.425
49 2.472 109 2.425
50 2.470 110 2.425
51 2.469 111 2.424
52 2.467 112 2.424
53 2.466 113 2.424
54 2.464 114 2.423
55 2.463 115 2.423
56 2.461 116 2.423
57 2.460 117 2.422
58 2.459 118 2.422
59 2.457 119 2.422
60 2.456 120 2.422
Critical Values of the Normal PPCC
Distribution
How to
Use This
Table
This table contains the critical values of the normal probability
plot correlation coefficient (PPCC) distribution that are
appropriate for determining whether or not a data set came
from a population with approximately a normal distribution. It
is used in conjuction with a normal probability plot. The test
statistic is the correlation coefficient of the points that make up
a normal probability plot. This test statistic is compared with
the critical value below. If the test statistic is less than the
tabulated value, the null hypothesis that the data came from a
population with a normal distribution is rejected.
For example, suppose a set of 50 data points had a correlation
coefficient of 0.985 from the normal probability plot. At the
5% significance level, the critical value is 0.9761. Since 0.985
is greater than 0.9761, we cannot reject the null hypothesis that
the data came from a population with a normal distribution.
Since perferct normality implies perfect correlation (i.e., a
correlation value of 1), we are only interested in rejecting
normality for correlation values that are too low. That is, this is
a lower one-tailed test.
The values in this table were determined from simulation
studies by Filliben and Devaney.

Critical values of the normal PPCC for testing if data come from
a normal distribution
N 0.01 0.05
3 0.8687 0.8790
4 0.8234 0.8666
5 0.8240 0.8786
6 0.8351 0.8880
7 0.8474 0.8970
8 0.8590 0.9043
9 0.8689 0.9115
10 0.8765 0.9173
11 0.8838 0.9223
12 0.8918 0.9267
13 0.8974 0.9310
14 0.9029 0.9343
15 0.9080 0.9376
16 0.9121 0.9405
17 0.9160 0.9433
18 0.9196 0.9452
19 0.9230 0.9479
20 0.9256 0.9498
21 0.9285 0.9515
22 0.9308 0.9535
23 0.9334 0.9548
24 0.9356 0.9564
25 0.9370 0.9575
26 0.9393 0.9590
27 0.9413 0.9600
28 0.9428 0.9615
29 0.9441 0.9622
30 0.9462 0.9634
31 0.9476 0.9644
32 0.9490 0.9652
33 0.9505 0.9661
34 0.9521 0.9671
35 0.9530 0.9678
36 0.9540 0.9686
37 0.9551 0.9693
38 0.9555 0.9700
39 0.9568 0.9704
40 0.9576 0.9712
41 0.9589 0.9719
42 0.9593 0.9723
43 0.9609 0.9730
44 0.9611 0.9734
45 0.9620 0.9739
46 0.9629 0.9744
47 0.9637 0.9748
48 0.9640 0.9753
49 0.9643 0.9758
50 0.9654 0.9761
55 0.9683 0.9781
60 0.9706 0.9797
65 0.9723 0.9809
70 0.9742 0.9822
75 0.9758 0.9831
80 0.9771 0.9841
85 0.9784 0.9850
90 0.9797 0.9857
95 0.9804 0.9864
100 0.9814 0.9869
110 0.9830 0.9881
120 0.9841 0.9889
130 0.9854 0.9897
140 0.9865 0.9904
150 0.9871 0.9909
160 0.9879 0.9915
170 0.9887 0.9919
180 0.9891 0.9923
190 0.9897 0.9927
200 0.9903 0.9930
210 0.9907 0.9933
220 0.9910 0.9936
230 0.9914 0.9939
240 0.9917 0.9941
250 0.9921 0.9943
260 0.9924 0.9945
270 0.9926 0.9947
280 0.9929 0.9949
290 0.9931 0.9951
300 0.9933 0.9952

310 0.9936 0.9954
320 0.9937 0.9955
330 0.9939 0.9956
340 0.9941 0.9957
350 0.9942 0.9958
360 0.9944 0.9959
370 0.9945 0.9960
380 0.9947 0.9961
390 0.9948 0.9962
400 0.9949 0.9963
410 0.9950 0.9964
420 0.9951 0.9965
430 0.9953 0.9966
440 0.9954 0.9966
450 0.9954 0.9967
460 0.9955 0.9968
470 0.9956 0.9968
480 0.9957 0.9969
490 0.9958 0.9969
500 0.9959 0.9970
525 0.9961 0.9972
550 0.9963 0.9973
575 0.9964 0.9974
600 0.9965 0.9975
625 0.9967 0.9976
650 0.9968 0.9977
675 0.9969 0.9977
700 0.9970 0.9978
725 0.9971 0.9979
750 0.9972 0.9980
775 0.9973 0.9980
800 0.9974 0.9981
825 0.9975 0.9981
850 0.9975 0.9982
875 0.9976 0.9982
900 0.9977 0.9983
925 0.9977 0.9983
950 0.9978 0.9984
975 0.9978 0.9984
1000 0.9979 0.9984
EDA Case Studies
Summary This section presents a series of case studies that demonstrate
the application of EDA methods to specific problems. In some
cases, we have focused on just one EDA technique that
uncovers virtually all there is to know about the data. For other
case studies, we need several EDA techniques, the selection of
which is dictated by the outcome of the previous step in the
analaysis sequence. Note in these case studies how the flow of
the analysis is motivated by the focus on underlying
assumptions and general EDA principles.
Table of
Contents
for Section
4
1. Introduction
2. By Problem Category
Case Studies Introduction
Purpose The purpose of the first eight case studies is to show how
EDA graphics and quantitative measures and tests are
applied to data from scientific processes and to critique
those data with regard to the following assumptions that
typically underlie a measurement process; namely, that the
data behave like:
random drawings
from a fixed distribution
with a fixed location
with a fixed standard deviation

Case studies 9 and 10 show the use of EDA techniques in
distributional modeling and the analysis of a designed
experiment, respectively.
Yi = C + Ei If the above assumptions are satisfied, the process is said to
be statistically "in control" with the core characteristic of
having "predictability". That is, probability statements can
be made about the process, not only in the past, but also in
the future.
An appropriate model for an "in control" process is
Yi = C + Ei
where C is a constant (the "deterministic" or "structural"
component), and where Ei is the error term (or "random"
component).
The constant C is the average value of the process--it is the
primary summary number which shows up on any report.
Although C is (assumed) fixed, it is unknown, and so a
primary analysis objective of the engineer is to arrive at an
estimate of C.
This goal partitions into 4 sub-goals:
1. Is the most common estimator of C, (bar{Y}), the
best estimator for C? What does "best" mean?
2. If (bar{Y}) is best, what is the uncertainty
(s_{bar{Y}}) for (bar{Y}). In particular, is the
usual formula for the uncertainty of (bar{Y}):
( s_{bar{Y}} = s/sqrt{N} )
valid? Here, s is the standard deviation of the data
and N is the sample size.
3. If (bar{Y}) is not the best estimator for C, what is
a better estimator for C (for example, median,
midrange, midmean)?
4. If there is a better estimator, (hat{C}), what is its
uncertainty? That is, what is (s_{hat{C}})?
EDA and the routine checking of underlying assumptions
provides insight into all of the above.
1. Location and variation checks provide information as
to whether C is really constant.
2. Distributional checks indicate whether (bar{Y}) is
the best estimator. Techniques for distributional
checking include histograms, normal probability
plots, and probability plot correlation coefficient
plots.
3. Randomness checks ascertain whether the usual
( s_{bar{Y}} = s/sqrt{N} )
is valid.
4. Distributional tests assist in determining a better
estimator, if needed.
5. Simulator tools (namely bootstrapping) provide
values for the uncertainty of alternative estimators.
Assumptions
not satisfied
If one or more of the above assumptions is not satisfied,
then we use EDA techniques, or some mix of EDA and

classical techniques, to find a more appropriate model for
the data. That is,
Yi = D + Ei
where D is the deterministic part and E is an error
component.
If the data are not random, then we may investigate fitting
some simple time series models to the data. If the constant
location and scale assumptions are violated, we may need
to investigate the measurement process to see if there is an
explanation.
The assumptions on the error term are still quite relevant in
the sense that for an appropriate model the error component
should follow the assumptions. The criterion for validating
the model, or comparing competing models, is framed in
terms of these assumptions.
Multivariable
data
Although the case studies in this chapter utilize univariate
data, the assumptions above are relevant for multivariable
data as well.
If the data are not univariate, then we are trying to find a
model
Yi = F(X1, ..., Xk) + Ei
where F is some function based on one or more variables.
The error component, which is a univariate data set, of a
good model should satisfy the assumptions given above.
The criterion for validating and comparing models is based
on how well the error component follows these
assumptions.
The load cell calibration case study in the process modeling
chapter shows an example of this in the regression context.
First three
case studies
utilize data
with known
characteristics
The first three case studies utilize data that are randomly
generated from the following distributions:
normal distribution with mean 0 and standard
deviation 1
uniform distribution with mean 0 and standard
deviation (sqrt{1/12}) (uniform over the interval
(0,1))
random walk
The other univariate case studies utilize data from scientific
processes. The goal is to determine if
Yi = C + Ei
is a reasonable model. This is done by testing the
underlying assumptions. If the assumptions are satisfied,
then an estimate of C and an estimate of the uncertainty of
C are computed. If the assumptions are not satisfied, we
attempt to find a model where the error component does
satisfy the underlying assumptions.
Graphical
methods that
are applied to
the data
To test the underlying assumptions, each data set is
analyzed using four graphical methods that are particularly
suited for this purpose:
1. run sequence plot which is useful for detecting shifts
of location or scale
2. lag plot which is useful for detecting non-randomness
in the data

3. histogram which is useful for trying to determine the
underlying distribution
4. normal probability plot for deciding whether the data
follow the normal distribution
There are a number of other techniques for addressing the
underlying assumptions. However, the four plots listed
above provide an excellent opportunity for addressing all of
the assumptions on a single page of graphics.
Additional graphical techniques are used in certain case
studies to develop models that do have error components
that satisfy the underlying assumptions.
Quantitative
methods that
are applied to
the data
The normal and uniform random number data sets are also
analyzed with the following quantitative techniques, which
are explained in more detail in an earlier section:
1. Summary statistics which include:
mean
standard deviation
autocorrelation coefficient to test for
randomness
normal and uniform probability plot correlation
coefficients (ppcc) to test for a normal or
uniform distribution, respectively
Wilk-Shapiro test for a normal distribution
2. Linear fit of the data as a function of time to assess
drift (test for fixed location)
3. Bartlett test for fixed variance
4. Autocorrelation plot and coefficient to test for
randomness
5. Runs test to test for lack of randomness
6. Anderson-Darling test for a normal distribution
7. Grubbs test for outliers
8. Summary report
Although the graphical methods applied to the normal and
uniform random numbers are sufficient to assess the
validity of the underlying assumptions, the quantitative
techniques are used to show the different flavor of the
graphical and quantitative approaches.
The remaining case studies intermix one or more of these
quantitative techniques into the analysis where appropriate.
Case Studies
Univariate
Yi = C +
Ei
Normal Random
Numbers
Uniform Random
Numbers
Random Walk
Josephson Junction
Cryothermometry
Beam Deflections Filter
Transmittance

Standard Resistor Heat Flow Meter
1
Reliability
Fatigue Life of
Aluminum Alloy
Specimens
Multi-
Factor
Ceramic Strength
Normal Random Numbers
Normal
Random
Numbers
This example illustrates the univariate analysis of a set of
normal random numbers.
1. Background and Data
2. Graphical Output and Interpretation
3. Quantitative Output and Interpretation
4. Work This Example Yourself
Background and Data
Generation The normal random numbers used in this case study are from a
Rand Corporation publication.
The motivation for studying a set of normal random numbers is
to illustrate the ideal case where all four underlying
assumptions hold.
Software The analyses used in this case study can be generated using
both Dataplot code and R code. The reader can download the
data as a text file.
Data The following is the set of normal random numbers used for
this case study.
-1.2760 -1.2180 -0.4530 -0.3500 0.7230
0.6760 -1.0990 -0.3140 -0.3940 -0.6330
-0.3180 -0.7990 -1.6640 1.3910 0.3820
0.7330 0.6530 0.2190 -0.6810 1.1290
-1.3770 -1.2570 0.4950 -0.1390 -0.8540
0.4280 -1.3220 -0.3150 -0.7320 -1.3480
2.3340 -0.3370 -1.9550 -0.6360 -1.3180
-0.4330 0.5450 0.4280 -0.2970 0.2760
-1.1360 0.6420 3.4360 -1.6670 0.8470
-1.1730 -0.3550 0.0350 0.3590 0.9300
0.4140 -0.0110 0.6660 -1.1320 -0.4100
-1.0770 0.7340 1.4840 -0.3400 0.7890
-0.4940 0.3640 -1.2370 -0.0440 -0.1110
-0.2100 0.9310 0.6160 -0.3770 -0.4330
1.0480 0.0370 0.7590 0.6090 -2.0430
-0.2900 0.4040 -0.5430 0.4860 0.8690
0.3470 2.8160 -0.4640 -0.6320 -1.6140
0.3720 -0.0740 -0.9160 1.3140 -0.0380
0.6370 0.5630 -0.1070 0.1310 -1.8080

-1.1260 0.3790 0.6100 -0.3640 -2.6260
2.1760 0.3930 -0.9240 1.9110 -1.0400
-1.1680 0.4850 0.0760 -0.7690 1.6070
-1.1850 -0.9440 -1.6040 0.1850 -0.2580
-0.3000 -0.5910 -0.5450 0.0180 -0.4850
0.9720 1.7100 2.6820 2.8130 -1.5310
-0.4900 2.0710 1.4440 -1.0920 0.4780
1.2100 0.2940 -0.2480 0.7190 1.1030
1.0900 0.2120 -1.1850 -0.3380 -1.1340
2.6470 0.7770 0.4500 2.2470 1.1510
-1.6760 0.3840 1.1330 1.3930 0.8140
0.3980 0.3180 -0.9280 2.4160 -0.9360
1.0360 0.0240 -0.5600 0.2030 -0.8710
0.8460 -0.6990 -0.3680 0.3440 -0.9260
-0.7970 -1.4040 -1.4720 -0.1180 1.4560
0.6540 -0.9550 2.9070 1.6880 0.7520
-0.4340 0.7460 0.1490 -0.1700 -0.4790
0.5220 0.2310 -0.6190 -0.2650 0.4190
0.5580 -0.5490 0.1920 -0.3340 1.3730
-1.2880 -0.5390 -0.8240 0.2440 -1.0700
0.0100 0.4820 -0.4690 -0.0900 1.1710
1.3720 1.7690 -1.0570 1.6460 0.4810
-0.6000 -0.5920 0.6100 -0.0960 -1.3750
0.8540 -0.5350 1.6070 0.4280 -0.6150
0.3310 -0.3360 -1.1520 0.5330 -0.8330
-0.1480 -1.1440 0.9130 0.6840 1.0430
0.5540 -0.0510 -0.9440 -0.4400 -0.2120
-1.1480 -1.0560 0.6350 -0.3280 -1.2210
0.1180 -2.0450 -1.9770 -1.1330 0.3380
0.3480 0.9700 -0.0170 1.2170 -0.9740
-1.2910 -0.3990 -1.2090 -0.2480 0.4800
0.2840 0.4580 1.3070 -1.6250 -0.6290
-0.5040 -0.0560 -0.1310 0.0480 1.8790
-1.0160 0.3600 -0.1190 2.3310 1.6720
-1.0530 0.8400 -0.2460 0.2370 -1.3120
1.6030 -0.9520 -0.5660 1.6000 0.4650
1.9510 0.1100 0.2510 0.1160 -0.9570
-0.1900 1.4790 -0.9860 1.2490 1.9340
0.0700 -1.3580 -1.2460 -0.9590 -1.2970
-0.7220 0.9250 0.7830 -0.4020 0.6190
1.8260 1.2720 -0.9450 0.4940 0.0500
-1.6960 1.8790 0.0630 0.1320 0.6820
0.5440 -0.4170 -0.6660 -0.1040 -0.2530
-2.5430 -1.3330 1.9870 0.6680 0.3600
1.9270 1.1830 1.2110 1.7650 0.3500
-0.3590 0.1930 -1.0230 -0.2220 -0.6160
-0.0600 -1.3190 0.7850 -0.4300 -0.2980
0.2480 -0.0880 -1.3790 0.2950 -0.1150
-0.6210 -0.6180 0.2090 0.9790 0.9060
-0.0990 -1.3760 1.0470 -0.8720 -2.2000
-1.3840 1.4250 -0.8120 0.7480 -1.0930
-0.4630 -1.2810 -2.5140 0.6750 1.1450
1.0830 -0.6670 -0.2230 -1.5920 -1.2780
0.5030 1.4340 0.2900 0.3970 -0.8370
-0.9730 -0.1200 -1.5940 -0.9960 -1.2440
-0.8570 -0.3710 -0.2160 0.1480 -2.1060
-1.4530 0.6860 -0.0750 -0.2430 -0.1700
-0.1220 1.1070 -1.0390 -0.6360 -0.8600
-0.8950 -1.4580 -0.5390 -0.1590 -0.4200
1.6320 0.5860 -0.4680 -0.3860 -0.3540
0.2030 -1.2340 2.3810 -0.3880 -0.0630
2.0720 -1.4450 -0.6800 0.2240 -0.1200
1.7530 -0.5710 1.2230 -0.1260 0.0340
-0.4350 -0.3750 -0.9850 -0.5850 -0.2030
-0.5560 0.0240 0.1260 1.2500 -0.6150
0.8760 -1.2270 -2.6470 -0.7450 1.7970
-1.2310 0.5470 -0.6340 -0.8360 -0.7190
0.8330 1.2890 -0.0220 -0.4310 0.5820
0.7660 -0.5740 -1.1530 0.5200 -1.0180
-0.8910 0.3320 -0.4530 -1.1270 2.0850
-0.7220 -1.5080 0.4890 -0.4960 -0.0250
0.6440 -0.2330 -0.1530 1.0980 0.7570
-0.0390 -0.4600 0.3930 2.0120 1.3560
0.1050 -0.1710 -0.1100 -1.1450 0.8780
-0.9090 -0.3280 1.0210 -1.6130 1.5600
-1.1920 1.7700 -0.0030 0.3690 0.0520
0.6470 1.0290 1.5260 0.2370 -1.3280
-0.0420 0.5530 0.7700 0.3240 -0.4890
-0.3670 0.3780 0.6010 -1.9960 -0.7380
0.4980 1.0720 1.5670 0.3020 1.1570
-0.7200 1.4030 0.6980 -0.3700 -0.5510

Graphical Output and Interpretation
Goal The goal of this analysis is threefold:
1. Determine if the univariate model:
( Y_{i} = C + E_{i} )
is appropriate and valid.
2. Determine if the typical underlying assumptions for an
"in control" measurement process are valid. These
assumptions are:
1. random drawings;
3. with the distribution having a fixed location;
and
4. the distribution having a fixed scale.
3. Determine if the confidence interval
( bar{Y} pm 2s/sqrt{N} )
is appropriate and valid where s is the standard
deviation of the original data.
4-Plot of
Data
Interpretation The assumptions are addressed by the graphics shown
above:
1. The run sequence plot (upper left) indicates that the
data do not have any significant shifts in location or
scale over time. The run sequence plot does not show
any obvious outliers.
2. The lag plot (upper right) does not indicate any non-
random pattern in the data.
3. The histogram (lower left) shows that the data are
reasonably symmetric, there do not appear to be
significant outliers in the tails, and that it is reasonable
to assume that the data are from approximately a
4. The normal probability plot (lower right) verifies that
an assumption of normality is in fact reasonable.
From the above plots, we conclude that the underlying
assumptions are valid and the data follow approximately a
normal distribution. Therefore, the confidence interval form
given previously is appropriate for quantifying the
uncertainty of the population mean. The numerical values
for this model are given in the Quantitative Output and
Interpretation section.

Individual
Plots
Although it is usually not necessary, the plots can be
generated individually to give more detail.
Run
Sequence
Plot
Lag Plot
Histogram
(with
overlaid
Normal
PDF)
Normal
Probability
Plot

Quantitative Output and Interpretation
Summary
Statistics
As a first step in the analysis, common summary statistics are computed from the
data.
Sample size = 500
Mean = -0.2935997E-02
Median = -0.9300000E-01
Minimum = -0.2647000E+01
Maximum = 0.3436000E+01
Range = 0.6083000E+01
Stan. Dev. = 0.1021041E+01
Location One way to quantify a change in location over time is to fit a straight line to the data
using an index variable as the independent variable in the regression. For our data,
we assume that data are in sequential run order and that the data were collected at
equally spaced time intervals. In our regression, we use the index variable X = 1, 2,
..., N, where N is the number of observations. If there is no significant drift in the
location over time, the slope parameter should be zero.
Coefficient Estimate Stan. Error t-Value
B0 0.699127E-02 0.9155E-01 0.0764
B1 -0.396298E-04 0.3167E-03 -0.1251
Residual Standard Deviation = 1.02205
Residual Degrees of Freedom = 498
The absolute value of the t-value for the slope parameter is smaller than the critical
value of t0.975,498 = 1.96. Thus, we conclude that the slope is not different from
zero at the 0.05 significance level.
Variation One simple way to detect a change in variation is with Bartlett's test, after dividing
the data set into several equal-sized intervals. The choice of the number of intervals
is somewhat arbitrary, although values of four or eight are reasonable. We will
divide our data into four intervals.
H0: σ1
2 = σ2
2 = σ3
2 = σ4
2
Critical value: Χ2
1-α,k-1 = 7.814728
In this case, Bartlett's test indicates that the variances are not significantly different
in the four intervals.
Randomness There are many ways in which data can be non-random. However, most common
forms of non-randomness can be detected with a few simple tests including the lag
plot shown on the previous page.
Another check is an autocorrelation plot that shows the autocorrelations for various
lags. Confidence bands can be plotted at the 95 % and 99 % confidence levels.
Points outside this band indicate statistically significant values (lag 0 is always 1).

The lag 1 autocorrelation, which is generally the one of most interest, is 0.045. The
critical values at the 5% significance level are -0.087 and 0.087. Since 0.045 is
within the critical region, the lag 1 autocorrelation is not statistically significant, so
there is no evidence of non-randomness.
A common test for randomness is the runs test.
Test statistic: Z = -1.0744
Critical value: Z1-α/2 = 1.96
The runs test fails to reject the null hypothesis that the data were produced in a
random manner.
Distributional
Analysis
Probability plots are a graphical test for assessing if a particular distribution
provides an adequate fit to a data set.
A quantitative enhancement to the probability plot is the correlation coefficient of
the points on the probability plot, or PPCC. For this data set the PPCC based on a
normal distribution is 0.996. Since the PPCC is greater than the critical value of
0.987 (this is a tabulated value), the normality assumption is not rejected.
Chi-square and Kolmogorov-Smirnov goodness-of-fit tests are alternative methods
for assessing distributional adequacy. The Wilk-Shapiro and Anderson-Darling tests
can be used to test for normality. The results of the Anderson-Darling test follow.
Adjusted test statistic: A2 = 1.0612
The Anderson-Darling test rejects the normality assumption at the 0.05 significance
level.
Outlier
Analysis
A test for outliers is the Grubbs test.
For this data set, Grubbs' test does not detect any outliers at the 0.05 significance
level.
Model Since the underlying assumptions were validated both graphically and analytically,
we conclude that a reasonable model for the data is:
Yi = C + Ei
where C is the estimated value of the mean, -0.00294. We can express the
uncertainty for C as a 95 % confidence interval (-0.09266, 0.08678).
Univariate
Report
It is sometimes useful and convenient to summarize the above results in a report.
Analysis of 500 normal random numbers
1: Sample Size = 500
2: Location
Mean = -0.00294
Standard Deviation of Mean = 0.045663
95% Confidence Interval for Mean = (-0.09266,0.086779)
Drift with respect to location? = NO
3: Variation
Standard Deviation = 1.021042

95% Confidence Interval for SD = (0.961437,1.088585)
Drift with respect to variation?
(based on Bartletts test on quarters
of the data) = NO
4: Data are Normal?
(as tested by Normal PPCC) = YES
(as tested by Anderson-Darling) = NO
5: Randomness
Autocorrelation = 0.045059
Data are Random?
(as measured by autocorrelation) = YES
6: Statistical Control
(i.e., no drift in location or scale,
data are random, distribution is
fixed, here we are testing only for
fixed normal)
Data Set is in Statistical Control? = YES
7: Outliers?
(as determined by Grubbs' test) = NO
Work This Example Yourself
View
Dataplot
Macro for
this Case
Study
This page allows you to repeat the analysis outlined in the case
study description on the previous page using Dataplot . It is
required that you have already downloaded and installed
Dataplot and configured your browser. to run Dataplot. Output
from each analysis step below will be displayed in one or more
of the Dataplot windows. The four main windows are the
Output window, the Graphics window, the Command History
window, and the data sheet window. Across the top of the main
windows there are menus for executing Dataplot commands.
Across the bottom is a command entry window where
commands can be typed in.
Data Analysis Steps Results and Conclusions
Click on the links below to start Dataplot and run this
case study yourself. Each step may use results from
previous steps, so please be patient. Wait until the
software verifies that the current step is complete before
clicking on the next step.
The links in this column will connect you with more
detailed information about each analysis step from the
case study description.
1. Invoke Dataplot and read data.
1. Read in the data. 1. You have read 1 column of numbers
into Dataplot, variable Y.
2. 4-plot of the data.
1. 4-plot of Y. 1. Based on the 4-plot, there are no shifts
in location or scale, and the data seem to
follow a normal distribution.
3. Generate the individual plots.
1. Generate a run sequence plot.
2. Generate a lag plot.
3. Generate a histogram with an
overlaid normal pdf.
1. The run sequence plot indicates that
there are no shifts of location or
scale.
2. The lag plot does not indicate any
significant patterns (which would
show the data were not random).
3. The histogram indicates that a
normal distribution is a good
distribution for these data.

4. Generate a normal probability
plot.
4. The normal probability plot verifies
that the normal distribution is a
reasonable distribution for these data.
4. Generate summary statistics, quantitative
analysis, and print a univariate report.
1. Generate a table of summary
statistics.
2. Generate the mean, a confidence
interval for the mean, and compute
a linear fit to detect drift in
location.
3. Generate the standard deviation, a
confidence interval for the standard
deviation, and detect drift in variation
by dividing the data into quarters and
computing Barltett's test for equal
4. Check for randomness by generating an
autocorrelation plot and a runs test.
5. Check for normality by computing the
normal probability plot correlation
coefficient.
6. Check for outliers using Grubbs' test.
7. Print a univariate report (this assumes
steps 2 thru 6 have already been run).
1. The summary statistics table displays
25+ statistics.
2. The mean is -0.00294 and a 95%
confidence interval is (-0.093,0.087).
The linear fit indicates no drift in
location since the slope parameter is
statistically not significant.
3. The standard deviation is 1.02 with
a 95% confidence interval of (0.96,1.09).
Bartlett's test indicates no significant
change in variation.
4. The lag 1 autocorrelation is 0.04.
From the autocorrelation plot, this is
within the 95% confidence interval
bands.
5. The normal probability plot correlation
coefficient is 0.996. At the 5% level,
we cannot reject the normality assumption.
6. Grubbs' test detects no outliers at the
5% level.
7. The results are summarized in a
convenient report.
Uniform Random Numbers
Uniform
Random
Numbers
uniform random numbers.
Background and Data
Generation The uniform random numbers used in this case study are from
a Rand Corporation publication.
The motivation for studying a set of uniform random numbers
is to illustrate the effects of a known underlying non-normal
distribution.
Data The following is the set of uniform random numbers used for
this case study.

.100973 .253376 .520135 .863467 .354876
.809590 .911739 .292749 .375420 .480564
.894742 .962480 .524037 .206361 .040200
.822916 .084226 .895319 .645093 .032320
.902560 .159533 .476435 .080336 .990190
.252909 .376707 .153831 .131165 .886767
.439704 .436276 .128079 .997080 .157361
.476403 .236653 .989511 .687712 .171768
.660657 .471734 .072768 .503669 .736170
.658133 .988511 .199291 .310601 .080545
.571824 .063530 .342614 .867990 .743923
.403097 .852697 .760202 .051656 .926866
.574818 .730538 .524718 .623885 .635733
.213505 .325470 .489055 .357548 .284682
.870983 .491256 .737964 .575303 .529647
.783580 .834282 .609352 .034435 .273884
.985201 .776714 .905686 .072210 .940558
.609709 .343350 .500739 .118050 .543139
.808277 .325072 .568248 .294052 .420152
.775678 .834529 .963406 .288980 .831374
.670078 .184754 .061068 .711778 .886854
.020086 .507584 .013676 .667951 .903647
.649329 .609110 .995946 .734887 .517649
.699182 .608928 .937856 .136823 .478341
.654811 .767417 .468509 .505804 .776974
.730395 .718640 .218165 .801243 .563517
.727080 .154531 .822374 .211157 .825314
.385537 .743509 .981777 .402772 .144323
.600210 .455216 .423796 .286026 .699162
.680366 .252291 .483693 .687203 .766211
.399094 .400564 .098932 .050514 .225685
.144642 .756788 .962977 .882254 .382145
.914991 .452368 .479276 .864616 .283554
.947508 .992337 .089200 .803369 .459826
.940368 .587029 .734135 .531403 .334042
.050823 .441048 .194985 .157479 .543297
.926575 .576004 .088122 .222064 .125507
.374211 .100020 .401286 .074697 .966448
.943928 .707258 .636064 .932916 .505344
.844021 .952563 .436517 .708207 .207317
.611969 .044626 .457477 .745192 .433729
.653945 .959342 .582605 .154744 .526695
.270799 .535936 .783848 .823961 .011833
.211594 .945572 .857367 .897543 .875462
.244431 .911904 .259292 .927459 .424811
.621397 .344087 .211686 .848767 .030711
.205925 .701466 .235237 .831773 .208898
.376893 .591416 .262522 .966305 .522825
.044935 .249475 .246338 .244586 .251025
.619627 .933565 .337124 .005499 .765464
.051881 .599611 .963896 .546928 .239123
.287295 .359631 .530726 .898093 .543335
.135462 .779745 .002490 .103393 .598080
.839145 .427268 .428360 .949700 .130212
.489278 .565201 .460588 .523601 .390922
.867728 .144077 .939108 .364770 .617429
.321790 .059787 .379252 .410556 .707007
.867431 .715785 .394118 .692346 .140620
.117452 .041595 .660000 .187439 .242397
.118963 .195654 .143001 .758753 .794041
.921585 .666743 .680684 .962852 .451551
.493819 .476072 .464366 .794543 .590479
.003320 .826695 .948643 .199436 .168108
.513488 .881553 .015403 .545605 .014511
.980862 .482645 .240284 .044499 .908896
.390947 .340735 .441318 .331851 .623241
.941509 .498943 .548581 .886954 .199437
.548730 .809510 .040696 .382707 .742015
.123387 .250162 .529894 .624611 .797524
.914071 .961282 .966986 .102591 .748522
.053900 .387595 .186333 .253798 .145065
.713101 .024674 .054556 .142777 .938919
.740294 .390277 .557322 .709779 .017119
.525275 .802180 .814517 .541784 .561180
.993371 .430533 .512969 .561271 .925536
.040903 .116644 .988352 .079848 .275938
.171539 .099733 .344088 .461233 .483247
.792831 .249647 .100229 .536870 .323075
.754615 .020099 .690749 .413887 .637919
.763558 .404401 .105182 .161501 .848769
.091882 .009732 .825395 .270422 .086304
.833898 .737464 .278580 .900458 .549751
.981506 .549493 .881997 .918707 .615068
.476646 .731895 .020747 .677262 .696229
.064464 .271246 .701841 .361827 .757687
.649020 .971877 .499042 .912272 .953750

.587193 .823431 .540164 .405666 .281310
.030068 .227398 .207145 .329507 .706178
.083586 .991078 .542427 .851366 .158873
.046189 .755331 .223084 .283060 .326481
.333105 .914051 .007893 .326046 .047594
.119018 .538408 .623381 .594136 .285121
.590290 .284666 .879577 .762207 .917575
.374161 .613622 .695026 .390212 .557817
.651483 .483470 .894159 .269400 .397583
.911260 .717646 .489497 .230694 .541374
.775130 .382086 .864299 .016841 .482774
.519081 .398072 .893555 .195023 .717469
.979202 .885521 .029773 .742877 .525165
.344674 .218185 .931393 .278817 .570568
( Y_{i} = C + E_{i} )
assumptions are:
1. random drawings;
and
4-Plot of
Data
above:
scale over time.
3. The histogram shows that the frequencies are
relatively flat across the range of the data. This

suggests that the uniform distribution might provide a
better distributional fit than the normal distribution.
4. The normal probability plot verifies that an
assumption of normality is not reasonable. In this
case, the 4-plot should be followed up by a uniform
probability plot to determine if it provides a better fit
to the data. This is shown below.
assumptions are valid. Therefore, the model Yi = C + Ei is
valid. However, since the data are not normally distributed,
using the mean as an estimate of C and the confidence
interval cited above for quantifying its uncertainty are not
valid or appropriate.
Individual
Plots
Although it is usually not necessary, the plots can be
Run
Sequence
Plot
Lag Plot
Histogram
(with
overlaid
Normal
PDF)
This plot shows that a normal distribution is a poor fit. The
flatness of the histogram suggests that a uniform distribution
might be a better fit.

Histogram
(with
overlaid
Uniform
PDF)
Since the histogram from the 4-plot suggested that the
uniform distribution might be a good fit, we overlay a
uniform distribution on top of the histogram. This indicates a
much better fit than a normal distribution.
Normal
Probability
Plot
As with the histogram, the normal probability plot shows
that the normal distribution does not fit these data well.
Uniform
Probability
Plot
Since the above plots suggested that a uniform distribution
might be appropriate, we generate a uniform probability
plot. This plot shows that the uniform distribution provides
an excellent fit to the data.
Better Model Since the data follow the underlying assumptions, but with a
uniform distribution rather than a normal distribution, we
would still like to characterize C by a typical value plus or
minus a confidence interval. In this case, we would like to
find a location estimator with the smallest variability.
The bootstrap plot is an ideal tool for this purpose. The
following plots show the bootstrap plot, with the

corresponding histogram, for the mean, median, mid-range,
and median absolute deviation.
Bootstrap
Plots
Mid-Range is
Best
From the above histograms, it is obvious that for these data,
the mid-range is far superior to the mean or median as an
estimate for location.
Using the mean, the location estimate is 0.507 and a 95%
confidence interval for the mean is (0.482,0.534). Using the
mid-range, the location estimate is 0.499 and the 95%
confidence interval for the mid-range is (0.497,0.503).
Although the values for the location are similar, the
difference in the uncertainty intervals is quite large.
Note that in the case of a uniform distribution it is known
theoretically that the mid-range is the best linear unbiased
estimator for location. However, in many applications, the
most appropriate estimator will not be known or it will be
mathematically intractable to determine a valid condfidence
interval. The bootstrap provides a method for determining
(and comparing) confidence intervals in these cases.
Summary
Statistics
As a first step in the analysis, common summary statistics are computed for
the data.
Sample size = 500
Mean = 0.5078304
Median = 0.5183650
Minimum = 0.0024900
Maximum = 0.9970800
Range = 0.9945900
Stan. Dev. = 0.2943252
Because the graphs of the data indicate the data may not be normally
distributed, we also compute two other statistics for the data, the normal
PPCC and the uniform PPCC.
Normal PPCC = 0.9771602
Uniform PPCC = 0.9995682
The uniform probability plot correlation coefficient (PPCC) value is larger
than the normal PPCC value. This is evidence that the uniform distribution
fits these data better than does a normal distribution.
Location One way to quantify a change in location over time is to fit a straight line to
the data using an index variable as the independent variable in the
regression. For our data, we assume that data are in sequential run order and
that the data were collected at equally spaced time intervals. In our
regression, we use the index variable X = 1, 2, ..., N, where N is the number
of observations. If there is no significant drift in the location over time, the
slope parameter should be zero.

B0 0.522923 0.2638E-01 19.82
B1 -0.602478E-04 0.9125E-04 -0.66
The t-value of the slope parameter, -0.66, is smaller than the critical value of
t0.975,498 = 1.96. Thus, we conclude that the slope is not different from zero
at the 0.05 significance level.
Variation One simple way to detect a change in variation is with a Bartlett test after
dividing the data set into several equal-sized intervals. However, the Bartlett
test is not robust for non-normality. Since we know this data set is not
approximated well by the normal distribution, we use the alternative Levene
test. In particular, we use the Levene test based on the median rather the
mean. The choice of the number of intervals is somewhat arbitrary, although
values of four or eight are reasonable. We will divide our data into four
intervals.
H0: σ1
2 = σ2
2 = σ3
2 = σ4
2
Critical value: Fα,k-1,N-k = 2.623
Critical region: Reject H0 if W > 2.623
In this case, the Levene test indicates that the variances are not significantly
different in the four intervals.
Randomness There are many ways in which data can be non-random. However, most
common forms of non-randomness can be detected with a few simple tests
including the lag plot shown on the previous page.
Another check is an autocorrelation plot that shows the autocorrelations for
various lags. Confidence bands can be plotted using 95% and 99%
confidence levels. Points outside this band indicate statistically significant
values (lag 0 is always 1).
The lag 1 autocorrelation, which is generally the one of most interest, is
0.03. The critical values at the 5 % significance level are -0.087 and 0.087.
This indicates that the lag 1 autocorrelation is not statistically significant, so
there is no evidence of non-randomness.

The runs test fails to reject the null hypothesis that the data were produced in
a random manner.
Distributional
Analysis
Probability plots are a graphical test of assessing whether a particular
distribution provides an adequate fit to a data set.
A quantitative enhancement to the probability plot is the correlation
coefficient of the points on the probability plot, or PPCC. For this data set
the PPCC based on a normal distribution is 0.977. Since the PPCC is less
than the critical value of 0.987 (this is a tabulated value), the normality
assumption is rejected.
Chi-square and Kolmogorov-Smirnov goodness-of-fit tests are alternative
methods for assessing distributional adequacy. The Wilk-Shapiro and
Anderson-Darling tests can be used to test for normality. The results of the
Anderson-Darling test follow.
The Anderson-Darling test rejects the normality assumption because the
value of the test statistic, 5.765, is larger than the critical value of 0.787 at
the 0.05 significance level.
Model Based on the graphical and quantitative analysis, we use the model
Yi = C + Ei
where C is estimated by the mid-range and the uncertainty interval for C is
based on a bootstrap analysis. Specifically,
C = 0.499
95% confidence limit for C = (0.497,0.503)
Univariate
Report
It is sometimes useful and convenient to summarize the above results in a
report.
Analysis for 500 uniform random numbers
2: Location
Mean = 0.50783
95% Confidence Interval for Mean = (0.48197,0.533692)
3: Variation
(based on Levene's test on quarters
of the data) = NO
4: Distribution
Normal Anderson-Darling = 5.7198390
Data are Normal?
(as tested by Normal PPCC) = NO
Uniform PPCC = 0.9995683
Uniform Anderson-Darling = 0.9082221
Data are Uniform?
(as tested by Uniform PPCC) = YES
(as tested by Anderson-Darling) = YES
5: Randomness
Autocorrelation = -0.03099
Data are Random?
(as measured by autocorrelation) = YES

data is random, distribution is
fixed uniform)
View
Dataplot
Macro for
this Case
Study
in location or scale, and the data do not
seem to follow a normal distribution.
overlaid uniform pdf.
plot.
6. Generate a uniform probability
plot.
scale.
normal distribution is not a good
uniform distribution is a good
that the normal distribution is not a
6. The uniform probability plot verifies
that the uniform distribution is a
4. Generate the bootstrap plot.
1. Generate a bootstrap plot. 1. The bootstrap plot clearly shows
the superiority of the mid-range

over the mean and median as the
location estimator of choice for
this problem.
statistics.
location.
computing Barltetts test for equal
coefficient.
25+ statistics.
2. The mean is 0.5078 and a 95%
confidence interval is (0.482,0.534).
statistically not significant.
Levene's test indicates no significant
drift in variation.
4. The lag 1 autocorrelation is -0.03.
within the 95% confidence interval
bands.
5. The uniform probability plot correlation
coefficient is 0.9995. This indicates that
the uniform distribution is a good fit.
convenient report.
Random Walk
Random
Walk
numbers derived from a random walk.
2. Test Underlying Assumptions
3. Develop Better Model
4. Validate New Model
Background and Data
Generation A random walk can be generated from a set of uniform random
numbers by the formula:
( R_{i} = sum_{j=1}^{i}(U_{j} - 0.5) )
where U is a set of uniform random numbers.
The motivation for studying a set of random walk data is to
illustrate the effects of a known underlying autocorrelation
structure (i.e., non-randomness) in the data.

Data The following is the set of random walk numbers used for this
case study.
-0.399027
-0.645651
-0.625516
-0.262049
-0.407173
-0.097583
0.314156
0.106905
-0.017675
-0.037111
0.357631
0.820111
0.844148
0.550509
0.090709
0.413625
-0.002149
0.393170
0.538263
0.070583
0.473143
0.132676
0.109111
-0.310553
0.179637
-0.067454
-0.190747
-0.536916
-0.905751
-0.518984
-0.579280
-0.643004
-1.014925
-0.517845
-0.860484
-0.884081
-1.147428
-0.657917
-0.470205
-0.798437
-0.637780
-0.666046
-1.093278
-1.089609
-0.853439
-0.695306
-0.206795
-0.507504
-0.696903
-1.116358
-1.044534
-1.481004
-1.638390
-1.270400
-1.026477
-1.123380
-0.770683
-0.510481
-0.958825
-0.531959
-0.457141
-0.226603
-0.201885
-0.078000
0.057733
-0.228762
-0.403292
-0.414237
-0.556689
-0.772007
-0.401024
-0.409768
-0.171804
-0.096501
-0.066854
0.216726
0.551008
0.660360
0.194795
-0.031321
0.453880
0.730594

1.136280
0.708490
1.149048
1.258757
1.102107
1.102846
0.720896
0.764035
1.072312
0.897384
0.965632
0.759684
0.679836
0.955514
1.290043
1.753449
1.542429
1.873803
2.043881
1.728635
1.289703
1.501481
1.888335
1.408421
1.416005
0.929681
1.097632
1.501279
1.650608
1.759718
2.255664
2.490551
2.508200
2.707382
2.816310
3.254166
2.890989
2.869330
3.024141
3.291558
3.260067
3.265871
3.542845
3.773240
3.991880
3.710045
4.011288
4.074805
4.301885
3.956416
4.278790
3.989947
4.315261
4.200798
4.444307
4.926084
4.828856
4.473179
4.573389
4.528605
4.452401
4.238427
4.437589
4.617955
4.370246
4.353939
4.541142
4.807353
4.706447
4.607011
4.205943
3.756457
3.482142
3.126784
3.383572
3.846550
4.228803
4.110948
4.525939
4.478307
4.457582
4.822199
4.605752
5.053262
5.545598
5.134798

5.438168
5.397993
5.838361
5.925389
6.159525
6.190928
6.024970
5.575793
5.516840
5.211826
4.869306
4.912601
5.339177
5.415182
5.003303
4.725367
4.350873
4.225085
3.825104
3.726391
3.301088
3.767535
4.211463
4.418722
4.554786
4.987701
4.993045
5.337067
5.789629
5.726147
5.934353
5.641670
5.753639
5.298265
5.255743
5.500935
5.434664
5.588610
6.047952
6.130557
5.785299
5.811995
5.582793
5.618730
5.902576
6.226537
5.738371
5.449965
5.895537
6.252904
6.650447
7.025909
6.770340
7.182244
6.941536
7.368996
7.293807
7.415205
7.259291
6.970976
7.319743
6.850454
6.556378
6.757845
6.493083
6.824855
6.533753
6.410646
6.502063
6.264585
6.730889
6.753715
6.298649
6.048126
5.794463
5.539049
5.290072
5.409699
5.843266
5.680389
5.185889
5.451353
5.003233
5.102844
5.566741
5.613668

5.352791
5.140087
4.999718
5.030444
5.428537
5.471872
5.107334
5.387078
4.889569
4.492962
4.591042
4.930187
4.857455
4.785815
5.235515
4.865727
4.855005
4.920206
4.880794
4.904395
4.795317
5.163044
4.807122
5.246230
5.111000
5.228429
5.050220
4.610006
4.489258
4.399814
4.606821
4.974252
5.190037
5.084155
5.276501
4.917121
4.534573
4.076168
4.236168
3.923607
3.666004
3.284967
2.980621
2.623622
2.882375
3.176416
3.598001
3.764744
3.945428
4.408280
4.359831
4.353650
4.329722
4.294088
4.588631
4.679111
4.182430
4.509125
4.957768
4.657204
4.325313
4.338800
4.720353
4.235756
4.281361
3.795872
4.276734
4.259379
3.999663
3.544163
3.953058
3.844006
3.684740
3.626058
3.457909
3.581150
4.022659
4.021602
4.070183
4.457137
4.156574
4.205304
4.514814
4.055510
3.938217
4.180232

3.803619
3.553781
3.583675
3.708286
4.005810
4.419880
4.881163
5.348149
4.950740
5.199262
4.753162
4.640757
4.327090
4.080888
3.725953
3.939054
3.463728
3.018284
2.661061
3.099980
3.340274
3.230551
3.287873
3.497652
3.014771
3.040046
3.342226
3.656743
3.698527
3.759707
4.253078
4.183611
4.196580
4.257851
4.683387
4.224290
3.840934
4.329286
3.909134
3.685072
3.356611
2.956344
2.800432
2.761665
2.744913
3.037743
2.787390
2.387619
2.424489
2.247564
2.502179
2.022278
2.213027
2.126914
2.264833
2.528391
2.432792
2.037974
1.699475
2.048244
1.640126
1.149858
1.475253
1.245675
0.831979
1.165877
1.403341
1.181921
1.582379
1.632130
2.113636
2.163129
2.545126
2.963833
3.078901
3.055547
3.287442
2.808189
2.985451
3.181679
2.746144
2.517390
2.719231
2.581058
2.838745
2.987765

3.459642
3.458684
3.870956
4.324706
4.411899
4.735330
4.775494
4.681160
4.462470
3.992538
3.719936
3.427081
3.256588
3.462766
3.046353
3.537430
3.579857
3.931223
3.590096
3.136285
3.391616
3.114700
2.897760
2.724241
2.557346
2.971397
2.479290
2.305336
1.852930
1.471948
1.510356
1.633737
1.727873
1.512994
1.603284
1.387950
1.767527
2.029734
2.447309
2.321470
2.435092
2.630118
2.520330
2.578147
2.729630
2.713100
3.107260
2.876659
2.774242
3.185503
3.403148
3.392646
3.123339
3.164713
3.439843
3.321929
3.686229
3.203069
3.185843
3.204924
3.102996
3.496552
3.191575
3.409044
3.888246
4.273767
3.803540
4.046417
4.071581
3.916256
3.634441
4.065834
3.844651
3.915219
Test Underlying Assumptions

( Y_{i} = C + E_{i} )
2. Determine if the typical underlying assumptions for an "in control"
measurement process are valid. These assumptions are:
1. random drawings;
3. with the distribution having a fixed location; and
is appropriate and valid, with s denoting the standard deviation of
the original data.
4-Plot of Data
Interpretation The assumptions are addressed by the graphics shown above:
1. The run sequence plot (upper left) indicates significant shifts in
location over time.
2. The lag plot (upper right) indicates significant non-randomness in
the data.
3. When the assumptions of randomness and constant location and
scale are not satisfied, the distributional assumptions are not
meaningful. Therefore we do not attempt to make any interpretation
of the histogram (lower left) or the normal probability plot (lower
right).
From the above plots, we conclude that the underlying assumptions are
seriously violated. Therefore the Yi = C + Ei model is not valid.
When the randomness assumption is seriously violated, a time series
model may be appropriate. The lag plot often suggests a reasonable
model. For example, in this case the strongly linear appearance of the lag
plot suggests a model fitting Yi versus Yi-1 might be appropriate. When
the data are non-random, it is helpful to supplement the lag plot with an
autocorrelation plot and a spectral plot. Although in this case the lag plot
is enough to suggest an appropriate model, we provide the autocorrelation
and spectral plots for comparison.
Autocorrelation
Plot
When the lag plot indicates significant non-randomness, it can be helpful
to follow up with a an autocorrelation plot.

This autocorrelation plot shows significant autocorrelation at lags 1
through 100 in a linearly decreasing fashion.
Spectral Plot Another useful plot for non-random data is the spectral plot.
This spectral plot shows a single dominant low frequency peak.
Quantitative
Output
Although the 4-plot above clearly shows the violation of the assumptions,
we supplement the graphical output with some quantitative measures.
Summary
Statistics
As a first step in the analysis, common summary statistics are computed
from the data.
Sample size = 500
Mean = 3.216681
Median = 3.612030
Minimum = -1.638390
Maximum = 7.415205
Range = 9.053595
Stan. Dev. = 2.078675
We also computed the autocorrelation to be 0.987, which is evidence of a
very strong autocorrelation.
Location One way to quantify a change in location over time is to fit a straight line
to the data using an index variable as the independent variable in the
regression. For our data, we assume that data are in sequential run order
and that the data were collected at equally spaced time intervals. In our
regression, we use the index variable X = 1, 2, ..., N, where N is the
number of observations. If there is no significant drift in the location over
time, the slope parameter should be zero.
B0 1.83351 0.1721 10.650
B1 0.552164E-02 0.5953E-03 9.275
The t-value of the slope parameter, 9.275, is larger than the critical value
of t0.975,498 = 1.96. Thus, we conclude that the slope is different from

zero at the 0.05 significance level.
dividing the data set into several equal-sized intervals. However, the
Bartlett test is not robust for non-normality. Since we know this data set is
not approximated well by the normal distribution, we use the alternative
Levene test. In particular, we use the Levene test based on the median
rather the mean. The choice of the number of intervals is somewhat
arbitrary, although values of four or eight are reasonable. We will divide
our data into four intervals.
H0: σ1
2 = σ2
2 = σ3
2 = σ4
2
In this case, the Levene test indicates that the variances are significantly
different in the four intervals since the test statistic of 10.459 is greater
than the 95 % critical value of 2.623. Therefore we conclude that the scale
is not constant.
Randomness Although the lag 1 autocorrelation coefficient above clearly shows the
non-randomness, we show the output from a runs test as well.
The runs test rejects the null hypothesis that the data were produced in a
random manner at the 0.05 significance level.
Distributional
Assumptions
Since the quantitative tests show that the assumptions of randomness and
constant location and scale are not met, the distributional measures will
not be meaningful. Therefore these quantitative tests are omitted.
Develop A Better Model
Lag Plot
Suggests
Better
Model
Since the underlying assumptions did not hold, we need to develop
a better model.
The lag plot showed a distinct linear pattern. Given the definition of
the lag plot, Yi versus Yi-1, a good candidate model is a model of
the form
( Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} )
Fit
Output
The results of a linear fit of this model generated the following
results.
A0 0.050165 0.024171 2.075
A1 0.987087 0.006313 156.350
The slope parameter, A1, has a t value of 156.350 which is
statistically significant. Also, the residual standard deviation is
0.2931. This can be compared to the standard deviation shown in
the summary table, which is 2.078675. That is, the fit to the
autoregressive model has reduced the variability by a factor of 7.

Time
Series
Model
This model is an example of a time series model. More extensive
discussion of time series is given in the Process Monitoring chapter.
Validate New Model
Plot
Predicted
with Original
Data
The first step in verifying the model is to plot the predicted
values from the fit with the original data.
This plot indicates a reasonably good fit.
Test
Underlying
Assumptions
on the
Residuals
In addition to the plot of the predicted values, the residual
standard deviation from the fit also indicates a significant
improvement for the model. The next step is to validate the
underlying assumptions for the error component, or
residuals, from this model.
4-Plot of
Residuals
above:
1. The run sequence plot (upper left) indicates no
significant shifts in location or scale over time.
2. The lag plot (upper right) exhibits a random
appearance.
3. The histogram shows a relatively flat appearance. This
indicates that a uniform probability distribution may
be an appropriate model for the error component (or
residuals).
4. The normal probability plot clearly shows that the
normal distribution is not an appropriate model for the
error component.

A uniform probability plot can be used to further test the
suggestion that a uniform distribution might be a good
model for the error component.
Uniform
Probability
Plot of
Residuals
Since the uniform probability plot is nearly linear, this
verifies that a uniform distribution is a good model for the
error component.
Conclusions Since the residuals from our model satisfy the underlying
assumptions, we conlude that
( Y_{i} = 0.0502 + 0.987*Y_{i-1} + E_{i} )
where the Ei follow a uniform distribution is a good model
for this data set. We could simplify this model to
( Y_{i} = 1.0*Y_{i-1} + E_{i} )
This has the advantage of simplicity (the current point is
simply the previous point plus a uniformly distributed error
term).
Using
Scientific and
Engineering
Knowledge
In this case, the above model makes sense based on our
definition of the random walk. That is, a random walk is the
cumulative sum of uniformly distributed data points. It
makes sense that modeling the current point as the previous
point plus a uniformly distributed error term is about as
good as we can do. Although this case is a bit artificial in
that we knew how the data were constructed, it is common
and desirable to use scientific and engineering knowledge of
the process that generated the data in formulating and testing
models for the data. Quite often, several competing models
will produce nearly equivalent mathematical results. In this
case, selecting the model that best approximates the
scientific understanding of the process is a reasonable
choice.
Time Series
Model
This model is an example of a time series model. More
extensive discussion of time series is given in the Process
Monitoring chapter.
View
Dataplot
Macro for
this Case
Study

Click on the links below to start Dataplot and run
this case study yourself. Each step may use results
from previous steps, so please be patient. Wait
until the software verifies that the current step is
complete before clicking on the next step.
2. Validate assumptions.
1. 4-plot of Y.
statistics.
3. Generate a linear fit to detect
drift in location.
4. Detect drift in variation by
dividing the data into quarters and
computing Levene's test for equal
5. Check for randomness by generating
a runs test.
1. Based on the 4-plot, there are shifts
in location and scale and the data are not
random.
25+ statistics.
3. The linear fit indicates drift in
location since the slope parameter
is statistically significant.
4. Levene's test indicates significant
drift in variation.
5. The runs test indicates significant
non-randomness.
3. Generate the randomness plots.
1. Generate an autocorrelation plot.
2. Generate a spectral plot.
1. The autocorrelation plot shows
significant autocorrelation at lag 1.
2. The spectral plot shows a single dominant
low frequency peak.
4. Fit Yi = A0 + A1*Yi-1 + Ei
and validate.
1. Generate the fit.
2. Plot fitted line with original data.
3. Generate a 4-plot of the residuals
from the fit.
4. Generate a uniform probability plot
of the residuals.
1. The residual standard deviation from the
fit is 0.29 (compared to the standard
deviation of 2.08 from the original
data).
2. The plot of the predicted values with
the original data indicates a good fit.
3. The 4-plot indicates that the assumptions
of constant location and scale are valid.
The lag plot indicates that the data are
random. However, the histogram and normal
probability plot indicate that the uniform
disribution might be a better model for
the residuals than the normal
distribution.
4. The uniform probability plot verifies
that the residuals can be fit by a
uniform distribution.

Josephson Junction Cryothermometry
Josephson
Junction
Cryothermometry
This example illustrates the univariate analysis of
Josephson junction cyrothermometry.
Background and Data
Generation This data set was collected by Bob Soulen of NIST in October,
1971 as a sequence of observations collected equi-spaced in
time from a volt meter to ascertain the process temperature in a
Josephson junction cryothermometry (low temperature)
experiment. The response variable is voltage counts.
Motivation The motivation for studying this data set is to illustrate the case
where there is discreteness in the measurements, but the
underlying assumptions hold. In this case, the discreteness is
due to the data being integers.
Data The following are the data used for this case study.
2899 2898 2898 2900 2898
2901 2899 2901 2900 2898
2898 2898 2898 2900 2898
2897 2899 2897 2899 2899
2900 2897 2900 2900 2899
2898 2898 2899 2899 2899
2899 2899 2898 2899 2899
2899 2902 2899 2900 2898
2899 2899 2899 2899 2899
2899 2900 2899 2900 2898
2901 2900 2899 2899 2899
2899 2899 2900 2899 2898
2898 2898 2900 2896 2897
2899 2899 2900 2898 2900
2901 2898 2899 2901 2900
2898 2900 2899 2899 2897
2899 2898 2899 2899 2898
2899 2897 2899 2899 2897
2899 2897 2899 2897 2897
2899 2897 2898 2898 2899
2897 2898 2897 2899 2899
2898 2898 2897 2898 2895
2897 2898 2898 2896 2898
2898 2897 2896 2898 2898
2897 2897 2898 2898 2896
2898 2898 2896 2899 2898
2898 2898 2899 2899 2898
2898 2899 2899 2899 2900
2900 2901 2899 2898 2898
2900 2899 2898 2901 2897
2898 2898 2900 2899 2899
2898 2898 2899 2898 2901
2900 2897 2897 2898 2898
2900 2898 2899 2898 2898
2898 2896 2895 2898 2898
2898 2898 2897 2897 2895
2897 2897 2900 2898 2896
2897 2898 2898 2899 2898
2897 2898 2898 2896 2900
2899 2898 2896 2898 2896
2896 2896 2897 2897 2896
2897 2897 2896 2898 2896

2898 2896 2897 2896 2897
2897 2898 2897 2896 2895
2898 2896 2896 2898 2896
2898 2898 2897 2897 2898
2897 2899 2896 2897 2899
2900 2898 2898 2897 2898
2899 2899 2900 2900 2900
2900 2899 2899 2899 2898
2900 2901 2899 2898 2900
2901 2901 2900 2899 2898
2901 2899 2901 2900 2901
2898 2900 2900 2898 2900
2900 2898 2899 2901 2900
2899 2899 2900 2900 2899
2900 2901 2899 2898 2898
2899 2896 2898 2897 2898
2898 2897 2897 2897 2898
2897 2899 2900 2899 2897
2898 2900 2900 2898 2898
2899 2900 2898 2900 2900
2898 2900 2898 2898 2898
2898 2898 2899 2898 2900
2897 2899 2898 2899 2898
2897 2900 2901 2899 2898
2898 2901 2898 2899 2897
2899 2897 2896 2898 2898
2899 2900 2896 2897 2897
2898 2899 2899 2898 2898
2897 2897 2898 2897 2897
2898 2898 2898 2896 2895
2898 2898 2898 2896 2898
2898 2898 2897 2897 2899
2896 2900 2897 2897 2898
2896 2897 2898 2898 2898
2897 2897 2898 2899 2897
2898 2899 2897 2900 2896
2899 2897 2898 2897 2900
2899 2900 2897 2897 2898
2897 2899 2899 2898 2897
2901 2900 2898 2901 2899
2900 2899 2898 2900 2900
2899 2898 2897 2900 2898
2898 2897 2899 2898 2900
2899 2898 2899 2897 2900
2898 2902 2897 2898 2899
2899 2899 2898 2897 2898
2897 2898 2899 2900 2900
2899 2898 2899 2900 2899
2900 2899 2899 2899 2899
2899 2898 2899 2899 2900
2902 2899 2900 2900 2901
2899 2901 2899 2899 2902
2898 2898 2898 2898 2899
2899 2900 2900 2900 2898
2899 2899 2900 2899 2900
2899 2900 2898 2898 2898
2900 2898 2899 2900 2899
2899 2900 2898 2898 2899
2899 2899 2899 2898 2898
2897 2898 2899 2897 2897
2901 2898 2897 2898 2899
2898 2897 2899 2898 2897
2898 2898 2897 2898 2899
2899 2899 2899 2900 2899
2899 2897 2898 2899 2900
2898 2897 2901 2899 2901
2898 2899 2901 2900 2900
2899 2900 2900 2900 2900
2901 2900 2901 2899 2897
2900 2900 2901 2899 2898
2900 2899 2899 2900 2899
2900 2899 2900 2899 2901
2900 2900 2899 2899 2898
2899 2900 2898 2899 2899
2901 2898 2898 2900 2899
2899 2898 2897 2898 2897
2899 2899 2899 2898 2898
2897 2898 2899 2897 2897
2899 2898 2898 2899 2899
2901 2899 2899 2899 2897
2900 2896 2898 2898 2900
2897 2899 2897 2896 2898
2897 2898 2899 2896 2899
2901 2898 2898 2896 2897
2899 2897 2898 2899 2898
2898 2898 2898 2898 2898

2899 2900 2899 2901 2898
2899 2899 2898 2900 2898
2899 2899 2901 2900 2901
2899 2901 2899 2901 2899
2900 2902 2899 2898 2899
2900 2899 2900 2900 2901
2900 2899 2901 2901 2899
2898 2901 2897 2898 2901
2900 2902 2899 2900 2898
2900 2899 2900 2899 2899
2899 2898 2900 2898 2899
2899 2899 2899 2898 2900
( Y_{i} = C + E_{i} )
assumptions are:
1. random drawings;
and
4-Plot of
Data
above:
scale over time.
reasonably symmetric, there does not appear to be
significant outliers in the tails, and that it is reasonable

to assume that the data can be fit with a normal
distribution.
4. The normal probability plot (lower right) is difficult to
interpret due to the fact that there are only a few
distinct values with many repeats.
The integer data with only a few distinct values and many
repeats accounts for the discrete appearance of several of the
plots (e.g., the lag plot and the normal probability plot). In
this case, the nature of the data makes the normal probability
plot difficult to interpret, especially since each number is
repeated many times. However, the histogram indicates that
a normal distribution should provide an adequate model for
the data.
assumptions are valid and the data can be reasonably
approximated with a normal distribution. Therefore, the
commonly used uncertainty standard is valid and
appropriate. The numerical values for this model are given
in the Quantitative Output and Interpretation section.
Individual
Plots
Although it is normally not necessary, the plots can be
Run
Sequence
Plot
Lag Plot

Histogram
(with
overlaid
Normal
PDF)
Normal
Probability
Plot
Summary
Statistics
As a first step in the analysis, common summary statistics were computed
from the data.
Sample size = 700
Mean = 2898.562
Median = 2899.000
Minimum = 2895.000
Maximum = 2902.000
Range = 7.000
Stan. Dev. = 1.305
Because of the discrete nature of the data, we also compute the normal
PPCC.
B0 2.898E+03 9.745E-02 29739.288
B1 1.071E-03 2.409e-04 4.445
The slope parameter, B1, has a t value of 4.445 which is statistically
significant (the critical value is 1.96). However, the value of the slope is
1.071E-03. Given that the slope is nearly zero, the assumption of constant
location is not seriously violated even though it is statistically significant.

test is not robust for non-normality. Since the nature of the data (a few
distinct points repeated many times) makes the normality assumption
questionable, we use the alternative Levene test. In particular, we use the
Levene test based on the median rather the mean. The choice of the number
of intervals is somewhat arbitrary, although values of four or eight are
reasonable. We will divide our data into four intervals.
H0: σ1
2 = σ2
2 = σ3
2 = σ4
2
Since the Levene test statistic value of 1.43 is less than the 95 % critical
value of 2.618, we conclude that the variances are not significantly different
in the four intervals.
common forms of non-randomness can be detected with a few simple tests.
The lag plot in the previous section is a simple graphical technique.
various lags. Confidence bands can be plotted at the 95 % and 99 %
0.31. The critical values at the 5 % level of significance are -0.087 and
0.087. This indicates that the lag 1 autocorrelation is statistically significant,
so there is some evidence for non-randomness.
The runs test indicates non-randomness.
Although the runs test and lag 1 autocorrelation indicate some mild non-
randomness, it is not sufficient to reject the Yi = C + Ei model. At least part
of the non-randomness can be explained by the discrete nature of the data.
Distributional
Analysis

coefficient of the points on the probability plot, or PPCC. For this data set
the PPCC based on a normal distribution is 0.975. Since the PPCC is less
than the critical value of 0.987 (this is a tabulated value), the normality
assumption is rejected.
The Anderson-Darling test rejects the normality assumption because the test
statistic, 16.858, is greater than the 95 % critical value 0.787.
Although the data are not strictly normal, the violation of the normality
assumption is not severe enough to conclude that the Yi = C + Ei model is
unreasonable. At least part of the non-normality can be explained by the
discrete nature of the data.
Outlier
Analysis
A test for outliers is the Grubbs test.
Critical value for a one-tailed test: 3.950619
For this data set, Grubbs' test does not detect any outliers at the 0.05
significance level.
Model Although the randomness and normality assumptions were mildly violated,
we conclude that a reasonable model for the data is:
( Y_{i} = 2898.7 + E_{i} )
In addition, a 95 % confidence interval for the mean value is (2898.515,
2898.928).
Univariate
Report
report.
Analysis for Josephson Junction Cryothermometry Data
2: Location
Mean = 2898.562
Drift with respect to location? = YES
(Further analysis indicates that
the drift, while statistically
significant, is not practically
significant)
3: Variation
of the data) = NO
4: Distribution
Normal Anderson-Darling = 16.7634
Data are Normal?
(as tested by Normal PPCC) = NO

5: Randomness
Data are Random?
(as measured by autocorrelation) = NO
fixed normal)
Data Set is in Statistical Control? = NO
Note: Although we have violations of
the assumptions, they are mild enough,
and at least partially explained by the
discrete nature of the data, so we may model
the data as if it were in statistical
control
7: Outliers?
(as determined by Grubbs test) = NO
View
Dataplot
Macro for
this Case
Study
in location or scale. Due to the nature
of the data (a few distinct points with
many repeats), the normality assumption is
questionable.
plot.
scale.
4. The discrete nature of the data masks
the normality or non-normality of the

data somewhat. The plot indicates that
a normal distribution provides a rough
approximation for the data.
statistics.
location.
computing Levene's test for equal
coefficient.
25+ statistics.
The linear fit indicates no meaningful drift
in location since the value of the slope
parameter is near zero.
3. The standard devaition is 1.30 with
Levene's test indicates no significant
drift in variation.
This indicates some mild non-randomness.
we reject the normality assumption.
5% level.
convenient report.
Beam Deflections
Beam
Deflection
This example illustrates the univariate analysis of beam
deflection data.
2. Test Underlying Assumptions
3. Develop a Better Model
4. Validate New Model
Background and Data
Generation This data set was collected by H. S. Lew of NIST in 1969 to
measure steel-concrete beam deflections. The response variable
is the deflection of a beam from the center point.
The motivation for studying this data set is to show how the
underlying assumptions are affected by periodic data.
Data The following are the data used for this case study. The reader
can download the data as a text file.
-213
-564
-35
-15
141
115
-420

-360
203
-338
-431
194
-220
-513
154
-125
-559
92
-21
-579
-52
99
-543
-175
162
-457
-346
204
-300
-474
164
-107
-572
-8
83
-541
-224
180
-420
-374
201
-236
-531
83
27
-564
-112
131
-507
-254
199
-311
-495
143
-46
-579
-90
136
-472
-338
202
-287
-477
169
-124
-568
17
48
-568
-135
162
-430
-422
172
-74
-577
-13
92
-534
-243
194
-355
-465
156
-81
-578
-64
139
-449
-384
193
-198
-538

110
-44
-577
-6
66
-552
-164
161
-460
-344
205
-281
-504
134
-28
-576
-118
156
-437
-381
200
-220
-540
83
11
-568
-160
172
-414
-408
188
-125
-572
-32
139
-492
-321
205
-262
-504
142
-83
-574
0
48
-571
-106
137
-501
-266
190
-391
-406
194
-186
-553
83
-13
-577
-49
103
-515
-280
201
300
-506
131
-45
-578
-80
138
-462
-361
201
-211
-554
32
74
-533
-235
187
-372
-442
182
-147
-566

25
68
-535
-244
194
-351
-463
174
-125
-570
15
72
-550
-190
172
-424
-385
198
-218
-536
96
Test Underlying Assumptions
( Y_{i} = C + E_{i} )
2. Determine if the typical underlying assumptions for an "in control"
measurement process are valid. These assumptions are:
1. random drawings;
3. with the distribution having a fixed location; and
is appropriate and valid where s is the standard deviation of the
original data.
4-Plot of Data
1. The run sequence plot (upper left) indicates that the data do not
have any significant shifts in location or scale over time.
2. The lag plot (upper right) shows that the data are not random. The
lag plot further indicates the presence of a few outliers.
3. When the randomness assumption is thus seriously violated, the
histogram (lower left) and normal probability plot (lower right) are

ignored since determining the distribution of the data is only
meaningful when the data are random.
From the above plots we conclude that the underlying randomness
assumption is not valid. Therefore, the model
Yi = C + Ei
is not appropriate.
We need to develop a better model. Non-random data can frequently be
modeled using time series mehtodology. Specifically, the circular pattern
in the lag plot indicates that a sinusoidal model might be appropriate. The
sinusoidal model will be developed in the next section.
Individual
Plots
The plots can be generated individually for more detail. In this case, only
the run sequence plot and the lag plot are drawn since the distributional
plots are not meaningful.
Run Sequence
Plot
Lag Plot
We have drawn some lines and boxes on the plot to better isolate the
outliers. The following data points appear to be outliers based on the lag
plot.
INDEX Y(i-1) Y(i)
158 -506.00 300.00
157 300.00 201.00
3 -15.00 -35.00
5 115.00 141.00
That is, the third, fifth, 157th, and 158th points appear to be outliers.
Autocorrelation
Plot
When the lag plot indicates significant non-randomness, it can be helpful
to follow up with a an autocorrelation plot.

This autocorrelation plot shows a distinct cyclic pattern. As with the lag
plot, this suggests a sinusoidal model.
Spectral Plot Another useful plot for non-random data is the spectral plot.
This spectral plot shows a single dominant peak at a frequency of 0.3.
This frequency of 0.3 will be used in fitting the sinusoidal model in the
next section.
Quantitative
Results
Although the lag plot, autocorrelation plot, and spectral plot clearly show
the violation of the randomness assumption, we supplement the graphical
output with some quantitative measures.
Summary
Statistics
As a first step in the analysis, summary statistics are computed from the
data.
Sample size = 200
Mean = -177.4350
Median = -162.0000
Minimum = -579.0000
Maximum = 300.0000
Range = 879.0000
Stan. Dev. = 277.3322
Location One way to quantify a change in location over time is to fit a straight line
to the data set using the index variable X = 1, 2, ..., N, with N denoting
the number of observations. If there is no significant drift in the location,
the slope parameter should be zero.
A0 -178.175 39.47 -4.514
A1 0.7366E-02 0.34 0.022
The slope parameter, A1, has a t value of 0.022 which is statistically not
significant. This indicates that the slope can in fact be considered zero.
dividing the data set into several equal-sized intervals. However, the

Bartlett the non-randomness of this data does not allows us to assume
normality, we use the alternative Levene test. In partiuclar, we use the
Levene test based on the median rather the mean. The choice of the
number of intervals is somewhat arbitrary, although values of 4 or 8 are
reasonable.
H0: σ1
2 = σ2
2 = σ3
2 = σ4
2
Sample size: N = 200
In this case, the Levene test indicates that the variances are not
significantly different in the four intervals since the test statistic value,
0.9378, is less than the critical value of 2.651.
Randomness A runs test is used to check for randomness
The absolute value of the test statistic is larger than the critical value at
the 5 % significance level, so we conclude that the data are not random.
Distributional
Assumptions
Since the quantitative tests show that the assumptions of constant scale
and non-randomness are not met, the distributional measures will not be
meaningful. Therefore these quantitative tests are omitted.
Develop a Better Model
Sinusoidal
Model
The lag plot and autocorrelation plot in the previous section strongly
suggested a sinusoidal model might be appropriate. The basic
sinusoidal model is:
( Y_{i} = C + alphasin{(2piomega T_{i} + phi)} + E_i )
where C is constant defining a mean level, α is an amplitude for the
sine function, ω is the frequency, Ti is a time variable, and (phi) is
the phase. This sinusoidal model can be fit using non-linear least
squares.
To obtain a good fit, sinusoidal models require good starting values for
C, the amplitude, and the frequency.
Good Starting
Value for C
A good starting value for C can be obtained by calculating the mean of
the data. If the data show a trend, i.e., the assumption of constant
location is violated, we can replace C with a linear or quadratic least
squares fit. That is, the model becomes
( Y_{i} = (B_0 + B_1*T_{i}) + alphasin{(2piomega T_{i}
+ phi)} + E_i )
or
( Y_{i} = (B_0 + B_1*T_{i} + B2*T^{2}_{i}) +
alphasin{(2piomega T_{i} + phi)} + E_i )
Since our data did not have any meaningful change of location, we can
fit the simpler model with C equal to the mean. From the summary
output in the previous page, the mean is -177.44.

Good Starting
Value for
Frequency
The starting value for the frequency can be obtained from the spectral
plot, which shows the dominant frequency is about 0.3.
Complex
Demodulation
Phase Plot
The complex demodulation phase plot can be used to refine this initial
estimate for the frequency.
For the complex demodulation plot, if the lines slope from left to
right, the frequency should be increased. If the lines slope from right
to left, it should be decreased. A relatively flat (i.e., horizontal) slope
indicates a good frequency. We could generate the demodulation phase
plot for 0.3 and then use trial and error to obtain a better estimate for
the frequency. To simplify this, we generate 16 of these plots on a
single page starting with a frequency of 0.28, increasing in increments
of 0.0025, and stopping at 0.3175.
Interpretation The plots start with lines sloping from left to right but gradually
change to a right to left slope. The relatively flat slope occurs for
frequency 0.3025 (third row, second column). The complex
demodulation phase plot restricts the range from (pi)/2 to -(pi)/2.
This is why the plot appears to show some breaks.
Good Starting
Values for
Amplitude
The complex demodulation amplitude plot is used to find a good
starting value for the amplitude. In addition, this plot indicates
whether or not the amplitude is constant over the entire range of the
data or if it varies. If the plot is essentially flat, i.e., zero slope, then it
is reasonable to assume a constant amplitude in the non-linear model.
However, if the slope varies over the range of the plot, we may need to
adjust the model to be:
( Y_{i} = C + (B_0 + B_1*T_{i})sin{(2piomega T_{i} +
phi)} + E_i )
That is, we replace α with a function of time. A linear fit is specified in
the model above, but this can be replaced with a more elaborate
function if needed.
Complex
Demodulation
Amplitude
Plot
The complex demodulation amplitude plot for this data shows that:

1. The amplitude is fixed at approximately 390.
2. There is a short start-up effect.
3. There is a change in amplitude at around x=160 that should be
investigated for an outlier.
In terms of a non-linear model, the plot indicates that fitting a single
constant for α should be adequate for this data set.
Fit Results Using starting estimates of 0.3025 for the frequency, 390 for the
amplitude, and -177.44 for C, the following parameters were
estimated.
C -178.786 11.02 -16.22
AMP -361.766 26.19 -13.81
FREQ 0.302596 0.1510E-03 2005.00
PHASE 1.46536 0.4909E-01 29.85
Model From the fit results, our proposed model is:
( hat{Y}_i = -178.786 - 361.766[2pi(0.302596)T_i +
1.46536] )
We will evaluate the adequacy of this model in the next section.
Validate New Model
4-Plot of
Residuals
The first step in evaluating the fit is to generate a 4-plot of the
residuals.
1. The run sequence plot (upper left) indicates that the data do not
have any significant shifts in location. There does seem to be
some shifts in scale. A start-up effect was detected previously by
the complex demodulation amplitude plot. There does appear to
be a few outliers.
2. The lag plot (upper right) shows that the data are random. The
outliers also appear in the lag plot.
3. The histogram (lower left) and the normal probability plot
(lower right) do not show any serious non-normality in the
residuals. However, the bend in the left portion of the normal
probability plot shows some cause for concern.
The 4-plot indicates that this fit is reasonably good. However, we will
attempt to improve the fit by removing the outliers.

Fit Results
with Outliers
Removed
The following parameter estimates were obtained after removing three
outliers.
C -178.788 10.57 -16.91
AMP -361.759 25.45 -14.22
FREQ 0.302597 0.1457E-03 2077.00
PHASE 1.46533 0.4715E-01 31.08
New Fit to
Edited Data
The original fit, with a residual standard deviation of 155.84, was:
( hat{Y}_i = -178.786 - 361.766[2pi(0.302596)T_i +
1.46536] )
The new fit, with a residual standard deviation of 148.34, is:
( hat{Y}_i = -178.788 - 361.759[2pi(0.302597)T_i +
1.46533] )
There is minimal change in the parameter estimates and about a 5 %
reduction in the residual standard deviation. In this case, removing the
residuals has a modest benefit in terms of reducing the variability of
the model.
4-Plot for
New Fit
This plot shows that the underlying assumptions are satisfied and
therefore the new fit is a good descriptor of the data.
In this case, it is a judgment call whether to use the fit with or without
the outliers removed.
View
Dataplot
Macro for
this Case
Study

Click on the links below to start Dataplot and run
this case study yourself. Each step may use results
from previous steps, so please be patient. Wait until
the software verifies that the current step is
complete before clicking on the next step.
2. Validate assumptions.
1. 4-plot of Y.
4. Generate an autocorrelation plot.
5. Generate a spectral plot.
statistics.
7. Generate a linear fit to detect
drift in location.
8. Detect drift in variation by
dividing the data into quarters and
computing Levene's test statistic for
equal standard deviations.
9. Check for randomness by generating
a runs test.
1. Based on the 4-plot, there are no
obvious shifts in location and scale,
but the data are not random.
2. Based on the run sequence plot, there
are no obvious shifts in location and
scale.
3. Based on the lag plot, the data
are not random.
4. The autocorrelation plot shows
significant autocorrelation at lag 1.
5. The spectral plot shows a single dominant
low frequency peak.
25+ statistics.
7. The linear fit indicates no drift in
is not statistically significant.
8. Levene's test indicates no
significant drift in variation.
9. The runs test indicates significant
non-randomness.
3. Fit
Yi = C + A*SIN(2*PI*omega*ti+phi).
1. Generate a complex demodulation
phase plot.
2. Generate a complex demodulation
amplitude plot.
3. Fit the non-linear model.
1. Complex demodulation phase plot
indicates a starting frequency
of 0.3025.
2. Complex demodulation amplitude
plot indicates an amplitude of
390 (but there is a short start-up
effect).
3. Non-linear fit generates final
parameter estimates. The
residual standard deviation from
the fit is 155.85 (compared to the
standard deviation of 277.73 from

the original data).
4. Validate fit.
from the fit.
2. Generate a nonlinear fit with
outliers removed.
from the fit with the outliers
removed.
1. The 4-plot indicates that the assumptions
of constant location and scale are valid.
The lag plot indicates that the data are
random. The histogram and normal
probability plot indicate that the residuals
that the normality assumption for the
residuals are not seriously violated,
although there is a bend on the probablity
plot that warrants attention.
2. The fit after removing 3 outliers shows
some marginal improvement in the model
(a 5% reduction in the residual standard
deviation).
3. The 4-plot of the model fit after
3 outliers removed shows marginal
improvement in satisfying model
assumptions.
Filter Transmittance
Filter
Transmittance
This example illustrates the univariate analysis of filter
transmittance data.
Background and Data
Generation This data set was collected by NIST chemist Radu
Mavrodineaunu in the 1970's from an automatic data
acquisition system for a filter transmittance experiment. The
response variable is transmittance.
The motivation for studying this data set is to show how the
underlying autocorrelation structure in a relatively small data
set helped the scientist detect problems with his automatic data
acquisition system.
2.00180
2.00170
2.00180
2.00190
2.00180
2.00170
2.00150
2.00140
2.00150
2.00150
2.00170
2.00180
2.00180

2.00190
2.00190
2.00210
2.00200
2.00160
2.00140
2.00130
2.00130
2.00150
2.00150
2.00160
2.00150
2.00140
2.00130
2.00140
2.00150
2.00140
2.00150
2.00160
2.00150
2.00160
2.00190
2.00200
2.00200
2.00210
2.00220
2.00230
2.00240
2.00250
2.00270
2.00260
2.00260
2.00260
2.00270
2.00260
2.00250
2.00240
( Y_{i} = C + E_{i} )
assumptions are:
1. random drawings;
and

4-Plot of
Data
above:
1. The run sequence plot (upper left) indicates a
significant shift in location around x=35.
2. The linear appearance in the lag plot (upper right)
indicates a non-random pattern in the data.
3. Since the lag plot indicates significant non-
randomness, we do not make any interpretation of
either the histogram (lower left) or the normal
probability plot (lower right).
The serious violation of the non-randomness assumption
means that the univariate model
( Y_{i} = C + E_{i} )
is not valid. Given the linear appearance of the lag plot, the
first step might be to consider a model of the type
( Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} )
However, in this case discussions with the scientist revealed
that non-randomness was entirely unexpected. An
examination of the experimental process revealed that the
sampling rate for the automatic data acquisition system was
too fast. That is, the equipment did not have sufficient time
to reset before the next sample started, resulting in the
current measurement being contaminated by the previous
measurement. The solution was to rerun the experiment
allowing more time between samples.
Simple graphical techniques can be quite effective in
revealing unexpected results in the data. When this occurs, it
is important to investigate whether the unexpected result is
due to problems in the experiment and data collection or is
indicative of unexpected underlying structure in the data.
This determination cannot be made on the basis of statistics
alone. The role of the graphical and statistical analysis is to
detect problems or unexpected results in the data. Resolving
the issues requires the knowledge of the scientist or
engineer.
Individual
Plots
Although it is generally unnecessary, the plots can be
generated individually to give more detail. Since the lag plot
indicates significant non-randomness, we omit the
distributional plots.

Run
Sequence
Plot
Lag Plot
Summary
Statistics
from the data.
Sample size = 50
Mean = 2.0019
Median = 2.0018
Minimum = 2.0013
Maximum = 2.0027
Range = 0.0014
Stan. Dev. = 0.0004
B0 2.00138 0.9695E-04 0.2064E+05
B1 0.185E-04 0.3309E-05 5.582
Residual Standard Deviation = 0.3376404E-03
The slope parameter, B1, has a t value of 5.582, which is statistically
significant. Although the estimated slope, 0.185E-04, is nearly zero, the
range of data (2.0013 to 2.0027) is also very small. In this case, we conclude
that there is drift in location, although it is relatively small.
dividing the data set into several equal sized intervals. However, the Bartlett
test is not robust for non-normality. Since the normality assumption is
questionable for these data, we use the alternative Levene test. In particular,

we use the Levene test based on the median rather the mean. The choice of
the number of intervals is somewhat arbitrary, although values of four or
eight are reasonable. We will divide our data into four intervals.
H0: σ1
2 = σ2
2 = σ3
2 = σ4
2
In this case, since the Levene test statistic value of 0.971 is less than the
critical value of 2.806 at the 5 % level, we conclude that there is no evidence
of a change in variation.
The lag plot in the 4-plot in the previous seciton is a simple graphical
technique.
One check is an autocorrelation plot that shows the autocorrelations for
0.93. The critical values at the 5 % level are -0.277 and 0.277. This indicates
that the lag 1 autocorrelation is statistically significant, so there is strong
evidence of non-randomness.
Because the test statistic is outside of the critical region, we reject the null
hypothesis and conclude that the data are not random.
Distributional
Analysis
Since we rejected the randomness assumption, the distributional tests are not
meaningful. Therefore, these quantitative tests are omitted. We also omit
Grubbs' outlier test since it also assumes the data are approximately
normally distributed.
Univariate
Report
report.
Analysis for filter transmittance data
1: Sample Size = 50

2: Location
Mean = 2.001857
3: Variation
Change in variation?
of the data) = NO
4: Distribution
Distributional tests omitted due to
non-randomness of the data
5: Randomness
Lag One Autocorrelation = 0.937998
Data are Random?
normal)
7: Outliers?
(Grubbs' test omitted) = NO
View
Dataplot
Macro for
this Case
Study
software verifies that the current step is complete
before clicking on the next step.
1. Read in the data. >
1. You have read 1 column of numbers
1. 4-plot of Y. 1. Based on the 4-plot, there is a shift
in location and the data are not random.
1. Generate a run sequence plot. 1. The run sequence plot indicates that
there is a shift in location.

2. Generate a lag plot. 2. The strong linear pattern of the lag
plot indicates significant
non-randomness.
statistics.
2. Compute a linear fit based on
quarters of the data to detect
drift in location.
3. Compute Levene's test based on
quarters of the data to detect
changes in variation.
25+ statistics.
2. The linear fit indicates a slight drift in
statistically significant, but small.
3. Levene's test indicates no significant
drift in variation.
This is outside the 95% confidence
interval bands which indicates significant
non-randomness.
convenient report.
Standard Resistor
Standard
Resistor
This example illustrates the univariate analysis of standard
resistor data.
Background and Data
Generation This data set was collected by Ron Dziuba of NIST over a 5-
year period from 1980 to 1985. The response variable is
resistor values.
The motivation for studying this data set is to illustrate data
that violate the assumptions of constant location and scale.
27.8680
27.8929
27.8773
27.8530
27.8876
27.8725
27.8743
27.8879
27.8728
27.8746
27.8863
27.8716
27.8818
27.8872
27.8885

27.8945
27.8797
27.8627
27.8870
27.8895
27.9138
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27.8852
27.8788
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27.8939
27.8558
27.8814
27.8479
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27.9210
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27.9820

27.9560
27.9670
27.9520
27.9470
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27.9610
27.9437
27.9660
27.9580
27.9660
27.9700
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27.9690
27.9698
27.9616
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27.9964
27.9842
27.9667
27.9610
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27.9616
27.9397
27.9799
28.0086
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28.0096
27.9762
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28.0030
27.9678
28.0146
27.9945
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27.9785
27.9791
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27.9805
27.9782
27.9753
27.9792
27.9704
27.9794
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27.9794
27.9795
27.9881
27.9772
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27.9736
27.9772
27.9960

27.9795
27.9779
27.9829
27.9829
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27.9811
27.9773
27.9778
27.9724
27.9756
27.9699
27.9724
27.9666
27.9666
27.9739
27.9684
27.9861
27.9901
27.9879
27.9865
27.9876
27.9814
27.9842
27.9868
27.9834
27.9892
27.9864
27.9843
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27.9860
27.9872
27.9869
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27.9843
27.9802
27.9863
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27.9881
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27.9888
27.9841
27.9863
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27.9878
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27.9997
28.0006
27.9999
28.0004
28.0020
28.0029
28.0008
28.0040
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28.0065
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28.0073
28.0017
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28.0036
28.0055
28.0007
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28.0011
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28.0092

28.0092
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28.0111
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28.0137
28.0236
28.0171
28.0224
28.0184
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28.0190
28.0204
28.0170
28.0183
28.0201
28.0182
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28.0127
28.0211
28.0057
28.0180
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28.0185
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28.0213
28.0216
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28.0162
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28.0179
28.0215
28.0194
28.0115
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28.0202
28.0240
28.0198
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28.0101
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28.0025
28.0097
28.0066
28.0072

28.0066
28.0068
28.0067
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28.0088
28.0091
28.0091
28.0115
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28.0139
28.0095
28.0115
28.0101
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28.0168
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28.0219
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28.0169
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28.0165
28.0144
28.0182
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28.0155
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28.0204
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28.0185
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28.0290
28.0240
28.0302
28.0243
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28.0287
28.0301
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28.0308
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28.0358
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28.0308
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28.0411
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28.0361
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28.0359
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28.0371
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28.0376
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28.0345
28.0333

28.0429
28.0379
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28.0177
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28.0484
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28.0490
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28.0494
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28.0443
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28.0566
28.0585

28.0486
28.0427
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28.0616
28.0298
28.0726
28.0695
28.0629
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28.0590
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28.0696
28.0581
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28.0572
28.0529
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28.0432
28.0211
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28.0619
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28.0474
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28.0558
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28.0444
28.0429
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28.0610
28.0461
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28.0734
28.0565
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28.0581
28.0519
28.0625
28.0583
28.0645
28.0642
28.0535
28.0510
28.0542
28.0677
28.0416
28.0676
28.0596

28.0635
28.0558
28.0623
28.0718
28.0585
28.0552
28.0684
28.0646
28.0590
28.0465
28.0594
28.0303
28.0533
28.0561
28.0585
28.0497
28.0582
28.0507
28.0562
28.0715
28.0468
28.0411
28.0587
28.0456
28.0705
28.0534
28.0558
28.0536
28.0552
28.0461
28.0598
28.0598
28.0650
28.0423
28.0442
28.0449
28.0660
28.0506
28.0655
28.0512
28.0407
28.0475
28.0411
28.0512
28.1036
28.0641
28.0572
28.0700
28.0577
28.0637
28.0534
28.0461
28.0701
28.0631
28.0575
28.0444
28.0592
28.0684
28.0593
28.0677
28.0512
28.0644
28.0660
28.0542
28.0768
28.0515
28.0579
28.0538
28.0526
28.0833
28.0637
28.0529
28.0535
28.0561
28.0736
28.0635
28.0600
28.0520
28.0695
28.0608
28.0608
28.0590
28.0290
28.0939
28.0618
28.0551

28.0757
28.0698
28.0717
28.0529
28.0644
28.0613
28.0759
28.0745
28.0736
28.0611
28.0732
28.0782
28.0682
28.0756
28.0857
28.0739
28.0840
28.0862
28.0724
28.0727
28.0752
28.0732
28.0703
28.0849
28.0795
28.0902
28.0874
28.0971
28.0638
28.0877
28.0751
28.0904
28.0971
28.0661
28.0711
28.0754
28.0516
28.0961
28.0689
28.1110
28.1062
28.0726
28.1141
28.0913
28.0982
28.0703
28.0654
28.0760
28.0727
28.0850
28.0877
28.0967
28.1185
28.0945
28.0834
28.0764
28.1129
28.0797
28.0707
28.1008
28.0971
28.0826
28.0857
28.0984
28.0869
28.0795
28.0875
28.1184
28.0746
28.0816
28.0879
28.0888
28.0924
28.0979
28.0702
28.0847
28.0917
28.0834
28.0823
28.0917
28.0779
28.0852
28.0863
28.0942
28.0801
28.0817

28.0922
28.0914
28.0868
28.0832
28.0881
28.0910
28.0886
28.0961
28.0857
28.0859
28.1086
28.0838
28.0921
28.0945
28.0839
28.0877
28.0803
28.0928
28.0885
28.0940
28.0856
28.0849
28.0955
28.0955
28.0846
28.0871
28.0872
28.0917
28.0931
28.0865
28.0900
28.0915
28.0963
28.0917
28.0950
28.0898
28.0902
28.0867
28.0843
28.0939
28.0902
28.0911
28.0909
28.0949
28.0867
28.0932
28.0891
28.0932
28.0887
28.0925
28.0928
28.0883
28.0946
28.0977
28.0914
28.0959
28.0926
28.0923
28.0950
28.1006
28.0924
28.0963
28.0893
28.0956
28.0980
28.0928
28.0951
28.0958
28.0912
28.0990
28.0915
28.0957
28.0976
28.0888
28.0928
28.0910
28.0902
28.0950
28.0995
28.0965
28.0972
28.0963
28.0946
28.0942
28.0998
28.0911

28.1043
28.1002
28.0991
28.0959
28.0996
28.0926
28.1002
28.0961
28.0983
28.0997
28.0959
28.0988
28.1029
28.0989
28.1000
28.0944
28.0979
28.1005
28.1012
28.1013
28.0999
28.0991
28.1059
28.0961
28.0981
28.1045
28.1047
28.1042
28.1146
28.1113
28.1051
28.1065
28.1065
28.0985
28.1000
28.1066
28.1041
28.0954
28.1090
( Y_{i} = C + E_{i} )
assumptions are:
1. random drawings;
and

4-Plot of
Data
above:
1. The run sequence plot (upper left) indicates significant
shifts in both location and variation. Specifically, the
location is increasing with time. The variability seems
greater in the first and last third of the data than it
does in the middle third.
2. The lag plot (upper right) shows a significant non-
random pattern in the data. Specifically, the strong
linear appearance of this plot is indicative of a model
that relates Yt to Yt-1.
3. The distributional plots, the histogram (lower left) and
the normal probability plot (lower right), are not
interpreted since the randomness assumption is so
clearly violated.
The serious violation of the non-randomness assumption
means that the univariate model
( Y_{i} = C + E_{i} )
is not valid. Given the linear appearance of the lag plot, the
first step might be to consider a model of the type
( Y_{i} = A_0 + A_1*Y_{i-1} + E_{i} )
However, discussions with the scientist revealed the
following:
1. the drift with respect to location was expected.
2. the non-constant variability was not expected.
The scientist examined the data collection device and
determined that the non-constant variation was a seasonal
effect. The high variability data in the first and last thirds
was collected in winter while the more stable middle third
was collected in the summer. The seasonal effect was
determined to be caused by the amount of humidity affecting
the measurement equipment. In this case, the solution was to
modify the test equipment to be less sensitive to
enviromental factors.
Simple graphical techniques can be quite effective in
revealing unexpected results in the data. When this occurs, it
is important to investigate whether the unexpected result is
due to problems in the experiment and data collection, or is
it in fact indicative of an unexpected underlying structure in
the data. This determination cannot be made on the basis of
statistics alone. The role of the graphical and statistical
analysis is to detect problems or unexpected results in the
data. Resolving the issues requires the knowledge of the
scientist or engineer.

Individual
Plots
generated individually to give more detail. Since the lag plot
indicates significant non-randomness, we omit the
distributional plots.
Run
Sequence
Plot
Lag Plot
Summary
Statistics
from the data.
Sample size = 1000
Mean = 28.01634
Median = 28.02910
Minimum = 27.82800
Maximum = 28.11850
Range = 0.29050
Stan. Dev. = 0.06349
B0 27.9114 0.1209E-02 0.2309E+05
B1 0.20967E-03 0.2092E-05 100.2
The slope parameter, B1, has a t value of 100.2 which is statistically
significant. The value of the slope parameter estimate is 0.00021. Although
this number is nearly zero, we need to take into account that the original

scale of the data is from about 27.8 to 28.2. In this case, we conclude that
there is a drift in location.
test is not robust for non-normality. Since the normality assumption is
questionable for these data, we use the alternative Levene test. In particular,
we use the Levene test based on the median rather the mean. The choice of
the number of intervals is somewhat arbitrary, although values of four or
eight are reasonable. We will divide our data into four intervals.
H0: σ1
2 = σ2
2 = σ3
2 = σ4
2
In this case, since the Levene test statistic value of 140.85 is greater than the
5 % significance level critical value of 2.614, we conclude that there is
significant evidence of nonconstant variation.
The lag plot in the 4-plot in the previous section is a simple graphical
technique.
One check is an autocorrelation plot that shows the autocorrelations for
The lag 1 autocorrelation, which is generally the one of greatest interest, is
This indicates that the lag 1 autocorrelation is statistically significant, so
there is strong evidence of non-randomness.
Because the test statistic is outside of the critical region, we reject the null
hypothesis and conclude that the data are not random.
Distributional
Analysis
Since we rejected the randomness assumption, the distributional tests are not
meaningful. Therefore, these quantitative tests are omitted. Since the
Grubbs' test for outliers also assumes the approximate normality of the data,
we omit Grubbs' test as well.

Univariate
Report
report.
Analysis for resistor case study
2: Location
Mean = 28.01635
3: Variation
Change in variation?
of the data) = YES
4: Randomness
Data Are Random?
5: Distribution
Distributional test omitted due to
non-randomness of the data
fixed)
7: Outliers?
(Grubbs' test omitted due to
non-randomness of the data)
View
Dataplot
Macro for
this Case
Study
NOTE: This case study has 1,000 points. For better
performance, it is highly recommended that you check
the "No Update" box on the Spreadsheet window for
this case study. This will suppress subsequent updating
of the Spreadsheet window as the data are created or
modified.
The links in this column will connect you with more detailed
information about each analysis step from the case study
description.

1. 4-plot of Y. 1. Based on the 4-plot, there are shifts
in location and variation and the data
are not random.
there are shifts of location and
variation.
2. The lag plot shows a strong linear
pattern, which indicates significant
non-randomness.
statistics.
2. Generate the sample mean, a confidence
interval for the population mean, and
compute a linear fit to detect drift in
location.
3. Generate the sample standard deviation,
a confidence interval for the population
standard deviation, and detect drift in
variation by dividing the data into
quarters and computing Levene's test for
equal standard deviations.
25+ statistics.
The linear fit indicates drift in
estimate is statistically significant.
Levene's test indicates significant
outside the 95% confidence interval
bands, indicating significant non-randomness.
convenient report.
Heat Flow Meter 1
Heat Flow
Meter
Calibration
and
Stability
This example illustrates the univariate analysis of standard
resistor data.
Background and Data
Generation This data set was collected by Bob Zarr of NIST in January,
1990 from a heat flow meter calibration and stability analysis.
The response variable is a calibration factor.
The motivation for studying this data set is to illustrate a well-
behaved process where the underlying assumptions hold and
the process is in statistical control.

9.206343
9.299992
9.277895
9.305795
9.275351
9.288729
9.287239
9.260973
9.303111
9.275674
9.272561
9.288454
9.255672
9.252141
9.297670
9.266534
9.256689
9.277542
9.248205
9.252107
9.276345
9.278694
9.267144
9.246132
9.238479
9.269058
9.248239
9.257439
9.268481
9.288454
9.258452
9.286130
9.251479
9.257405
9.268343
9.291302
9.219460
9.270386
9.218808
9.241185
9.269989
9.226585
9.258556
9.286184
9.320067
9.327973
9.262963
9.248181
9.238644
9.225073
9.220878
9.271318
9.252072
9.281186
9.270624
9.294771
9.301821
9.278849
9.236680
9.233988
9.244687
9.221601
9.207325
9.258776
9.275708
9.268955
9.257269
9.264979
9.295500
9.292883
9.264188
9.280731
9.267336
9.300566
9.253089
9.261376
9.238409
9.225073
9.235526

9.239510
9.264487
9.244242
9.277542
9.310506
9.261594
9.259791
9.253089
9.245735
9.284058
9.251122
9.275385
9.254619
9.279526
9.275065
9.261952
9.275351
9.252433
9.230263
9.255150
9.268780
9.290389
9.274161
9.255707
9.261663
9.250455
9.261952
9.264041
9.264509
9.242114
9.239674
9.221553
9.241935
9.215265
9.285930
9.271559
9.266046
9.285299
9.268989
9.267987
9.246166
9.231304
9.240768
9.260506
9.274355
9.292376
9.271170
9.267018
9.308838
9.264153
9.278822
9.255244
9.229221
9.253158
9.256292
9.262602
9.219793
9.258452
9.267987
9.267987
9.248903
9.235153
9.242933
9.253453
9.262671
9.242536
9.260803
9.259825
9.253123
9.240803
9.238712
9.263676
9.243002
9.246826
9.252107
9.261663
9.247311
9.306055
9.237646
9.248937
9.256689
9.265777
9.299047
9.244814
9.287205
9.300566

9.256621
9.271318
9.275154
9.281834
9.253158
9.269024
9.282077
9.277507
9.284910
9.239840
9.268344
9.247778
9.225039
9.230750
9.270024
9.265095
9.284308
9.280697
9.263032
9.291851
9.252072
9.244031
9.283269
9.196848
9.231372
9.232963
9.234956
9.216746
9.274107
9.273776
( Y_{i} = C + E_{i} )
assumptions are:
1. random drawings;
and
4-Plot of
Data

above:
scale over time.
reasonably symmetric, there does not appear to be
significant outliers in the tails, and it seems reasonable
to assume that the data are from approximately a
4. The normal probability plot (lower right) verifies that
an assumption of normality is in fact reasonable.
Individual
Plots
Run
Sequence
Plot
Lag Plot
Histogram
(with
overlaid
Normal
PDF)

Normal
Probability
Plot
Summary
Statistics
from the data.
Sample size = 195
Mean = 9.261460
Median = 9.261952
Minimum = 9.196848
Maximum = 9.327973
Range = 0.131126
Stan. Dev. = 0.022789
B0 9.26699 0.3253E-02 2849.
B1 -0.56412E-04 0.2878E-04 -1.960
The slope parameter, B1, has a t value of -1.96 which is (barely) statistically
significant since it is essentially equal to the 95 % level cutoff of -1.96.
However, notice that the value of the slope parameter estimate is -0.00056.
This slope, even though statistically significant, can essentially be
considered zero.
dividing the data set into several equal-sized intervals. The choice of the
number of intervals is somewhat arbitrary, although values of four or eight
are reasonable. We will divide our data into four intervals.
H0: σ1
2 = σ2
2 = σ3
2 = σ4
2
1-α,k-1 = 7.815
In this case, since the Bartlett test statistic of 3.147 is less than the critical
value at the 5 % significance level of 7.815, we conclude that the variances
are not significantly different in the four intervals. That is, the assumption of
constant scale is valid.

The lag plot in the previous section is a simple graphical technique.
The lag 1 autocorrelation, which is generally the one of greatest interest, is
This indicates that the lag 1 autocorrelation is statistically significant, so
there is evidence of non-randomness.
The value of the test statistic is less than -1.96, so we reject the null
hypothesis at the 0.05 significant level and conclude that the data are not
random.
Although the autocorrelation plot and the runs test indicate some mild non-
randomness, the violation of the randomness assumption is not serious
enough to warrant developing a more sophisticated model. It is common in
practice that some of the assumptions are mildly violated and it is a
judgement call as to whether or not the violations are serious enough to
warrant developing a more sophisticated model for the data.
Distributional
Analysis
coefficient of the points on the probability plot. For this data set the
correlation coefficient is 0.996. Since this is greater than the critical value of
0.987 (this is a tabulated value), the normality assumption is not rejected.
Adjusted test statistic: A 2 = 0.129
Critical region: Reject H0 if A 2 > 0.787

The Anderson-Darling test also does not reject the normality assumption
because the test statistic, 0.129, is less than the critical value at the 5 %
significance level of 0.787.
Outlier
Analysis
A test for outliers is the Grubbs' test.
For this data set, Grubbs' test does not detect any outliers at the 0.05
significance level.
Model Since the underlying assumptions were validated both graphically and
analytically, with a mild violation of the randomness assumption, we
conclude that a reasonable model for the data is:
( Y_{i} = 9.26146 + E_{i} )
We can express the uncertainty for C, here estimated by 9.26146, as the 95
% confidence interval (9.258242,9.26479).
Univariate
Report
report. The report for the heat flow meter data follows.
Analysis for heat flow meter data
2: Location
Mean = 9.26146
95 % Confidence Interval for Mean = (9.258242,9.264679)
3: Variation
95 % Confidence Interval for SD = (0.02073,0.025307)
(based on Bartlett's test on quarters
of the data) = NO
4: Randomness
Data are Random?
5: Data are Normal?
(as tested by Anderson-Darling) = YES
(as tested by Normal PPCC) = YES
fixed normal)
7: Outliers?
(as determined by Grubbs' test) = NO
View
Dataplot
Macro for
this Case
Study

detailed information about each analysis step from the case
study description.
in location or scale, and the data seem to
follow a normal distribution.
plot.
scale.
that the normal distribution is a
statistics.
location.
computing Bartlett's test for equal
coefficient.
25+ statistics.
estimate is essentially zero.
Bartlett's test indicates no significant
statistically significant at the 95%
level.
we cannot reject the normality assumption.
5% level.

convenient report.
Fatigue Life of Aluminum Alloy Specimens
Fatigue
Life of
Aluminum
Alloy
Specimens
This example illustrates the univariate analysis of the fatigue
life of aluminum alloy specimens.
Background and Data
Generation This data set comprises measurements of fatigue life
(thousands of cycles until rupture) of rectangular strips of
6061-T6 aluminum sheeting, subjected to periodic loading with
maximum stress of 21,000 psi (pounds per square inch), as
reported by Birnbaum and Saunders (1958).
Purpose of
Analysis
The goal of this case study is to select a probabilistic model,
from among several reasonable alternatives, to describe the
dispersion of the resulting measured values of life-length.
The original study, in the field of statistical reliability analysis,
was concerned with the prediction of failure times of a material
subjected to a load varying in time. It was well-known that a
structure designed to withstand a particular static load may fail
sooner than expected under a dynamic load.
If a realistic model for the probability distribution of lifetime
can be found, then it can be used to estimate the time by which
a part or structure needs to be replaced to guarantee that the
probability of failure does not exceed some maximum
acceptable value, for example 0.1 %, while it is in service.
The chapter of this eHandbook that is concerned with the
assessment of product reliability contains additional material
on statistical methods used in reliability analysis. This case
study is meant to complement that chapter by showing the use
of graphical and other techniques in the model selection stage
of such analysis.
When there is no cogent reason to adopt a particular model, or
when none of the models under consideration seems adequate
for the purpose, one may opt for a non-parametric statistical
method, for example to produce tolerance bounds or
confidence intervals.
A non-parametric method does not rely on the assumption that
the data are like a sample from a particular probability
distribution that is fully specified up to the values of some
adjustable parameters. For example, the Gaussian probability
distribution is a parametric model with two adjustable
parameters.
The price to be paid when using non-parametric methods is
loss of efficiency, meaning that they may require more data for
statistical inference than a parametric counterpart would, if
applicable. For example, non-parametric confidence intervals
for model parameters may be considerably wider than what a
confidence interval would need to be if the underlying

distribution could be identified correctly. Such identification is
what we will attempt in this case study.
It should be noted --- a point that we will stress later in the
development of this case study --- that the very exercise of
selecting a model often contributes substantially to the
uncertainty of the conclusions derived after the selection has
been made.
Software The analyses used in this case study can be generated using R
code. The reader can download the data as a text file.
Data The following data are used for this case study.
370 1016 1235 1419 1567 1820
706 1018 1238 1420 1578 1868
716 1020 1252 1420 1594 1881
746 1055 1258 1450 1602 1890
785 1085 1262 1452 1604 1893
797 1102 1269 1475 1608 1895
844 1102 1270 1478 1630 1910
855 1108 1290 1481 1642 1923
858 1115 1293 1485 1674 1940
886 1120 1300 1502 1730 1945
886 1134 1310 1505 1750 2023
930 1140 1313 1513 1750 2100
960 1199 1315 1522 1763 2130
988 1200 1330 1522 1768 2215
990 1200 1355 1530 1781 2268
1000 1203 1390 1540 1782 2440
1010 1222 1416 1560 1792
Goal The goal of this analysis is to select a probabilistic model to describe the dispersion
of the measured values of fatigue life of specimens of an aluminum alloy described in
[1.4.2.9.1], from among several reasonable alternatives.
Initial Plots
of the Data
Simple diagrams can be very informative about location, spread, and to detect
possibly anomalous data values or particular patterns (clustering, for example). These
include dot-charts, boxplots, and histograms. Since building an effective histogram
requires that a choice be made of bin size, and this choice can be influential, one may
wish to examine a non-parametric estimate of the underlying probability density.

These several plots variously show that the measurements range from a value slightly
greater than 350,000 to slightly less than 2,500,000 cycles. The boxplot suggests that
the largest measured value may be an outlier.
A recommended first step is to check consistency between the data and what is to be
expected if the data were a sample from a particular probability distribution.
Knowledge about the underlying properties of materials and of relevant industrial
processes typically offer clues as to the models that should be entertained. Graphical
diagnostic techniques can be very useful at this exploratory stage: foremost among
these, for univariate data, is the quantile-quantile plot, or QQ-plot (Wilk and
Gnanadesikan, 1968).
Each data point is represented by one point in the QQ-plot. The ordinate of each of
these points is one data value; if this data value happens to be the kth order statistic in
the sample (that is, the kth largest value), then the corresponding abscissa is the
"typical" value that the kth largest value should have in a sample of the same size as
the data, drawn from a particular distribution. If F denotes the cumulative probability
distribution function of interest, and the sample comprises n values, then F -1 [(k -
1/2) / (n + 1/2)] is a reasonable choice for that "typical" value, because it is an
approximation to the median of the kth order statistic in a sample of size n from this
distribution.
The following figure shows a QQ-plot of our data relative to the Gaussian (or,
normal) probability distribution. If the data matched expectations perfectly, then the
points would all fall on a straight line.

In practice, one needs to gauge whether the deviations from such perfect alignment
are commensurate with the natural variability associated with sampling. This can
easily be done by examining how variable QQ-plots of samples from the target
distribution may be.
The following figure shows, superimposed on the QQ-plot of the data, the QQ-plots
of 99 samples of the same size as the data, drawn from a Gaussian distribution with
the same mean and standard deviation as the data.
The fact that the cloud of QQ-plots corresponding to 99 samples from the Gaussian
distribution effectively covers the QQ-plot for the data, suggests that the chances are
better than 1 in 100 that our data are inconsistent with the Gaussian model.

This proves nothing, of course, because even the rarest of events may happen.
However, it is commonly taken to be indicative of an acceptable fit for general
purposes. In any case, one may naturally wonder if an alternative model might not
provide an even better fit.
Knowing the provenance of the data, that they portray strength of a material, strongly
suggests that one may like to examine alternative models, because in many studies of
reliability non-Gaussian models tend to be more appropriate than Gaussian models.
Candidate
Distributions
There are many probability distributions that could reasonably be entertained as
candidate models for the data. However, we will restrict ourselves to consideration of
the following because these have proven to be useful in reliability studies.
Normal distribution
Gamma distribution
Birnbaum-Saunders distribution
3-parameter Weibull distribution
Approach A very simple approach amounts to comparing QQ-plots of the data for the candidate
models under consideration. This typically involves first fitting the models to the
data, for example employing the method of maximum likelihood [1.3.6.5.2].
The maximum likelihood estimates are the following:
Gaussian: mean 1401, standard deviation 389
Gamma: shape 11.85, rate 0.00846
Birnbaum-Saunders: shape 0.310, scale 1337
3-parameter Weibull: location 181, shape 3.43, scale 1357
The following figure shows how close (or how far) the best fitting probability
densities of the four distributions approximate the non-parametric probability density
estimate. This comparison, however, takes into account neither the fact that our
sample is fairly small (101 measured values), nor that the fitted models themselves
have been estimated from the same data that the non-parametric estimate was derived
from.
These limitations notwithstanding, it is worth examining the corresponding QQ-plots,
shown below, which suggest that the Gaussian and the 3-parameter Weibull may be
the best models.

Model
Selection
A more careful comparison of the merits of the alternative models needs to take into
account the fact that the 3-parameter Weibull model (precisely because it has three
parameters), may be intrinsically more flexible than the others, which all have two
adjustable parameters only.
Two criteria can be employed for a formal comparison: Akaike's Information
Criterion (AIC), and the Bayesian Information Criterion (BIC) (Hastie et. al., 2001).
The smaller the value of either model selection criterion, the better the model:
AIC BIC
GAU 1495 1501
GAM 1499 1504
BS 1507 1512
WEI 1498 1505
On this basis (and according both to AIC and BIC), there seems to be no cogent
reason to replace the Gaussian model by any of the other three. The values of BIC
can also be used to derive an approximate answer to the question of how strongly the
data may support each of these models. Doing this involves the application of
Bayesian statistical methods [8.1.10].
We start from an a priori assignment of equal probabilities to all four models,
indicating that we have no reason to favor one over another at the outset, and then
update these probabilities based on the measured values of lifetime. The updated
probabilities of the four models, called their posterior probabilities, are
approximately proportional to exp(-BIC(GAU)/2), exp(-BIC(GAM)/2), exp(-
BIC(BS)/2), and exp(-BIC(WEI)/2). The values are 76 % for GAU, 16 % for GAM,
0.27 % for BS, and 7.4 % for WEI.
One possible use for the selected model is to answer the question of the age in service
by which a part or structure needs to be replaced to guarantee that the probability of
failure does not exceed some maximum acceptable value, for example 0.1 %.The
answer to this question is the 0.1st percentile of the fitted distribution, that is G -1
(0.001) = 198 thousand cycles, where, in this case, G -1 denotes the inverse of the
fitted, Gaussian probability distribution.
To assess the uncertainty of this estimate one may employ the statistical bootstrap
[1.3.3.4]. In this case, this involves drawing a suitably large number of bootstrap
samples from the data, and for each of them applying the model fitting and model
selection exercise described above, ending with the calculation of G -1 (0.001) for the
best model (which may vary from sample to sample).

The bootstrap samples should be of the same size as the data, with each being drawn
uniformly at random from the data, with replacement. This process, based on 5,000
bootstrap samples, yielded a 95 % confidence interval for the 0.1st percentile ranging
from 40 to 366 thousands of cycles. The large uncertainty is not surprising given that
we are attempting to estimate the largest value that is exceeded with probability 99.9
%, based on a sample comprising only 101 measured values.
Prediction
Intervals
One more application in this analysis is to evaluate prediction intervals for the fatigue
life of the aluminum alloy specimens. For example, if we were to test three new
specimens using the same process, we would want to know (with 95 % confidence)
the minimum number of cycles for these three specimens. That is, we need to find a
statistical interval [L, ∞] that contains the fatigue life of all three future specimens
with 95 % confidence. The desired interval is a one-sided, lower 95 % prediction
interval. Since tables of factors for constructing L, are widely available for normal
models, we use the results corresponding to the normal model here for illustration.
Specifically, L is computed as
where factor r is given in Table A.14 of Hahn and Meeker (1991) or can be obtained
from an R program.
Ceramic Strength
Ceramic
Strength
This case study analyzes the effect of machining factors on the
strength of ceramics.
2. Analysis of the Response Variable
3. Analysis of Batch Effect
4. Analysis of Lab Effect
5. Analysis of Primary Factors
Background and Data
Generation The data for this case study were collected by Said Jahanmir of
the NIST Ceramics Division in 1996 in connection with a
NIST/industry ceramics consortium for strength optimization
of ceramic strength
The motivation for studying this data set is to illustrate the
analysis of multiple factors from a designed experiment
This case study will utilize only a subset of a full study that
was conducted by Lisa Gill and James Filliben of the NIST
Statistical Engineering Division
The response variable is a measure of the strength of the
ceramic material (bonded Si nitrate). The complete data set
contains the following variables:
1. Factor 1 = Observation ID, i.e., run number (1 to 960)
2. Factor 2 = Lab (1 to 8)
3. Factor 3 = Bar ID within lab (1 to 30)
4. Factor 4 = Test number (1 to 4)
5. Response Variable = Strength of Ceramic
6. Factor 5 = Table speed (2 levels: 0.025 and 0.125)
7. Factor 6 = Down feed rate (2 levels: 0.050 and 0.125)
8. Factor 7 = Wheel grit size (2 levels: 150 and 80)
9. Factor 8 = Direction (2 levels: longitudinal and
transverse)

10. Factor 9 = Treatment (1 to 16)
11. Factor 10 = Set of 15 within lab (2 levels: 1 and 2)
12. Factor 11 = Replication (2 levels: 1 and 2)
13. Factor 12 = Bar Batch (1 and 2)
The four primary factors of interest are:
1. Table speed (X1)
2. Down feed rate (X2)
3. Wheel grit size (X3)
4. Direction (X4)
For this case study, we are using only half the data.
Specifically, we are using the data with the direction
longitudinal. Therefore, we have only three primary factors
In addition, we are interested in the nuisance factors
1. Lab
2. Batch
Purpose of
Analysis
The goals of this case study are:
1. Determine which of the four primary factors has the
strongest effect on the strength of the ceramic material
2. Estimate the magnitude of the effects
3. Determine the optimal settings for the primary factors
4. Determine if the nuisance factors (lab and batch) have an
effect on the ceramic strength
This case study is an example of a designed experiment. The
Process Improvement chapter contains a detailed discussion of
the construction and analysis of designed experiments. This
case study is meant to complement the material in that chapter
by showing how an EDA approach (emphasizing the use of
graphical techniques) can be used in the analysis of designed
experiments
Data The following are the data used for this case study
Run Lab Batch Y X1 X2 X3
1 1 1 608.781 -1 -1 -1
2 1 2 569.670 -1 -1 -1
3 1 1 689.556 -1 -1 -1
4 1 2 747.541 -1 -1 -1
5 1 1 618.134 -1 -1 -1
6 1 2 612.182 -1 -1 -1
7 1 1 680.203 -1 -1 -1
8 1 2 607.766 -1 -1 -1
9 1 1 726.232 -1 -1 -1
10 1 2 605.380 -1 -1 -1
11 1 1 518.655 -1 -1 -1
12 1 2 589.226 -1 -1 -1
13 1 1 740.447 -1 -1 -1
14 1 2 588.375 -1 -1 -1
15 1 1 666.830 -1 -1 -1
16 1 2 531.384 -1 -1 -1
17 1 1 710.272 -1 -1 -1
18 1 2 633.417 -1 -1 -1
19 1 1 751.669 -1 -1 -1
20 1 2 619.060 -1 -1 -1
21 1 1 697.979 -1 -1 -1
22 1 2 632.447 -1 -1 -1
23 1 1 708.583 -1 -1 -1
24 1 2 624.256 -1 -1 -1
25 1 1 624.972 -1 -1 -1
26 1 2 575.143 -1 -1 -1
27 1 1 695.070 -1 -1 -1
28 1 2 549.278 -1 -1 -1
29 1 1 769.391 -1 -1 -1
30 1 2 624.972 -1 -1 -1
61 1 1 720.186 -1 1 1
62 1 2 587.695 -1 1 1

63 1 1 723.657 -1 1 1
64 1 2 569.207 -1 1 1
65 1 1 703.700 -1 1 1
66 1 2 613.257 -1 1 1
67 1 1 697.626 -1 1 1
68 1 2 565.737 -1 1 1
69 1 1 714.980 -1 1 1
70 1 2 662.131 -1 1 1
71 1 1 657.712 -1 1 1
72 1 2 543.177 -1 1 1
73 1 1 609.989 -1 1 1
74 1 2 512.394 -1 1 1
75 1 1 650.771 -1 1 1
76 1 2 611.190 -1 1 1
77 1 1 707.977 -1 1 1
78 1 2 659.982 -1 1 1
79 1 1 712.199 -1 1 1
80 1 2 569.245 -1 1 1
81 1 1 709.631 -1 1 1
82 1 2 725.792 -1 1 1
83 1 1 703.160 -1 1 1
84 1 2 608.960 -1 1 1
85 1 1 744.822 -1 1 1
86 1 2 586.060 -1 1 1
87 1 1 719.217 -1 1 1
88 1 2 617.441 -1 1 1
89 1 1 619.137 -1 1 1
90 1 2 592.845 -1 1 1
151 2 1 753.333 1 1 1
152 2 2 631.754 1 1 1
153 2 1 677.933 1 1 1
154 2 2 588.113 1 1 1
155 2 1 735.919 1 1 1
156 2 2 555.724 1 1 1
157 2 1 695.274 1 1 1
158 2 2 702.411 1 1 1
159 2 1 504.167 1 1 1
160 2 2 631.754 1 1 1
161 2 1 693.333 1 1 1
162 2 2 698.254 1 1 1
163 2 1 625.000 1 1 1
164 2 2 616.791 1 1 1
165 2 1 596.667 1 1 1
166 2 2 551.953 1 1 1
167 2 1 640.898 1 1 1
168 2 2 636.738 1 1 1
169 2 1 720.506 1 1 1
170 2 2 571.551 1 1 1
171 2 1 700.748 1 1 1
172 2 2 521.667 1 1 1
173 2 1 691.604 1 1 1
174 2 2 587.451 1 1 1
175 2 1 636.738 1 1 1
176 2 2 700.422 1 1 1
177 2 1 731.667 1 1 1
178 2 2 595.819 1 1 1
179 2 1 635.079 1 1 1
180 2 2 534.236 1 1 1
181 2 1 716.926 1 -1 -1
182 2 2 606.188 1 -1 -1
183 2 1 759.581 1 -1 -1
184 2 2 575.303 1 -1 -1
185 2 1 673.903 1 -1 -1
186 2 2 590.628 1 -1 -1
187 2 1 736.648 1 -1 -1
188 2 2 729.314 1 -1 -1
189 2 1 675.957 1 -1 -1
190 2 2 619.313 1 -1 -1
191 2 1 729.230 1 -1 -1
192 2 2 624.234 1 -1 -1
193 2 1 697.239 1 -1 -1
194 2 2 651.304 1 -1 -1
195 2 1 728.499 1 -1 -1
196 2 2 724.175 1 -1 -1
197 2 1 797.662 1 -1 -1
198 2 2 583.034 1 -1 -1
199 2 1 668.530 1 -1 -1
200 2 2 620.227 1 -1 -1
201 2 1 815.754 1 -1 -1
202 2 2 584.861 1 -1 -1
203 2 1 777.392 1 -1 -1
204 2 2 565.391 1 -1 -1
205 2 1 712.140 1 -1 -1
206 2 2 622.506 1 -1 -1
207 2 1 663.622 1 -1 -1
208 2 2 628.336 1 -1 -1

209 2 1 684.181 1 -1 -1
210 2 2 587.145 1 -1 -1
271 3 1 629.012 1 -1 1
272 3 2 584.319 1 -1 1
273 3 1 640.193 1 -1 1
274 3 2 538.239 1 -1 1
275 3 1 644.156 1 -1 1
276 3 2 538.097 1 -1 1
277 3 1 642.469 1 -1 1
278 3 2 595.686 1 -1 1
279 3 1 639.090 1 -1 1
280 3 2 648.935 1 -1 1
281 3 1 439.418 1 -1 1
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283 3 1 614.664 1 -1 1
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285 3 1 537.161 1 -1 1
286 3 2 569.858 1 -1 1
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288 3 2 617.246 1 -1 1
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294 3 2 598.853 1 -1 1
295 3 1 687.665 1 -1 1
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297 3 1 710.858 1 -1 1
298 3 2 605.694 1 -1 1
299 3 1 701.716 1 -1 1
300 3 2 627.516 1 -1 1
301 3 1 382.133 1 1 -1
302 3 2 574.522 1 1 -1
303 3 1 719.744 1 1 -1
304 3 2 582.682 1 1 -1
305 3 1 756.820 1 1 -1
306 3 2 563.872 1 1 -1
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308 3 2 715.962 1 1 -1
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310 3 2 616.430 1 1 -1
311 3 1 670.308 1 1 -1
312 3 2 778.011 1 1 -1
313 3 1 660.062 1 1 -1
314 3 2 604.255 1 1 -1
315 3 1 790.382 1 1 -1
316 3 2 571.906 1 1 -1
317 3 1 714.750 1 1 -1
318 3 2 625.925 1 1 -1
319 3 1 716.959 1 1 -1
320 3 2 682.426 1 1 -1
321 3 1 603.363 1 1 -1
322 3 2 707.604 1 1 -1
323 3 1 713.796 1 1 -1
324 3 2 617.400 1 1 -1
325 3 1 444.963 1 1 -1
326 3 2 689.576 1 1 -1
327 3 1 723.276 1 1 -1
328 3 2 676.678 1 1 -1
329 3 1 745.527 1 1 -1
330 3 2 563.290 1 1 -1
361 4 1 778.333 -1 -1 1
362 4 2 581.879 -1 -1 1
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364 4 2 447.701 -1 -1 1
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382 4 2 494.122 -1 -1 1
383 4 1 725.663 -1 -1 1
384 4 2 620.948 -1 -1 1

385 4 1 698.818 -1 -1 1
386 4 2 615.903 -1 -1 1
387 4 1 760.000 -1 -1 1
388 4 2 606.667 -1 -1 1
389 4 1 775.272 -1 -1 1
390 4 2 579.167 -1 -1 1
421 4 1 708.885 -1 1 -1
422 4 2 662.510 -1 1 -1
423 4 1 727.201 -1 1 -1
424 4 2 436.237 -1 1 -1
425 4 1 642.560 -1 1 -1
426 4 2 644.223 -1 1 -1
427 4 1 690.773 -1 1 -1
428 4 2 586.035 -1 1 -1
429 4 1 688.333 -1 1 -1
430 4 2 620.833 -1 1 -1
431 4 1 743.973 -1 1 -1
432 4 2 652.535 -1 1 -1
433 4 1 682.461 -1 1 -1
434 4 2 593.516 -1 1 -1
435 4 1 761.430 -1 1 -1
436 4 2 587.451 -1 1 -1
437 4 1 691.542 -1 1 -1
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444 4 2 707.117 -1 1 -1
445 4 1 746.467 -1 1 -1
446 4 2 621.779 -1 1 -1
447 4 1 744.819 -1 1 -1
448 4 2 585.777 -1 1 -1
449 4 1 655.029 -1 1 -1
450 4 2 703.980 -1 1 -1
541 5 1 715.224 -1 -1 -1
542 5 2 698.237 -1 -1 -1
543 5 1 614.417 -1 -1 -1
544 5 2 757.120 -1 -1 -1
545 5 1 761.363 -1 -1 -1
546 5 2 621.751 -1 -1 -1
547 5 1 716.106 -1 -1 -1
548 5 2 472.125 -1 -1 -1
549 5 1 659.502 -1 -1 -1
550 5 2 612.700 -1 -1 -1
551 5 1 730.781 -1 -1 -1
552 5 2 583.170 -1 -1 -1
553 5 1 546.928 -1 -1 -1
554 5 2 599.771 -1 -1 -1
555 5 1 734.203 -1 -1 -1
556 5 2 549.227 -1 -1 -1
557 5 1 682.051 -1 -1 -1
558 5 2 605.453 -1 -1 -1
559 5 1 701.341 -1 -1 -1
560 5 2 569.599 -1 -1 -1
561 5 1 759.729 -1 -1 -1
562 5 2 637.233 -1 -1 -1
563 5 1 689.942 -1 -1 -1
564 5 2 621.774 -1 -1 -1
565 5 1 769.424 -1 -1 -1
566 5 2 558.041 -1 -1 -1
567 5 1 715.286 -1 -1 -1
568 5 2 583.170 -1 -1 -1
569 5 1 776.197 -1 -1 -1
570 5 2 345.294 -1 -1 -1
571 5 1 547.099 1 -1 1
572 5 2 570.999 1 -1 1
573 5 1 619.942 1 -1 1
574 5 2 603.232 1 -1 1
575 5 1 696.046 1 -1 1
576 5 2 595.335 1 -1 1
577 5 1 573.109 1 -1 1
578 5 2 581.047 1 -1 1
579 5 1 638.794 1 -1 1
580 5 2 455.878 1 -1 1
581 5 1 708.193 1 -1 1
582 5 2 627.880 1 -1 1
583 5 1 502.825 1 -1 1
584 5 2 464.085 1 -1 1
585 5 1 632.633 1 -1 1
586 5 2 596.129 1 -1 1
587 5 1 683.382 1 -1 1
588 5 2 640.371 1 -1 1
589 5 1 684.812 1 -1 1
590 5 2 621.471 1 -1 1

591 5 1 738.161 1 -1 1
592 5 2 612.727 1 -1 1
593 5 1 671.492 1 -1 1
594 5 2 606.460 1 -1 1
595 5 1 709.771 1 -1 1
596 5 2 571.760 1 -1 1
597 5 1 685.199 1 -1 1
598 5 2 599.304 1 -1 1
599 5 1 624.973 1 -1 1
600 5 2 579.459 1 -1 1
601 6 1 757.363 1 1 1
602 6 2 761.511 1 1 1
603 6 1 633.417 1 1 1
604 6 2 566.969 1 1 1
605 6 1 658.754 1 1 1
606 6 2 654.397 1 1 1
607 6 1 664.666 1 1 1
608 6 2 611.719 1 1 1
609 6 1 663.009 1 1 1
610 6 2 577.409 1 1 1
611 6 1 773.226 1 1 1
612 6 2 576.731 1 1 1
613 6 1 708.261 1 1 1
614 6 2 617.441 1 1 1
615 6 1 739.086 1 1 1
616 6 2 577.409 1 1 1
617 6 1 667.786 1 1 1
618 6 2 548.957 1 1 1
619 6 1 674.481 1 1 1
620 6 2 623.315 1 1 1
621 6 1 695.688 1 1 1
622 6 2 621.761 1 1 1
623 6 1 588.288 1 1 1
624 6 2 553.978 1 1 1
625 6 1 545.610 1 1 1
626 6 2 657.157 1 1 1
627 6 1 752.305 1 1 1
628 6 2 610.882 1 1 1
629 6 1 684.523 1 1 1
630 6 2 552.304 1 1 1
631 6 1 717.159 -1 1 -1
632 6 2 545.303 -1 1 -1
633 6 1 721.343 -1 1 -1
634 6 2 651.934 -1 1 -1
635 6 1 750.623 -1 1 -1
636 6 2 635.240 -1 1 -1
637 6 1 776.488 -1 1 -1
638 6 2 641.083 -1 1 -1
639 6 1 750.623 -1 1 -1
640 6 2 645.321 -1 1 -1
641 6 1 600.840 -1 1 -1
642 6 2 566.127 -1 1 -1
643 6 1 686.196 -1 1 -1
644 6 2 647.844 -1 1 -1
645 6 1 687.870 -1 1 -1
646 6 2 554.815 -1 1 -1
647 6 1 725.527 -1 1 -1
648 6 2 620.087 -1 1 -1
649 6 1 658.796 -1 1 -1
650 6 2 711.301 -1 1 -1
651 6 1 690.380 -1 1 -1
652 6 2 644.355 -1 1 -1
653 6 1 737.144 -1 1 -1
654 6 2 713.812 -1 1 -1
655 6 1 663.851 -1 1 -1
656 6 2 696.707 -1 1 -1
657 6 1 766.630 -1 1 -1
658 6 2 589.453 -1 1 -1
659 6 1 625.922 -1 1 -1
660 6 2 634.468 -1 1 -1
721 7 1 694.430 1 1 -1
722 7 2 599.751 1 1 -1
723 7 1 730.217 1 1 -1
724 7 2 624.542 1 1 -1
725 7 1 700.770 1 1 -1
726 7 2 723.505 1 1 -1
727 7 1 722.242 1 1 -1
728 7 2 674.717 1 1 -1
729 7 1 763.828 1 1 -1
730 7 2 608.539 1 1 -1
731 7 1 695.668 1 1 -1
732 7 2 612.135 1 1 -1
733 7 1 688.887 1 1 -1
734 7 2 591.935 1 1 -1
735 7 1 531.021 1 1 -1
736 7 2 676.656 1 1 -1

737 7 1 698.915 1 1 -1
738 7 2 647.323 1 1 -1
739 7 1 735.905 1 1 -1
740 7 2 811.970 1 1 -1
741 7 1 732.039 1 1 -1
742 7 2 603.883 1 1 -1
743 7 1 751.832 1 1 -1
744 7 2 608.643 1 1 -1
745 7 1 618.663 1 1 -1
746 7 2 630.778 1 1 -1
747 7 1 744.845 1 1 -1
748 7 2 623.063 1 1 -1
749 7 1 690.826 1 1 -1
750 7 2 472.463 1 1 -1
811 7 1 666.893 -1 1 1
812 7 2 645.932 -1 1 1
813 7 1 759.860 -1 1 1
814 7 2 577.176 -1 1 1
815 7 1 683.752 -1 1 1
816 7 2 567.530 -1 1 1
817 7 1 729.591 -1 1 1
818 7 2 821.654 -1 1 1
819 7 1 730.706 -1 1 1
820 7 2 684.490 -1 1 1
821 7 1 763.124 -1 1 1
822 7 2 600.427 -1 1 1
823 7 1 724.193 -1 1 1
824 7 2 686.023 -1 1 1
825 7 1 630.352 -1 1 1
826 7 2 628.109 -1 1 1
827 7 1 750.338 -1 1 1
828 7 2 605.214 -1 1 1
829 7 1 752.417 -1 1 1
830 7 2 640.260 -1 1 1
831 7 1 707.899 -1 1 1
832 7 2 700.767 -1 1 1
833 7 1 715.582 -1 1 1
834 7 2 665.924 -1 1 1
835 7 1 728.746 -1 1 1
836 7 2 555.926 -1 1 1
837 7 1 591.193 -1 1 1
838 7 2 543.299 -1 1 1
839 7 1 592.252 -1 1 1
840 7 2 511.030 -1 1 1
901 8 1 740.833 -1 -1 1
902 8 2 583.994 -1 -1 1
903 8 1 786.367 -1 -1 1
904 8 2 611.048 -1 -1 1
905 8 1 712.386 -1 -1 1
906 8 2 623.338 -1 -1 1
907 8 1 738.333 -1 -1 1
908 8 2 679.585 -1 -1 1
909 8 1 741.480 -1 -1 1
910 8 2 665.004 -1 -1 1
911 8 1 729.167 -1 -1 1
912 8 2 655.860 -1 -1 1
913 8 1 795.833 -1 -1 1
914 8 2 715.711 -1 -1 1
915 8 1 723.502 -1 -1 1
916 8 2 611.999 -1 -1 1
917 8 1 718.333 -1 -1 1
918 8 2 577.722 -1 -1 1
919 8 1 768.080 -1 -1 1
920 8 2 615.129 -1 -1 1
921 8 1 747.500 -1 -1 1
922 8 2 540.316 -1 -1 1
923 8 1 775.000 -1 -1 1
924 8 2 711.667 -1 -1 1
925 8 1 760.599 -1 -1 1
926 8 2 639.167 -1 -1 1
927 8 1 758.333 -1 -1 1
928 8 2 549.491 -1 -1 1
929 8 1 682.500 -1 -1 1
930 8 2 684.167 -1 -1 1
931 8 1 658.116 1 -1 -1
932 8 2 672.153 1 -1 -1
933 8 1 738.213 1 -1 -1
934 8 2 594.534 1 -1 -1
935 8 1 681.236 1 -1 -1
936 8 2 627.650 1 -1 -1
937 8 1 704.904 1 -1 -1
938 8 2 551.870 1 -1 -1
939 8 1 693.623 1 -1 -1
940 8 2 594.534 1 -1 -1
941 8 1 624.993 1 -1 -1
942 8 2 602.660 1 -1 -1

943 8 1 700.228 1 -1 -1
944 8 2 585.450 1 -1 -1
945 8 1 611.874 1 -1 -1
946 8 2 555.724 1 -1 -1
947 8 1 579.167 1 -1 -1
948 8 2 574.934 1 -1 -1
949 8 1 720.872 1 -1 -1
950 8 2 584.625 1 -1 -1
951 8 1 690.320 1 -1 -1
952 8 2 555.724 1 -1 -1
953 8 1 677.933 1 -1 -1
954 8 2 611.874 1 -1 -1
955 8 1 674.600 1 -1 -1
956 8 2 698.254 1 -1 -1
957 8 1 611.999 1 -1 -1
958 8 2 748.130 1 -1 -1
959 8 1 530.680 1 -1 -1
960 8 2 689.942 1 -1 -1
Analysis of the Response Variable
Numerical
Summary
As a first step in the analysis, common summary statistics are
computed for the response variable.
Sample size = 480
Mean = 650.0773
Median = 646.6275
Minimum = 345.2940
Maximum = 821.6540
Range = 476.3600
Stan. Dev. = 74.6383
4-Plot The next step is generate a 4-plot of the response variable.
This 4-plot shows:
1. The run sequence plot (upper left corner) shows that the
location and scale are relatively constant. It also shows a
few outliers on the low side. Most of the points are in the
range 500 to 750. However, there are about half a dozen
points in the 300 to 450 range that may require special
attention.
A run sequence plot is useful for designed experiments in
that it can reveal time effects. Time is normally a
nuisance factor. That is, the time order on which runs are
made should not have a significant effect on the
response. If a time effect does appear to exist, this means
that there is a potential bias in the experiment that needs
to be investigated and resolved.
2. The lag plot (the upper right corner) does not show any
significant structure. This is another tool for detecting
any potential time effect.

3. The histogram (the lower left corner) shows the response
appears to be reasonably symmetric, but with a bimodal
distribution.
4. The normal probability plot (the lower right corner)
shows some curvature indicating that distributions other
than the normal may provide a better fit.
Analysis of the Batch Effect
Batch is a
Nuisance
Factor
The two nuisance factors in this experiment are the batch
number and the lab. There are two batches and eight labs.
Ideally, these factors will have minimal effect on the response
variable.
We will investigate the batch factor first.
Bihistogram
This bihistogram shows the following.
1. There does appear to be a batch effect.
2. The batch 1 responses are centered at 700 while the batch
2 responses are centered at 625. That is, the batch effect is
approximately 75 units.
3. The variability is comparable for the 2 batches.
4. Batch 1 has some skewness in the lower tail. Batch 2 has
some skewness in the center of the distribution, but not as
much in the tails compared to batch 1.
5. Both batches have a few low-lying points.
Although we could stop with the bihistogram, we will show a
few other commonly used two-sample graphical techniques for
comparison.

Quantile-
Quantile
Plot
This q-q plot shows the following.
1. Except for a few points in the right tail, the batch 1 values
have higher quantiles than the batch 2 values. This
implies that batch 1 has a greater location value than
batch 2.
2. The q-q plot is not linear. This implies that the difference
between the batches is not explained simply by a shift in
location. That is, the variation and/or skewness varies as
well. From the bihistogram, it appears that the skewness
in batch 2 is the most likely explanation for the non-
linearity in the q-q plot.
Box Plot
This box plot shows the following.
1. The median for batch 1 is approximately 700 while the
median for batch 2 is approximately 600.
2. The spread is reasonably similar for both batches, maybe
slightly larger for batch 1.
3. Both batches have a number of outliers on the low side.
Batch 2 also has a few outliers on the high side. Box plots
are a particularly effective method for identifying the
presence of outliers.
Block Plots A block plot is generated for each of the eight labs, with "1" and
"2" denoting the batch numbers. In the first plot, we do not
include any of the primary factors. The next 3 block plots
include one of the primary factors. Note that each of the 3
primary factors (table speed = X1, down feed rate = X2, wheel
grit size = X3) has 2 levels. With 8 labs and 2 levels for the
primary factor, we would expect 16 separate blocks on these
plots. The fact that some of these blocks are missing indicates
that some of the combinations of lab and primary factor are
empty.

These block plots show the following.
1. The mean for batch 1 is greater than the mean for batch 2
in all of the cases above. This is strong evidence that the
batch effect is real and consistent across labs and primary
factors.
Quantitative
Techniques
We can confirm some of the conclusions drawn from the above
graphics by using quantitative techniques. The F-test can be
used to test whether or not the variances from the two batches
are equal and the two sample t-test can be used to test whether
or not the means from the two batches are equal. Summary
statistics for each batch are shown below.
Batch 1:
MEAN = 688.9987
VARIANCE = 4296.6845
Batch 2:
MEAN = 611.1559
VARIANCE = 3825.9544
F-Test The two-sided F-test indicates that the variances for the two
batches are not significantly different at the 5 % level.
H0: σ1
2 = σ2
2
Ha: σ1
2 ≠ σ2
2
Test statistic: F = 1.123
Numerator degrees of freedom: ν1 = 239
Denominator degrees of freedom: ν2 = 239
Critical values: F1-α/2,ν1,ν2
= 0.845
Fα/2,ν1,ν2
= 1.289
Critical region: Reject H0 if F < 0.845 or F > 1.289
Two Sample
t-Test
Since the F-test indicates that the two batch variances are equal,
we can pool the variances for the two-sided, two-sample t-test
to compare batch means.
H0: μ1 = μ2
Ha: μ1 ≠ μ2
Pooled standard deviation: sp = 63.7285
Critical value: t1-α/2,ν = 1.965
The t-test indicates that the mean for batch 1 is larger than the
mean for batch 2 at the 5 % significance level.

Conclusions We can draw the following conclusions from the above analysis.
1. There is in fact a significant batch effect. This batch effect
is consistent across labs and primary factors.
2. The magnitude of the difference is on the order of 75 to
100 (with batch 2 being smaller than batch 1). The
standard deviations do not appear to be significantly
different.
3. There is some skewness in the batches.
This batch effect was completely unexpected by the scientific
investigators in this study.
Note that although the quantitative techniques support the
conclusions of unequal means and equal standard deviations,
they do not show the more subtle features of the data such as the
presence of outliers and the skewness of the batch 2 data.
Analysis of the Lab Effect
Box Plot The next matter is to determine if there is a lab effect. The
first step is to generate a box plot for the ceramic strength
based on the lab.
1. There is minor variation in the medians for the 8 labs.
2. The scales are relatively constant for the labs.
3. Two of the labs (3 and 5) have outliers on the low side.
Box Plot for
Batch 1
Given that the previous section showed a distinct batch effect,
the next step is to generate the box plots for the two batches
separately.

1. Each of the labs has a median in the 650 to 700 range.
2. The variability is relatively constant across the labs.
3. Each of the labs has at least one outlier on the low side.
Box Plot for
Batch 2
1. The medians are in the range 550 to 600.
2. There is a bit more variability, across the labs, for
batch2 compared to batch 1.
3. Six of the eight labs show outliers on the high side.
Three of the labs show outliers on the low side.
Conclusions We can draw the following conclusions about a possible lab
effect from the above box plots.
1. The batch effect (of approximately 75 to 100 units) on
location dominates any lab effects.
2. It is reasonable to treat the labs as homogeneous.
Analysis of Primary Factors
Main effects The first step in analyzing the primary factors is to determine which factors
are the most significant. The DOE scatter plot, DOE mean plot, and the DOE
standard deviation plots will be the primary tools, with "DOE" being short for
"design of experiments".

Since the previous pages showed a significant batch effect but a minimal lab
effect, we will generate separate plots for batch 1 and batch 2. However, the
labs will be treated as equivalent.
DOE
Scatter Plot
for Batch 1
This DOE scatter plot shows the following for batch 1.
1. Most of the points are between 500 and 800.
2. There are about a dozen or so points between 300 and 500.
3. Except for the outliers on the low side (i.e., the points between 300 and
500), the distribution of the points is comparable for the 3 primary
factors in terms of location and spread.
DOE Mean
Plot for
Batch 1

This DOE mean plot shows the following for batch 1.
1. The table speed factor (X1) is the most significant factor with an effect,
the difference between the two points, of approximately 35 units.
2. The wheel grit factor (X3) is the next most significant factor with an
effect of approximately 10 units.
3. The feed rate factor (X2) has minimal effect.
DOE SD
Plot for
Batch 1
This DOE standard deviation plot shows the following for batch 1.
1. The table speed factor (X1) has a significant difference in variability
between the levels of the factor. The difference is approximately 20
units.
2. The wheel grit factor (X3) and the feed rate factor (X2) have minimal
differences in variability.

DOE
Scatter Plot
for Batch 2
This DOE scatter plot shows the following for batch 2.
1. Most of the points are between 450 and 750.
2. There are a few outliers on both the low side and the high side.
3. Except for the outliers (i.e., the points less than 450 or greater than
750), the distribution of the points is comparable for the 3 primary
factors in terms of location and spread.
DOE Mean
Plot for
Batch 2
This DOE mean plot shows the following for batch 2.

1. The feed rate (X2) and wheel grit (X3) factors have an approximately
equal effect of about 15 or 20 units.
2. The table speed factor (X1) has a minimal effect.
DOE SD
Plot for
Batch 2
This DOE standard deviation plot shows the following for batch 2.
1. The difference in the standard deviations is roughly comparable for the
three factors (slightly less for the feed rate factor).
Interaction
Effects
The above plots graphically show the main effects. An additonal concern is
whether or not there any significant interaction effects.
Main effects and 2-term interaction effects are discussed in the chapter on
Process Improvement.
In the following DOE interaction plots, the labels on the plot give the
variables and the estimated effect. For example, factor 1 is table speed and it
has an estimated effect of 30.77 (it is actually -30.77 if the direction is taken
into account).

DOE
Interaction
Plot for
Batch 1
The ranked list of factors for batch 1 is:
1. Table speed (X1) with an estimated effect of -30.77.
2. The interaction of table speed (X1) and wheel grit (X3) with an
estimated effect of -20.25.
3. The interaction of table speed (X1) and feed rate (X2) with an
estimated effect of 9.7.
4. Wheel grit (X3) with an estimated effect of -7.18.
5. Down feed (X2) and the down feed interaction with wheel grit (X3) are
essentially zero.

DOE
Interaction
Plot for
Batch 2
The ranked list of factors for batch 2 is:
1. Down feed (X2) with an estimated effect of 18.22.
2. The interaction of table speed (X1) and wheel grit (X3) with an
estimated effect of -16.71.
3. Wheel grit (X3) with an estimated effect of -14.71
4. Remaining main effect and 2-factor interaction effects are essentially
zero.
Conclusions From the above plots, we can draw the following overall conclusions.
1. The batch effect (of approximately 75 units) is the dominant primary
factor.
2. The most important factors differ from batch to batch. See the above
text for the ranked list of factors with the estimated effects.
View
Dataplot
Macro for
this Case
Study
This page allows you to use Dataplot to repeat the analysis
outlined in the case study description on the previous page. It is

Click on the links below to start Dataplot and
run this case study yourself. Each step may use
results from previous steps, so please be patient.
Wait until the software verifies that the current
step is complete before clicking on the next step.
The links in this column will connect you with
more detailed information about each analysis
step from the case study description.
2. Plot of the response variable
1. Numerical summary of Y.
2. 4-plot of Y.
1. The summary shows the mean strength
is 650.08 and the standard deviation
of the strength is 74.64.
2. The 4-plot shows no drift in
the location and scale and a
bimodal distribution.
3. Determine if there is a batch effect.
1. Generate a bihistogram based on
the 2 batches.
2. Generate a q-q plot.
3. Generate a box plot.
4. Generate block plots.
5. Perform a 2-sample t-test for
equal means.
6. Perform an F-test for equal
1. The bihistogram shows a distinct
batch effect of approximately
75 units.
2. The q-q plot shows that batch 1
and batch 2 do not come from a
common distribution.
3. The box plot shows that there is
a batch effect of approximately
75 to 100 units and there are
some outliers.
4. The block plot shows that the batch
effect is consistent across labs
and levels of the primary factor.
5. The t-test confirms the batch
effect with respect to the means.
6. The F-test does not indicate any
significant batch effect with
respect to the standard deviations.
4. Determine if there is a lab effect.
1. Generate a box plot for the labs
with the 2 batches combined.
for batch 1 only.
for batch 2 only.
1. The box plot does not show a
significant lab effect.
significant lab effect for batch 1.
significant lab effect for batch 2.
5. Analysis of primary factors.
1. Generate a DOE scatter plot for
batch 1.
2. Generate a DOE mean plot for
batch 1.
3. Generate a DOE sd plot for
batch 1.
1. The DOE scatter plot shows the
range of the points and the
presence of outliers.
2. The DOE mean plot shows that
table speed is the most
significant factor for batch 1.
3. The DOE sd plot shows that
table speed has the most
variability for batch 1.

4. Generate a DOE scatter plot for
batch 2.
5. Generate a DOE mean plot for
batch 2.
6. Generate a DOE sd plot for
batch 2.
7. Generate a DOE interaction
effects matrix plot for
batch 1.
8. Generate a DOE interaction
effects matrix plot for
batch 2.
4. The DOE scatter plot shows
the range of the points and
the presence of outliers.
5. The DOE mean plot shows that
feed rate and wheel grit are
the most significant factors
for batch 2.
6. The DOE sd plot shows that
the variability is comparable
for all 3 factors for batch 2.
7. The DOE interaction effects
matrix plot provides a ranked
list of factors with the
estimated effects.
8. The DOE interaction effects
matrix plot provides a ranked
list of factors with the
estimated effects.
References For Chapter 1: Exploratory Data
Analysis
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Statistician, pp. 195-199.
Anscombe, F. and Tukey, J. W. (1963), The Examination and Analysis of
Residuals, Technometrics, pp. 141-160.
Barnett and Lewis (1994), Outliers in Statistical Data, 3rd. Ed., John Wiley
and Sons.
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Bloomfield, Peter (1976), Fourier Analysis of Time Series, John Wiley and
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Box, G. E. P. and Cox, D. R. (1964), An Analysis of Transformations, Journal
of the Royal Statistical Society, pp. 211-243, discussion pp. 244-252.
Box, G. E. P., Hunter, W. G., and Hunter, J. S. (1978), Statistics for
Experimenters: An Introduction to Design, Data Analysis, and Model
Building, John Wiley and Sons.
Box, G. E. P., and Jenkins, G. (1976), Time Series Analysis: Forecasting and
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Bradley, (1968). Distribution-Free Statistical Tests, Chapter 12.
Brown, M. B. and Forsythe, A. B. (1974), Journal of the American Statistical
Association, 69, pp. 364-367.
Bury, Karl (1999). Statistical Distributions in Engineering, Cambridge
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Chakravarti, Laha, and Roy, (1967). Handbook of Methods of Applied
Statistics, Volume I, John Wiley and Sons, pp. 392-394.
Chambers, John, William Cleveland, Beat Kleiner, and Paul Tukey, (1983),
Graphical Methods for Data Analysis, Wadsworth.
Chatfield, C. (1989). The Analysis of Time Series: An Introduction, Fourth
Edition, Chapman & Hall, New York, NY.
Cleveland, William (1985), Elements of Graphing Data, Wadsworth.

Cleveland, William and Marylyn McGill, Editors (1988), Dynamic Graphics
for Statistics, Wadsworth.
Cleveland, William (1993), Visualizing Data, Hobart Press.
Devaney, Judy (1997), Equation Discovery Through Global Self-Referenced
Geometric Intervals and Machine Learning, Ph.d thesis, George Mason
University, Fairfax, VA.
Draper and Smith, (1981). Applied Regression Analysis, 2nd ed., John Wiley
and Sons.
du Toit, Steyn, and Stumpf (1986), Graphical Exploratory Data Analysis,
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Efron and Gong (February 1983), A Leisurely Look at the Bootstrap, the
Jackknife, and Cross Validation, The American Statistician.
Evans, Hastings, and Peacock (2000), Statistical Distributions, 3rd. Ed., John
Wiley and Sons.
Everitt, Brian (1978), Multivariate Techniques for Multivariate Data, North-
Holland.
Filliben, J. J. (February 1975), The Probability Plot Correlation Coefficient
Test for Normality, Technometrics, pp. 111-117.
Fuller Jr., E. R., Frieman, S. W., Quinn, J. B., Quinn, G. D., and Carter, W. C.
(1994), Fracture Mechanics Approach to the Design of Glass Aircraft
Windows: A Case Study, SPIE Proceedings, Vol. 2286, (Society of Photo-
Optical Instrumentation Engineers (SPIE), Bellingham, WA).
Gill, Lisa (April 1997), Summary Analysis: High Performance Ceramics
Experiment to Characterize the Effect of Grinding Parameters on Sintered
Reaction Bonded Silicon Nitride, Reaction Bonded Silicon Nitride, and
Sintered Silicon Nitride , presented at the NIST - Ceramic Machining
Consortium, 10th Program Review Meeting, April 10, 1997.
Granger and Hatanaka (1964), Spectral Analysis of Economic Time Series,
Princeton University Press.
Grubbs, Frank (1950), Sample Criteria for Testing Outlying Observations,
Annals of Mathematical Statistics, 21(1) pp. 27-58.
Grubbs, Frank (February 1969), Procedures for Detecting Outlying
Observations in Samples, Technometrics, 11(1), pp. 1-21.
Hahn, G. J. and Meeker, W. Q. (1991), Statistical Intervals, John Wiley and
Sons.
Harris, Robert L. (1996), Information Graphics, Management Graphics.
Hastie, T., Tibshirani, R. and Friedman, J. (2001), The Elements of Statistical
Learning: Data Mining, Inference, and Prediction, Springer-Verlag, New
York.
Hawkins, D. M. (1980), Identification of Outliers, Chapman and Hall.
Boris Iglewicz and David Hoaglin (1993), "Volume 16: How to Detect and
Handle Outliers", The ASQC Basic References in Quality Control: Statistical
Techniques, Edward F. Mykytka, Ph.D., Editor.
Jenkins and Watts, (1968), Spectral Analysis and Its Applications, Holden-
Day.
Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate
Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

Johnson, Kotz, and Kemp, (1992), Univariate Discrete Distributions, 2nd.
Ed., John Wiley and Sons.
Kuo, Way and Pierson, Marcia Martens, Eds. (1993), Quality Through
Engineering Design", specifically, the article Filliben, Cetinkunt, Yu, and
Dommenz (1993), Exploratory Data Analysis Techniques as Applied to a
High-Precision Turning Machine, Elsevier, New York, pp. 199-223.
Levene, H. (1960). In Contributions to Probability and Statistics: Essays in
Honor of Harold Hotelling, I. Olkin et al. eds., Stanford University Press, pp.
278-292.
McNeil, Donald (1977), Interactive Data Analysis, John Wiley and Sons.
Mendenhall, William and Reinmuth, James (1982), Statistics for Management
and Ecomonics, Fourth Edition, Duxbury Press.
Mosteller, Frederick and Tukey, John (1977), Data Analysis and Regression,
Addison-Wesley.
Natrella, Mary (1963), Experimental Statistics, National Bureau of Standards
Handbook 91.
Nelson, Wayne (1982), Applied Life Data Analysis, Addison-Wesley.
Nelson, Wayne and Doganaksoy, Necip (1992), A Computer Program
POWNOR for Fitting the Power-Normal and -Lognormal Models to Life or
Strength Data from Specimens of Various Sizes, NISTIR 4760, U.S.
Department of Commerce, National Institute of Standards and Technology.
Neter, Wasserman, and Kutner (1990), Applied Linear Statistical Models, 3rd
ed., Irwin.
Pepi, John W., (1994), Failsafe Design of an All BK-7 Glass Aircraft
Window, SPIE Proceedings, Vol. 2286, (Society of Photo-Optical
Instrumentation Engineers (SPIE), Bellingham, WA).
The RAND Corporation (1955), A Million Random Digits with 100,000
Normal Deviates, Free Press.
Rosner, Bernard (May 1983), Percentage Points for a Generalized ESD
Many-Outlier Procedure,Technometrics, 25(2), pp. 165-172.
Ryan, Thomas (1997), Modern Regression Methods, John Wiley.
Scott, David (1992), Multivariate Density Estimation: Theory, Practice, and
Visualization , John Wiley and Sons.
Snedecor, George W. and Cochran, William G. (1989), Statistical Methods,
Eighth Edition, Iowa State University Press.
Stefansky, W. (1972), Rejecting Outliers in Factorial Designs, Technometrics,
14, pp. 469-479.
Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some
Comparisons, Journal of the American Statistical Association, 69, pp. 730-
737.
Stephens, M. A. (1976). Asymptotic Results for Goodness-of-Fit Statistics
with Unknown Parameters, Annals of Statistics, 4, pp. 357-369.
Stephens, M. A. (1977). Goodness of Fit for the Extreme Value Distribution,
Biometrika, 64, pp. 583-588.
Stephens, M. A. (1977). Goodness of Fit with Special Reference to Tests for
Exponentiality , Technical Report No. 262, Department of Statistics, Stanford
University, Stanford, CA.

Stephens, M. A. (1979). Tests of Fit for the Logistic Distribution Based on the
Empirical Distribution Function, Biometrika, 66, pp. 591-595.
Tietjen and Moore (August 1972), Some Grubbs-Type Statistics for the
Detection of Outliers, Technometrics, 14(3), pp. 583-597.
Tufte, Edward (1983), The Visual Display of Quantitative Information,
Graphics Press.
Tukey, John (1977), Exploratory Data Analysis, Addison-Wesley.
Velleman, Paul and Hoaglin, David (1981), The ABC's of EDA: Applications,
Basics, and Computing of Exploratory Data Analysis, Duxbury.
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the Analysis of Data, Biometrika, 5(5), pp. 1-19.
1.4.4.
Dataplot Commands for EDA Techniques
This page documents the Dataplot commands that can be used for the graphical and analytical techniques discussed in this chapter. This is
only meant to guide you to the appropriate commands. The complete documentation for these commands is available in the Dataplot
Reference Manual.
Dataplot Commands for 1-Factor ANOVA The Dataplot command for a one way analysis of variance is
ANOVA Y X
where Y is a response variable and X is a group identifier variable.
Dataplot is currently limited to the balanced case (i.e., each level has the same number of observations) and it does not compute
interaction effect estimates.
Dataplot Commands for Multi-Factor ANOVA The Dataplot commands for generating multi-factor analysis of variance are:
ANOVA Y X1
ANOVA Y X1 X2
ANOVA Y X1 X2 X3
ANOVA Y X1 X2 X3 X4
ANOVA Y X1 X2 X3 X4 X5
where Y is the response variable and X1, X2, X3, X4, and X5 are factor variables. Dataplot allows up to 10 factor variables.
Dataplot is currently limited to the balanced case (i.e., each level has the same number of observations) and it does not compute
interaction effect estimates.
Dataplot Commands for the Anderson-Darling Test The Dataplot commands for the Anderson-Darling test are
ANDERSON DARLING NORMAL TEST Y
ANDERSON DARLING LOGNORMAL TEST Y
ANDERSON DARLING EXPONENTIAL TEST Y
ANDERSON DARLING WEIBULL TEST Y
ANDERSON DARLING EXTREME VALUE TYPE I TEST Y
where Y is the response variable.
Dataplot Commands for Autocorrelation To generate the lag 1 autocorrelation value in Dataplot, enter
LET A = AUTOCORRELATION Y
In Dataplot, the easiest way to generate the autocorrelations for lags greater than 1 is:

AUTOCORRELATION PLOT Y
LET AC = YPLOT
LET LAG = XPLOT
RETAIN AC LAG SUBSET TAGPLOT = 1
The AUTOCORRELATION PLOT command generates an autocorrelation plot for lags 0 to N/4. It also generates 95% and 99%
confidence limits for the autocorrelations. Dataplot stores the plot coordinates in the internal variables XPLOT, YPLOT, and TAGPLOT.
The 2 LET commands and the RETAIN command are used to extract the numerical values of the autocorrelations. The variable LAG
identifies the lag while the corresponding row of AC contains the autocorrelation value.
Dataplot Commands for Autocorrelation Plots The command to generate an autocorrelation plot is
AUTOCORRELATION PLOT Y
The appearance of the autocorrelation plot can be controlled by appropriate settings of the LINE, CHARACTER, and SPIKE commands.
Dataplot draws the following curves on the autocorrelation plot:
1. The auotocorrelations.
2. A reference line at zero.
3. A reference line at the upper 95% confidence limit.
4. A reference line at the lower 95% confidence limit.
For example, to draw the autocorrelations as spikes, the zero reference line as a solid line, the 95% lines as dashed lines, and the 99% line
as dotted lines, enter the command
LINE BLANK SOLID DASH DASH DOT DOT
CHARACTER BLANK ALL
SPIKE ON OFF OFF OFF OFF OFF
SPIKE BASE 0
By default, the confidence bands are fixed width. This is appropriate for testing for white noise (i.e., randomness). For Box-Jenkins
modeling, variable-width confidence bands are more appropriate. Enter the following command for variable-width confidence bands:
SET AUTOCORRELATION BAND BOX-JENKINS
To restore fixed-width confidence bands, enter
SET AUTOCORRELATION BAND WHITE-NOISE
Dataplot Commands for the Bartlett Test The Dataplot command for the Bartlett test is
BARTLETT TEST Y X
where Y is the response variable and X is the group id variable.
The above computes the standard form of Bartlett's test. To compute the Dixon-Massey form of Bartlett's test, the Dataplot command is
one of the following (these are synonyms, not distinct commands)
DIXON BARTLETT TEST Y X
DIXON MASSEY BARTLETT TEST Y X
DM BARTLETT TEST Y X
Dataplot Commands for Bihistograms The Dataplot command to generate a bihistogram is
BIHISTOGRAM Y1 Y2
As with the standard histogram, the class width, the lower class limit, and the upper class limit can be controlled with the commands
CLASS WIDTH <value>
CLASS LOWER <value>
CLASS UPPER <value>
In addition, relative bihistograms, cumulative bihistograms, and relative cumulative bihistograms can be generated with the commands
RELATIVE BIHISTOGRAM Y1 Y2
CUMULATIVE BIHISTOGRAM Y1 Y2
RELATIVE CUMULATIVE BIHISTOGRAM Y1 Y2
Dataplot Commands for the Binomial Probability Functions Dataplot can compute the probability functions for the binomial distribution

with the following commands.
cdf LET Y = BINCDF(X,P,N)
pdf LET Y = BINPDF(X,P,N)
ppf LET Y = BINPPF(F,P,N)
random numbers LET N = value
LET P = value
LET Y = BINOMIAL RANDOM NUMBERS FOR I = 1 1 1000
probability plot LET N = value
LET P = value
BINOMIAL PROBABILITY PLOT Y
where X can be a number, a parameter, or a variable. P and N are the shape parameters and are required. They can be a number, a
parameter, or a variable. They are typically a number or a parameter.
These functions can be used in the Dataplot PLOT and FIT commands as well. For example,
PLOT BINPDF(X,0.5,100) FOR X = 0 1 100
Return to the Binomial Distribution Page
Dataplot Commands for the Block Plot The Dataplot command for the block plot is
BLOCK PLOT Y X1 X2 X3 etc. XP
where
Y is the response variable,
X1, X2, X3, etc. are the one or more nuisance (= secondary) factors, and
XP is the primary factor of interest.
The following commands typically precede the block plot.
CHARACTER 1 2
LINE BLANK BLANK
These commands set the plot character for the primary factor. Although 1 and 2 are useful indicators, the choice of plot character is at the
discretion of the user.
Dataplot Commands for the Bootstrap Plot The Dataplot command for the bootstrap plot is
BOOTSTRAP <STAT> PLOT Y
where <STAT> is one of the following:
MEAN
MIDMEAN
MIDRANGE
MEDIAN
TRIMMED MEAN
WINSORIZED MEAN
GEOMETRIC MEAN
HARMONIC MEAN
SUM
PRODUCT
MINIMUM
MAXIMUM
STANDARD DEVIATION
VARIANCE
STANDARD DEVIATION OF MEAN
VARIANCE OF MEAN
RELATIVE STANDARD DEVIATION
RELATIVE VARIANCE
AVERAGE ABSOLUTE DEVIATION
MEDIAN ABSOLUTE DEVIATION
LOWER QUARTILE
LOWER HINGE
UPPER QUARTILE
UPPER HINGE

FIRST DECILE
SECOND DECILE
THIRD DECILE
FOURTH DECILE
FIFTH DECILE
SIXTH DECILE
SEVENTH DECILE
EIGHTH DECILE
NINTH DECILE
PERCENTILE
SKEWNESS
KURTOSIS
AUTOCORRELATION
AUTOCOVARIANCE
SINE FREQUENCY
COSINE FREQUENCY
TAGUCHI SN0
TAGUCHI SN+
TAGUCHI SN-
TAGUCHI SN00
The BOOTSTRAP PLOT command is almost always followed by a histogram or some other distributional plot.
Dataplot automatically stores the following internal parameters after a BOOTSTRAP PLOT command:
BMEAN - mean of the plotted bootstrap values
BSD - standard deviation of the plotted bootstrap values
B001 - the 0.1 percentile of the plotted bootstrap values
B10 - the 10 percentile of the plotted bootstrap values
These internal parameters are useful for generating confidence intervals and can be printed (PRINT BMEAN) or used as any user-defined
parameter could (e.g., LET UCL = B95).
To specify the number of bootstrap subsamples to use, enter the command
BOOTSTRAP SAMPLE <N>
where <N> is the number of samples you want. The default is 500 (it may be 100 in older implementations).
Dataplot can also generate bootstrap estimates for statistics that are not directly supported. The following example shows a bootstrap
calculation for the mean of 500 normal random numbers. Although we can do this directly in Dataplot, this demonstrates the steps
necessary for an unsupported statistic. The subsamples are generated with a loop. The BOOTSTRAP INDEX and BOOTSTRAP
SAMPLE commands generate a single subsample which is stored in Y2. The desired statistic is then calculated for Y2 and the result
stored in an array. After the loop, the array XMEAN contains the 100 mean values.
LET Y = NORMAL RANDOM NUMBERS FOR I = 1 1 500
LET N = SIZE Y
LOOP FOR K = 1 1 500
LET IND = BOOTSTRAP INDEX FOR I = 1 1 N
LET Y2 = BOOTSTRAP SAMPLE Y IND
LET A = MEAN Y2
LET XMEAN(K) = A
END OF LOOP
HISTOGRAM XMEAN
Dataplot Command for the Box-Cox Linearity Plot The Dataplot command to generate a Box-Cox linearity plot is
BOX-COX LINEARITY PLOT Y X

where Y and X are the response variables.
Dataplot Command for the Box-Cox Normality Plot The Dataplot command to generate a Box-Cox normality plot is
BOX-COX NORMALITY PLOT Y
Dataplot Commands for the Boxplot The Dataplot command to generate a boxplot is
BOX PLOT Y X
The BOX PLOT command is usually preceded by the commands
CHARACTER BOX PLOT
LINE BOX PLOT
These commands set the default line and character settings for the box plot. You can use the CHARACTER and LINE commands to
choose your own line and character settings if you prefer.
To show the outliers as circles, enter the command
FENCES ON
Dataplot Commands for the Cauchy Probability Functions Dataplot can compute the probability functions for the Cauchy distribution
with the following commands.
cdf LET Y = CAUCDF(X,A,B)
pdf LET Y = CAUPDF(X,A,B)
ppf LET Y = CAUPPF(X,A,B)
hazard LET Y = CAUHAZ(X,A,B)
cumulative hazard LET Y = CAUCHAZ(X,A,B)
survival LET Y = 1 - CAUCDF(X,A,B)
inverse survival LET Y = CAUPPF(1-X,A,B)
random numbers LET Y = CAUCHY RANDOM NUMBERS FOR I = 1 1 1000
probability plot CAUCHY PROBABILITY PLOT Y
where X can be a number, a parameter, or a variable. A and B are the location and scale parameters and they are optional (a location of 0
and scale of 1 are used if they are omitted). If given, A and B can be a number, a parameter, or a variable. However, they are typically
either a number or a parameter.
PLOT CAUPDF(X) FOR X = -5 0.01 5
Dataplot Commands for the Chi-Square Probability Functions Dataplot can compute the probability functions for the chi-square
distribution with the following commands.
cdf LET Y = CHSCDF(X,NU,NU2,A,B)
pdf LET Y = CHSPDF(X,NU,A,B)
ppf LET Y = CHSPPF(X,NU,A,B)
random numbers LET NU = value
LET Y = CHI-SQUARE RANDOM NUMBERS FOR I = 1 1 1000
probability plot LET NU = value
CHI-SQUARE PROBABILITY PLOT Y
ppcc plot LET NU = value
CHI-SQUARE PPCC PLOT Y
where X can be a number, a parameter, or a variable. NU is the shape parameter (number of degrees of freedom). NU can be a number, a
parameter, or a variable. However, it is typically either a number or a parameter. A and B are the location and scale parameters and they
are optional (a location of 0 and scale of 1 are used if they are omitted). If given, A and B can be a number, a parameter, or a variable.
However, they are typically either a number or a parameter.
PLOT CHSPDF(X,5) FOR X = 0 0.01 5

Dataplot Commands for the Chi-Square Goodness of Fit Test The Dataplot commands for the chi-square goodness of fit test are
<dist> CHI-SQUARE GOODNESS OF FIT TEST Y
<dist> CHI-SQUARE GOODNESS OF FIT TEST Y X
<dist> CHI-SQUARE GOODNESS OF FIT TEST Y XL XU
where <dist> is one of 70+ built-in distributions. Dataplot supports the chi-square goodness-of-fit test for all distributions that support the
cumulative distribution function. To see a list of supported distributions, enter the command LIST DISTRIBUTIONS. Some specific
examples are
NORMAL CHI-SQUARE GOODNESS OF FIT TEST Y
LOGISTIC CHI-SQUARE GOODNESS OF FIT TEST Y
DOUBLE EXPONENTIAL CHI-SQUARE GOODNESS OF FIT TEST Y
You can specify the location and scale parameters (for any of the supported distributions) by entering
LET CHSLOC = value
LET CHSSCAL = value
You may need to enter the values for 1 or more shape parameters for distributions that require them. For example, to specify the shape
parameter gamma for the gamma distribution, enter the commands
LET GAMMA = value
GAMMA CHI-SQUARE GOODNESS OF FIT TEST Y
Dataplot also allows you to control the class width, the lower limit (i.e., start of the first bin), and the upper limit (i.e., the end value for
the last bin). These commands are
CLASS WIDTH value
CLASS LOWER value
CLASS UPPER value
In most cases, the default Dataplot class intervals will be adequate.
If your data are already binned, you can enter the commands
NORMAL CHI-SQUARE GOODNESS OF FIT TEST Y X
NORMAL CHI-SQUARE GOODNESS OF FIT TEST Y XL XU
In both commands above, Y is the frequency variable. If one X variable is given, Dataplot assumes that it is the bin mid point and that
bins have equal width. If two X variables are given, Dataplot assumes that these are the bin end points and that the bin widths are not
necessarily of equal width. Unequal bin widths are typically used to combine classes with small frequencies since the chi-square
approximation for the test may not be accurate if there are frequency classes with less than five observations.
Dataplot Command for the Chi-Square Test for the Standard Deviation The Dataplot command for the chi-square test for the standard
deviation is
CHI-SQUARE TEST Y A
where Y is the response variable and A is the value being tested.
Dataplot Command for Complex Demodulation Amplitude Plot The Dataplot command for a complex demodulation amplitude plot is
COMPLEX DEMODULATION AMPLITUDE PLOT Y
Return to the Complex Demodulation Amplitude Plot Page
Dataplot Commands for Complex Demodulation Phase Plot The Dataplot commands for a complex demodulation phase plot are
DEMODULATION FREQUENCY <VALUE>
COMPLEX DEMODULATION PHASE PLOT Y
where Y is the response variable. The DEMODULATION FREQUENCY is used to specify the desired frequency for the COMPLEX
DEMODULATION PLOT. The value of the demodulation frequency is usually obtained from a spectral plot.
Return to the Complex Demodulation Phase Plot Page
Dataplot Commands for Conditioning Plot The Dataplot command to generate a conditioning plot is
CONDITION PLOT Y X COND

Y is the response variable, X is the independent variable, and COND is the conditioning variable. Dataplot expects COND to contain a
discrete number of distinct values. Dataplot provides a number of commands for creating a discrete variable from a continuous variable.
For example, suppose X2 is a continuous variable that we want to split into 4 regions. We could enter the following sequence of
commands to create a discrete variable from X2.
LET COND = X2
LET COND = 1 SUBSET X2 = 0 TO 99.99
LET COND = 4 SUBSET X2 = 300 TO 400
The SUBSET feature can be used as above to create whatever ranges we want. A simpler, more automatic way is to use the CODE
command in Dataplot. For example,
LET COND = CODE4 X2
splits the data into quartiles and assigns a value of 1 to 4 to COND based on what quartile the corresponding value of X2 is in.
The appearance of the plot can be controlled by appropriate settings of the CHARACTER and LINE commands and their various attribute
setting commands.
In addition, Dataplot provides a number of SET commands to control the appearance of the conditioning plot. In Dataplot, enter HELP
CONDITION PLOT for details.
Dataplot Commands for Confidence Limits and One Sample t-test The following commands can be used in Dataplot to generate a
confidence interval for the mean or to generate a one sample t-test, respectively.
CONFIDENCE LIMITS Y
T TEST Y U0
where Y is the response variable and U0 is a parameter or scalar value that defines the hypothesized value.
Return to the Confidence Limits for the Mean Page
Dataplot Commands for Contour Plots The Dataplot command for generating a contour plot is
CONTOUR PLOT Z X Y Z0
The variables X and Y define the grid, the Z variable is the response variable, and Z0 defines the desired contour levels. Currently,
Dataplot only supports contour plots over regular grids. Dataplot does provide 2D interpolation capabilities to form regular grids from
irregular data. Dataplot also does not support labels for the contour lines or solid fills between contour lines.
Dataplot Commands for Control Charts The Dataplot commands for generating control charts are
XBAR CONTROL CHART Y X
R CONTROL CHART Y X
S CONTROL CHART Y X
C CONTROL CHART Y X
U CONTROL CHART Y X
P CONTROL CHART Y X
NP CONTROL CHART Y X
CUSUM CONTROL CHART Y X
EWMA CONTROL CHART Y X
MOVING AVERAGE CONTROL CHART Y
MOVING AVERAGE CONTROL CHART Y X
MOVING RANGE CONTROL CHART Y
MOVING RANGE CONTROL CHART Y X
MOVING SD CONTROL CHART Y
MOVING SD CONTROL CHART Y X
where Y is the response variable and X is the group identifier variable.
Dataplot computes the control limits. In some cases, you may have pre determined values to put in as control limits (e.g., based on
historical data). Dataplot allows you to specify these limits by entering the following commands before the control chart command.
LET TARGET = <value>
LET LSL = <value>
LET USL = <value>
These allow you to specify the target, lower specification, and upper specification limit respectively.
The appearance of the plot can be controlled by appropriate settings of the LINE and CHARACTER commands. Specifically,
there are seven settings:

1. the response curve
2. the reference line at the Dataplot determined target value
3. the reference line at the Dataplot determined upper specification limit
4. the reference line at the Dataplot determined lower specification limit
5. the reference line at the user-specified target value
6. the reference line at the user-specified upper specification limit
7. the reference line at the user-specified lower specification limit
Dataplot Commands for DEX Contour Plots The Dataplot command for generating a linear dex contour plot is
DEX CONTOUR PLOT Y X1 X2 Y0
The variables X1 and X2 are the two factor variables, Y is the response variable, and Y0 defines the desired contour levels.
Dataplot does not have a built-in quadratic dex contour plot. However, the macro DEXCONTQ.DP will generate a quadratic
dex contour plot. Enter LIST DEXCONTQ.DP for more information.
Dataplot Commands for DEX Interaction Effects Plots The Dataplot command to generate a dex mean interaction effects plot
is
DEX MEAN INTERACTION EFFECTS PLOT Y X1 X2 X3 X4 X5
where Y is the response variable and X1, X2, X3, X4, and X5 are the factor variables. The number of factor variables can
vary, and is at least one.
Dataplot supports the following additional plots for other location statistics
DEX MEDIAN INTERACTION EFFECTS PLOT Y X1 X2 X3 X4 X5
DEX MIDMEAN INTERACTION EFFECTS PLOT Y X1 X2 X3 X4 X5
DEX TRIMMED MEAN INTERACTION EFFECTS PLOT Y X1 X2 X3 X4 X5
DEX WINSORIZED MEAN INTERACTION EFFECTS PLOT Y X1 X2 X3 X4 X5
If you want the raw data plotted rather than a statistic, enter
DEX INTERACTION EFFECTS PLOT Y X1 X2 X3 X4 X5
The LINE and CHARACTER commands can be used to control the appearance of the plot. For example, a typical sequence
of commands might be
LINE SOLID SOLID
CHARACTER CIRCLE BLANK
CHARACTER FILL ON
This draws the connecting line between the levels of a factor and the overall mean reference line as solid lines. In addition,
the level means are drawn with a solid fill circle.
This command is a variant of the SCATTER PLOT MATRIX command. There are a number of options to control the
appearance of these plots. In Dataplot, you can enter HELP SCATTER PLOT MATRIX for details.
Dataplot Commands for DEX Mean Plots The Dataplot command to generate a dex mean plot is
DEX MEAN PLOT Y X1 X2 X3 X4 X5
Dataplot supports the following additional plots for other location statistics
DEX MEDIAN PLOT Y X1 X2 X3 X4 X5
DEX MIDMEAN PLOT Y X1 X2 X3 X4 X5
DEX TRIMMED MEAN PLOT Y X1 X2 X3 X4 X5
DEX WINSORIZED MEAN PLOT Y X1 X2 X3 X4 X5
LINE SOLID SOLID
CHARACTER FILL ON
It is often desirable to provide alphabetic labels for the factors. For example, if there are 2 factors, time and temperature, the
following commands could be used to define alphabetic labels:
XLIMITS 1 2
XTIC OFFSET 0.5 0.5
MAJOR XTIC MARK NUMBER 2
MINOR XTIC MARK NUMBER 0
XTIC MARK LABEL FORMAT ALPHA
XTIC MARK LABEL CONTENT TIME TEMPERATURE

Dataplot Commands for a DEX Scatter Plot The Dataplot command for generating a dex scatter plot is
DEX SCATTER PLOT Y X1 X2 X3 X4 X5
The DEX SCATTER PLOT is typically preceded by the commands
CHARACTER X BLANK
LINE BLANK SOLID
However, you can set the plot character and line settings to whatever seems appropriate.
XLIMITS 1 2
XTIC OFFSET 0.5 0.5
Dataplot Commands for a DEX Standard Deviation Plot The Dataplot command to generate a dex standard deviation plot is
DEX STANDARD DEVIATION PLOT Y X1 X2 X3 X4 X5
Dataplot supports the following additional plots for other scale statistics.
DEX VARIANCE PLOT Y X1 X2 X3 X4 X5
DEX MEDIAN ABSOLUTE VALUE PLOT Y X1 X2 X3 X4 X5
DEX AVERAGE ABSOLUTE VALUE PLOT Y X1 X2 X3 X4 X5
DEX RANGE VALUE PLOT Y X1 X2 X3 X4 X5
DEX MIDRANGE VALUE PLOT Y X1 X2 X3 X4 X5
DEX MINIMUM PLOT Y X1 X2 X3 X4 X5
DEX MAXIMUM PLOT Y X1 X2 X3 X4 X5
LINE SOLID SOLID
CHARACTER FILL ON
XLIMITS 1 2
XTIC OFFSET 0.5 0.5
Dataplot Commands for the Double Exponential Probability Functions Dataplot can compute the probability functions for
the double exponential distribution with the following commands.
cdf LET Y = DEXCDF(X,A,B)
pdf LET Y = DEXPDF(X,A,B)
ppf LET Y = DEXPPF(X,A,B)
hazard LET Y = DEXHAZ(X,A,B)/(1 - DEXCDF(X,A,B))
cumulative hazard LET Y = -LOG(1 - DEXCHAZ(X,A,B))
survival LET Y = 1 - DEXCDF(X,A,B)
inverse survival LET Y = DEXPPF(1-X,A,B)
random numbers LET Y = DOUBLE EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 1000
probability plot DOUBLE EXPONENTIAL PROBABILITY PLOT Y
maximum likelihood LET MU = MEDIAN Y
LET BETA = MEDIAN ABSOLUTE DEVIATION Y
where X can be a number, a parameter, or a variable. A and B are the location and scale parameters and they are optional (a
location of 0 and scale of 1 are used if they are omitted). If given, A and B can be a number, a parameter, or a variable.

However, they are typically either a number or a parameter.
PLOT DEXPDF(X) FOR X = -5 0.01 5
Dataplot Command for Confidence Interval for the Difference Between Two Proportions The Dataplot command for a
confidence interval for the difference of two proportions is
DIFFERENCE OF PROPORTIONS CONFIDENCE INTERVAL Y1 Y2
where Y1 contains the data for sample 1 and Y2 contains the data for sample 2. For large samples, Dataplot uses the binomial
computation, not the normal approximation.
The following command sets the lower and upper bounds that define a success in the response variable
ANOP LIMITS <lower bound> <upper bound>
Dataplot Command for Duane Plot The Dataplot command for a Duane plot is
DUANE PLOT Y
where Y is a response variable containing failure times.
Dataplot Command for Starting Values for Rational Function Models Starting values for a rational function model can be
obtained by fitting an exact rational function to a subset of the original data. The number of points in the subset should equal
the number of parameters to be estimated in the rational function model. The EXACT RATIONAL FIT can be used to fit this
subset model and thus to provide starting values for the rational function model. For example, to fit a quadratic/quadratic
rational function model to data in X and Y, you might do something like the following.
LET X2 = DATA 12 17 22 34 56
LET Y2 = DATA 7 9 6 19 23
EXACT 2/2 FIT Y2 X2 Y X
FIT Y = (A0 + A1*X + A2*X**2)/(1 + B1*X + B2*X**2)
The DATA command is used to define the subset variables and EXACT 2/2 FIT is used to fit the exact rational function. The
"2/2" identifies the degree of the numerator as 2 and the degree of the denominator as 2. It provides values for A0, A1, A2,
B1, and B2, which are used to fit the rational function model for the full data set.
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Dataplot Commands for the Exponential Probability Functions Dataplot can compute the probability functions for the
exponential distribution with the following commands.
cdf LET Y = EXPCDF(X,A,B)
pdf LET Y = EXPPDF(X,A,B)
ppf LET Y = EXPPPF(X,A,B)
hazard LET Y = EXPHAZ(X,A,B)
cumulative hazard LET Y = EXPCHAZ(X,A,B)
survival LET Y = 1 - EXPCDF(X,A,B)
inverse survival LET Y = EXPPPF(1-X,A,B)
random numbers LET Y = EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 1000
probability plot EXPONENTIAL PROBABILITY PLOT Y
parameter estimation If your data are not censored, enter the commands
SET CENSORING TYPE NONE
EXPONENTIAL MLE Y
If your data have type 1 censoring at fixed time t0, enter the commands
LET TEND = censoring time
SET CENSORING TYPE 1
EXPONENTIAL MLE Y X
If your data have type 2 censoring, enter the commands
EXPONENTIAL MLE Y X
Y is the response variable and X is the censoring variable where a value of 1
indicates a failure time and a value of 0 indicates a censoring time. In addition to
the point estimates, confidence intervals for the parameters are generated.
In the above, X can be a number, a parameter, or a variable. A and B are the location and scale parameters and they are
optional (a location of 0 and scale of 1 are used if they are omitted). If given, A and B can be a number, a parameter, or a
variable. However, they are typically either a number or parameter.

PLOT EXPPDF(X) FOR X = 0 0.01 4
Dataplot Command for Generalized ESD Test The Dataplot command for the generalized ESD (Extreme Studentized
Deviate) test is
LET NOUTLIER = <value>
EXTREME STUDENTIZED DEVIATE TEST Y
where Y is the response variable and NOUTLIER specifies the upper bound on the number of outliers to test.
Dataplot Commands for the Extreme Value Type I (Gumbel) Distribution To specify the form of the Gumbel distribution
based on the smallest value, enter the command
SET MINMAX 1
To specify the form of the Gumbel distribution based on the largest value, enter the command
SET MINMAX 2
One of these commands must be entered before using the commands below.
Dataplot can compute the probability functions for the extreme value type I distribution with the following commands.
cdf LET Y = EV1CDF(X,A,B)
pdf LET Y = EV1PDF(X,A,B)
ppf LET Y = EV1PPF(X,A,B)
hazard LET Y = EV1HAZ(X,A,B)
cumulative hazard LET Y = EV1CHAZ(X,A,B)
survival LET Y = 1 - EV1CDF(X,A,B)
inverse survival LET Y = EV1PPF(1-X,A,B)
random numbers LET Y = EXTREME VALUE TYPE 1 RANDOM NUMBERS FOR I = 1 1 1000
probability plot EXTREME VALUE TYPE 1 PROBABILITY PLOT Y
maximum likelihood EV1 MLE Y
This returns a point estimate for the full sample case. It does not provide
confidence intervals for the parameters and it does not handle censored data.
In the above, X can be a number, a parameter, or a variable. A and B are the location and scale parameters and they are
variable. However, they are typically either a number or a parameter.
SET MINMAX 1
PLOT EV1PDF(X) FOR X = -4 0.01 4
Dataplot Commands for the F Distribution Probability Functions Dataplot can compute the probability functions for the F
cdf LET Y = FCDF(X,NU1,NU2,A,B)
pdf LET Y = FPDF(X,NU1,NU2,A,B)
ppf LET Y = FPPF(X,NU1,NU2,A,B)
random numbers LET NU1 = value
LET NU2 = value
LET Y = F RANDOM NUMBERS FOR I = 1 1 1000
probability plot LET NU1 = value
LET NU2 = value
F PROBABILITY PLOT Y
where X can be a number, a parameter, or a variable. NU1 and NU2 are the shape parameters (= number of degrees of
freedom). NU1 and NU2 can be a number, a parameter, or a variable. However, they are typically either a number or a
parameter. A and B are the location and scale parameters and they are optional (a location of 0 and scale of 1 are used if they
are omitted). If given, A and B can be a number, a parameter, or a variable. However, they are typically either a number or a
parameter.
PLOT FPDF(X,10,10) FOR X = 0 0.01 5
Dataplot Command for F Test for Equality of Two Standard Deviations The Datpalot command for the F test for the equality
of two standard deviations is
F TEST Y1 Y2
where Y1 is the data for sample one and Y2 is the data for sample two.
Dataplot Commands for the Histogram The Dataplot command to generate a histogram is
HISTOGRAM Y

where Y is the response variable. The different variants of the histogram can be generated with the commands
RELATIVE HISTOGRAM Y
CUMULATIVE HISTOGRAM Y
RELATIVE CUMULATIVE HISTOGRAM Y
The class width, the start of the first class, and the end of the last class can be specified with the commands
CLASS WIDTH <value>
CLASS LOWER <value>
CLASS UPPER <value>
By default, Dataplot uses a class width of 0.3*SD where SD is the standard deviation of the data. The lower class limit is the
sample mean minus 6 times the sample standard deviation. Similarly, the upper class limit is the sample mean plus 6 times
the sample standard deviation.
By default, Dataplot uses the probability normalization for relative histograms. If you want the relative counts to sum to one
instead, enter the command
SET RELATIVE HISTOGRAM PERCENT
To reset the probability interpretation, enter
SET RELATIVE HISTOGRAM AREA
Dataplot Commands for a Lag Plot The Dataplot command to generate a lag plot is
LAG PLOT Y
The appearance of the lag plot can be controlled with appropriate settings for the LINE and CHARACTER commands.
Typical settings for these commands would be
LINE BLANK
CHARACTER X
To generate a linear fit of the points on the lag plot when an autoregressive fit is suggested, enter the following commands
LAG PLOT Y
LINEAR FIT YPLOT XPLOT
The variables YPLOT and XPLOT are internal variables that store the coordinates of the most recent plot.
Dataplot Commands for the Fatigue Life Probability Functions Dataplot can compute the probability functions for the fatigue
life distribution with the following commands.
cdf LET Y = FLCDF(X,GAMMA,A,B)
pdf LET Y = FLPDF(X,GAMMA,A,B)
ppf LET Y = FLPPF(X,GAMMA,A,B)
hazard LET Y = FLHAZ(X,GAMMA,A,B)
cumulative hazard LET Y = FLCHAZ(X,GAMMA,A,B)
survival LET Y = 1 - FLCDF(X,GAMMA,A,B)
inverse survival LET Y = FLPPF(1-X,GAMMA,A,B)
random numbers LET GAMMA = value
LET Y = FATIGUE LIFE RANDOM NUMBERS FOR I = 1 1 1000
probability plot LET GAMMA = value
FATIGUE LIFE PROBABILITY PLOT Y
ppcc plot LET GAMMA = value
FATIGUE LIFE PPCC PLOT Y
where X can be a number, a parameter, or a variable. FLMA is the shape parameter and is required. It can be a number, a
parameter, or a variable. It is typically a number or a parameter. A and B are the location and scale parameters and they are
PLOT FLPDF(X,2) FOR X = 0.01 0.01 10
Dataplot Command for Fitting Dataplot can generate both linear and nonlinear fit commands.
For example, to generate a linear fit of Y versus X1, X2, and X3, the command is:
FIT Y X1 X2 X3
To generate quadratic and cubic fits of Y versus X, the commands are:
QUADRATIC FIT Y X
CUBIC FIT Y X
Nonlinear fits are generated by entering an equation. For example,
FIT Y = A*(EXP(-B*X/10) - EXP(-X/10))
FIT Y = C/(1+C*A*X**B)
FIT Y = A - B*X - ATAN(C/(X-D))/3.14159
In the above equations, there are variables (X and Y), parameters (A, B, C, and D), and constants (10 and 3.14159). The FIT
command estimates values for the parameters. If you have a parameter that you do not want estimated, enter it as a constant

or with the "^" (e.g., FIT Y = ^C/(1+^C*A*X**B). The "^" substitutes the value of a parameter into a command.
You can also define a function and then fit the function. For example,
LET FUNCTION F = C/(1+C*A*X**B)
FIT Y = F
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Dataplot Commands for the Gamma Probability Functions Dataplot can compute the probability functions for the gamma
cdf LET Y = GAMCDF(X,GAMMA,A,B)
pdf LET Y = GAMPDF(X,GAMMA,A,B)
ppf LET Y = GAMPPF(X,GAMMA,A,B)
hazard LET Y = GAMHAZ(X,GAMMA,A,B)
cumulative hazard LET Y = GAMCHAZ(X,GAMMA,A,B)
survival LET Y = 1 - GAMCDF(X,GAMMA,A,B)
inverse survival LET Y = GAMPPF(1-X,GAMMA,A,B)
LET Y = Gamma RANDOM NUMBERS FOR I = 1 1 1000
Gamma PROBABILITY PLOT Y
Gamma PPCC PLOT Y
maximum likelihood GAMMA MLE Y
This returns a point estimate for the full-sample case. It does not provide
confidence intervals for the parameters and it does not handle censored data.
where X can be a number, a parameter, or a variable. GAMMA is the shape parameter and is required. It can be a number, a
PLOT GAMPDF(X,2) FOR X = 0.01 0.01 10
Dataplot Command for Grubbs' Test The Dataplot command for Grubbs' test is
GRUBBS <MINIMUM/MAXIMUM> TEST Y
where Y is the response variable. Dataplot identifies one outlier at a time. The MINIMUM or MAXIMUM keyword is
optional. If omitted, the most extreme value will be checked (regardless of whether it is in the minimum or maximum
direction).
Dataplot Commands for Hazard Plots The Dataplot commands for hazard plots are
EXPONENTIAL HAZARD PLOT Y X
NORMAL HAZARD PLOT Y X
LOGNORMAL HAZARD PLOT Y X
WEIBULL HAZARD PLOT Y X
where Y is a response variable containing failure times and X is a censoring variable (0 means failure time, 1 means
censoring time).
Dataplot Command for Kruskal-Wallis Test The Dataplot command for a Kruskal-Wallis test is
KRUSKAL WALLIS TEST Y X
where Y is the response variable and X is the group identifier variable.
Dataplot Commands for the Kolmogorov Smirnov Goodness-of-Fit Test The Dataplot command for the Kolmogorov-Smirnov
goodness-of-fit test is
<dist> KOLMOGOROV-SMIRNOV GOODNESS OF FIT TEST Y
where <dist> is one of 60+ built-in distributions. The K-S goodness of fit test is supported for all Dataplot internal continuous
distributions that support the CDF (cumulative distribution function). The command LIST DISTRIBUTIONS shows the
currently supported distributions in Dataplot. Some specific examples are
NORMAL KOLM-SMIR GOODNESS OF FIT Y
LOGISTIC KOLM-SMIR GOODNESS OF FIT Y
DOUBLE EXPONENTIAL KOLM-SMIR GOODNESS OF FIT Y
You can specify the location and scale parameters by entering
LET KSLOC = value
LET KSSCALE = value
You may need to enter the values for 1 or more shape parameters for distributions that require them. For example, to specify

the shape parameter gamma for the gamma distribution, enter the commands
LET GAMMA = value
GAMMA KOLMOGOROV-SMIRNOV GOODNESS OF FIT TEST Y
Be aware that you should not use the same data to estimate these distributional parameters as you use to calculate the K-S test
as the critical values of the K-S test assume the distribution is fully specified.
The empirical cdf function can be plotted with the following command
EMPIRICAL CDF PLOT Y
Dataplot Commands for Least Squares Estimation of Distributional Parameters The following example shows how to use
Dataplot to obtain least squares estimates for data generated from a Weibull distribution.
. Generate some Weibull data
SET MINMAX MIN
LET GAMMA = 5
LET Y = WEIBULL RAND NUMB FOR I = 1 1 1000
. Bin the data
SET RELATIVE HISTOGRAM AREA
RELATIVE HISTOGRAM Y
LET ZY = YPLOT
LET ZX = XPLOT
RETAIN ZY ZX SUBSET YPLOT > 0
. Specify some starting values
LET SHAPE = 3
LET LOC = MINIMUM Y
LET SCALE = 1
. Now perform the least squares fit
FIT ZY = WEIPDF(ZX,SHAPE,LOC,SCALE)
The RELATIVE HISTOGRAM generates a relative histogram. The command SET RELATIVE HISTOGRAM specifies that
the relative histogram is created so that the area under the histogram is 1 (i.e., the integral is 1) rather than the sum of the bars
equaling 1. This effectively makes the relative histogram an estimator of the underlying density function. Dataplot saves the
coordinates of the histogram in the internal variables XPLOT and YPLOT. The SUBSET command eliminates zero frequency
classes. The FIT command then performs the least squares fit.
The same general approach can be used to compute least squares estimates for any distribution for which Dataplot has a pdf
function. The primary difficulty with the least squares fitting is that it can be quite sensitive to starting values. For
distributions with no shape parameters, the probability plot can be used to determine starting values for the location and scale
parameters. For distributions with a single shape parameter, the ppcc plot can be used to determine a starting value for the
shape parameter and the probability plot used to determine starting values for the location and scale parameters.
The approach above can be used in any statistical software package that provides non-linear least squares fitting and a
method for defining the probability density function (either built-in or user definable).
Dataplot Command for Levene's Test The Dataplot command for the Levene test is
LEVENE TEST Y X
Dataplot Command for the Linear Correlation Plot The Dataplot command to generate a linear correlation plot is
LINEAR CORRELATION PLOT Y X TAG
where Y is the response variable, X is the independent variable, and TAG is the group id variable.
The appearance of the plot can be controlled with appropriate settings for the LINE and CHARACTER commands. Typical
settings would be
CHARACTER X BLANK
LINE BLANK SOLID
Dataplot Command for the Linear Intercept Plot The Dataplot command to generate a linear intercept plot is
LINEAR INTERCEPT PLOT Y X TAG
settings would be
CHARACTER X BLANK
LINE BLANK SOLID
Dataplot Command for the Linear Slope Plot The Dataplot command to generate a linear slope plot is
LINEAR SLOPE PLOT Y X TAG

settings would be
CHARACTER X BLANK
LINE BLANK SOLID
Dataplot Command for the Linear Residual Standard Deviation Plot The Dataplot command to generate a linear residual
standard deviation plot is
LINEAR RESSD PLOT Y X TAG
settings would be
CHARACTER X BLANK
LINE BLANK SOLID
Dataplot Commands for Measures of Location Various measures of location can be computed in Dataplot as follows:
LET A = MEAN Y
LET A = MEDIAN Y
LET A = MIDMEAN Y
LET P1 = 10
LET P2 = 10
LET A = TRIMMED MEAN Y
LET P1 = 10
LET P2 = 10
LET A = WINSORIZED Y
In the above, P1 and P2 are used to set the percentage of values that are trimmed or Winsorized. Use P1 to set the percentage
for the lower tail and P2 the percentage for the upper tail.
Dataplot Commands for the Lognormal Probability Functions Dataplot can compute the probability functions for the
lognormal distribution with the following commands.
cdf LET Y = LGNCDF(X,SD,A,B)
pdf LET Y = LGNPDF(X,SD,A,B)
ppf LET Y = LGNPPF(X,SD,A,B)
hazard LET Y = LGNHAZ(X,SD,A,B)
cumulative hazard LET Y = LGNCHAZ(X,SD,A,B)
survival LET Y = 1 - LGNCDF(X,SD,A,B)
inverse survival LET Y = LGNPPF(1-X,SD,A,B)
random numbers LET SD = value
LET Y = LOGNORMAL RANDOM NUMBERS FOR I = 1 1 1000
probability plot LET SD = value
LOGNORMAL PROBABILITY PLOT Y
ppcc plot LET SD = value
LOGNORMAL PPCC PLOT Y
parameter estimation LOGNORMAL MLE Y
This returns point estimates for the shape and scale parameters. It does not handle
censored data and it does not generate confidence intervals for the parameters.
where X can be a number, a parameter, or a variable. SD is the shape parameter and is optional. It can be a number, a
PLOT LGNPDF(X,5) FOR X = 0.01 0.01 5
Dataplot Commands for Maximum Likelihood Estimation for Distributions Dataplot performs maximum likelihood
estimation for a few specific distributions as documented in the table below. Unless specified otherwise, censored data are not
supported and only point estimates are generated (i.e., no confidence intervals for the parameters). For censored data, create
an id variable that is equal to 1 for a failure time and equal to 0 for a censoring time. Type I censoring is censoring at a fixed

time t0. Type II censoring is censoring after a pre-determined number of units have failed.
Normal NORMAL MAXIMUM LIKELIHOOD Y
Exponential EXPONENTIAL MAXIMUM LIKELIHOOD Y
Confidence intervals are generated for the parameters and both type I and type II censoring are supported.
For type I censoring, enter the following commands
EXPONENTIAL MAXIMUM LIKELIHOOD Y ID
For type II censoring, enter the following commands
EXPONENTIAL MAXIMUM LIKELIHOOD Y ID
Weibull WEIBULL MAXIMUM LIKELIHOOD Y
Confidence intervals are generated for the parameters and both type I and type II censoring are supported.
For type I censoring, enter the following commands
WEIBULL MAXIMUM LIKELIHOOD Y ID
For type II censoring, enter the following commands
WEIBULL MAXIMUM LIKELIHOOD Y ID
Lognormal LOGNORMAL MAXIMUM LIKELIHOOD Y
Double
Exponential
DOUBLE EXPONENTIAL MAXIMUM LIKELIHOOD Y
Pareto PARETO MAXIMUM LIKELIHOOD Y
Gamma GAMMA MAXIMUM LIKELIHOOD Y
Inverse
Gaussian
INVERSE GAUSSIAN MAXIMUM LIKELIHOOD Y
Gumbel GUMBEL MAXIMUM LIKELIHOOD Y
Binomial BINOMIAL MAXIMUM LIKELIHOOD Y
Poisson POISSON MAXIMUM LIKELIHOOD Y
Dataplot Command for the Mean Plot The Dataplot command to generate a mean plot is
MEAN PLOT Y X
where Y is a response variable and X is a group id variable.
Dataplot supports this command for a number of other common location statistics. For example, MEDIAN PLOT Y X and
MID-RANGE PLOT Y X compute the median and mid-range instead of the mean for each group.
Dataplot Commands for Normal Probability Functions Dataplot can compute the various probability functions for the normal
cdf LET Y = NORCDF(X,A,B)
pdf LET Y = NORPDF(X,A,B)
ppf LET Y = NORPPF(X,A,B)
hazard LET Y = NORHAZ(X,A,B)
cumulative hazard LET Y = NORCHAZ(X,A,B)
survival LET Y = 1 - NORCDF(X,A,B)
inverse survival LET Y = NORPPF(1-X,A,B)
random numbers LET Y = NORMAL RANDOM NUMBERS FOR I = 1 1 1000
probability plot NORMAL PROBABILITY PLOT Y
parameter estimates LET YMEAN = MEAN Y
LET YSD = STANDARD DEVIATION Y
where X can be a number, a parameter, or a variable. A and B are the location and scale parameters and they are optional (a
location of 0 and scale of 1 are used if they are omitted). If given, A and B can be a number, a parameter, or a variable.
However, they are typically either a number or parameter.
PLOT NORPDF(X) FOR X = -4 0.01 4

Dataplot Commands for a Normal Probability Plot The Dataplot command to generate a normal probability plot is
NORMAL PROBABILITY PLOT Y
If your data are already grouped (i.e., Y contains counts for the groups identified by X), the Dataplot command is
NORMAL PROBABILITY PLOT Y X
Dataplot returns the following internal parameters when it generates a probability plot.
PPCC - the correlation coefficient of the fitted line on the probability plot. This is a measure of how well the straight
line fits the probability plot.
PPA0 - the intercept term for the fitted line on the probability plot. This is an estimate of the location parameter.
PPA1 - the slope term for the fitted line on the probability plot. This is an estimate of the scale parameter.
SDPPA0 - the standard deviation of the intercept term for the fitted line on the probability plot.
SDPPA1 - the standard deviation of the slope term for the fitted line on the probability plot.
PPRESSD - the residual standard deviation of the fitted line on the probability plot. This is a measure of the adequacy
of the fitted line.
PPRESDF - the residual degrees of freedom of the fitted line on the probability plot.
Dataplot Commands for the Generation of Normal Random Numbers The Dataplot commands to generate 1,000 normal
random numbers with a location of 50 and a scale of 20 are
LET LOC = 50
LET SCALE = 20
LET Y = NORM RAND NUMBERS FOR I = 1 1 1000
LET Y = LOC + SCALE*Y
Programs that automatically generate random numbers are typically controlled by a seed, which is usually an integer value.
The importance of the seed is that it allows the random numbers to be replicated. That is, giving the program the same seed
should generate the same sequence of random numbers. If the ability to replicate the set of random numbers is not important,
you can give any valid value for the seed.
In Dataplot, the seed is an odd integer with a minimum (and default) value of 305. Seeds less than 305 generate the same
sequence as 305 and even numbers generate the same sequence as the preceding odd number. To change the seed value to 401
in Dataplot, enter the command:
SEED 401
Dataplot Commands for Partial Autocorrelation Plots The command to generate a partial autocorrelation plot is
PARTIAL AUTOCORRELATION PLOT Y
The appearance of the partial autocorrelation plot can be controlled by appropriate settings of the LINE, CHARACTER, and
SPIKE commands. Dataplot draws the following curves on the autocorrelation plot:
1. The autocorrelations.
2. A reference line at zero.
For example, to draw the partial autocorrelations as spikes, the zero reference line as a solid line, the 95% lines as dashed
lines, and the 99% line as dotted lines, enter the command
LINE BLANK SOLID DASH DASH DOT DOT
CHARACTER BLANK ALL
SPIKE ON OFF OFF OFF OFF OFF
SPIKE BASE 0
Dataplot Commands for the Poisson Probability Functions Dataplot can compute the probability functions for the Poisson
cdf LET Y = POICDF(X,LAMBDA)
pdf LET Y = POIPDF(X,LAMBDA)
ppf LET Y = POIPPF(X,LAMBDA)
random numbers LET LAMBDA = value
LET Y = POISSON RANDOM NUMBERS FOR I = 1 1 1000
probability plot LET LAMBDA = value
POISSON PROBABILITY PLOT Y
ppcc plot POISSON PPCC PLOT Y
where X can be a number, a parameter, or a variable. LAMBDA is the shape parameter and is required. It can be a number, a
parameter, or a variable. It is typically a number or a parameter.
PLOT POIPDF(X,15) FOR X = 0 1 50

Dataplot Commands for the Power Lognormal Distribution Dataplot can compute the probability functions for the power
lognormal distribution with the following commands.
cdf LET Y = PLNCDF(X,P,SD,MU)
pdf LET Y = PLNPDF(X,P,SD,MU)
ppf LET Y = PLNPPF(X,P,SD,MU)
hazard LET Y = PLNHAZ(X,P,SD,MU)
cumulative hazard LET Y = PLNCHAZ(X,P,SD,MU)
survival LET Y = 1 - PLNCDF(X,P,SD,MU)
inverse survival LET Y = PLNPPF(1-X,P,SD,MU)
probability plot LET P = value
LET SD = value (defaults to 1)
POWER LOGNORMAL PROBABILITY PLOT Y
ppcc plot LET SD = value POWER LOGNORMAL PPCC PLOT Y
In the above, X can be a number, a parameter, or a variable. SD and MU are the scale and location parameters, respectively,
and they are optional (a location of 0 and scale of 1 are used if they are omitted). If given, SD and MU can be a number, a
parameter, or a variable. However, they are typically either a number or a parameter.
These functions can be used in the Dataplot PLOT and FIT commands as well. For example, the command
PLOT PLNPDF(X,5,1) FOR X = 0.01 0.01 5
Dataplot Commands for the Power Normal Probability Functions Dataplot can compute the probability functions for the
power normal distribution with the following commands.
cdf LET Y = PNRCDF(X,P,SD,MU)
pdf LET Y = PNRPDF(X,P,SD,MU)
ppf LET Y = PNRPPF(X,P,SD,MU)
hazard LET Y = PNRHAZ(X,P,SD,MU)
cumulative hazard LET Y = PNRCHAZ(X,P,SD,MU)
survival LET Y = 1 - PNRCDF(X,P,SD,MU)
inverse survival LET Y = PNRPPF(1-X,P,SD,MU)
probability plot LET P = value
LET SD = value (defaults to 1)
POWER NORMAL PROBABILITY PLOT Y
ppcc plot POWER NORMAL PPCC PLOT Y
In the above, X can be a number, a parameter, or a variable. SD and MU are the scale and location parameters, respectively,
and they are optional (a location of 0 and scale of 1 are used if they are omitted). If given, SD and MU can be a number, a
parameter, or a variable. However, they are typically either a number or a parameter.
PLOT PNRPDF(X,10,1) FOR X = -5 0.01 5
Dataplot Commands for Probability Plots The Dataplot command for a probability plot is
<dist> PROBABILITY PLOT Y
where <dist> is the name of the specific distribution. Dataplot currently supports probability plots for over 70 distributions.
For example,
NORMAL PROBABILITY PLOT Y
EXPONENTIAL PROBABILITY PLOT Y
DOUBLE EXPONENTIAL PROBABILITY PLOT Y
CAUCHY PROBABILITY PLOT Y
For some distributions, you may need to specify one or more shape parameters. For example, to specify the shape parameter
for the gamma distribution, you might enter the following commands:
LET GAMMA = 2
GAMMA PROBABILITY PLOT Y
Enter the command LIST DISTRIBUTIONS to see a list of distributions for which Dataplot supports probability plots (and to
see what parameters need to be specified).
Dataplot returns the following internal parameters when it generates a probability plot.
PPCC - the correlation coefficient of the fitted line on the probability plot. This is a measure of how well the straight
line fits the probability plot.
PPA0 - the intercept term for the fitted line on the probability plot. This is an estimate of the location parameter.
PPA1 - the slope term for the fitted line on the probability plot. This is an estimate of the scale parameter.

SDPPA0 - the standard deviation of the intercept term for the fitted line on the probability plot.
SDPPA1 - the standard deviation of the slope term for the fitted line on the probability plot.
PPRESSD - the residual standard deviation of the fitted line on the probability plot. This is a measure of the adequacy
of the fitted line.
PPRESDF - the residual degrees of freedom of the fitted line on the probability plot.
Dataplot Commands for the PPCC Plot The Dataplot command to generate a PPCC plot for unbinned data is:
<dist> PPCC PLOT Y
where <dist> identifies the distributional family and Y is the response variable.
The Dataplot command to generate a PPCC plot for binned data is:
<dist> PPCC PLOT Y X
where <dist> identifies the distributional family, Y is the counts variable, and X is the bin identifier variable.
Dataplot supports the PPCC plot for over 25 distributions. Some of the most common are WEIBULL, TUKEY LAMBDA,
GAMMA, PARETO, and INVERSE GAUSSIAN. Enter the command LIST DISTRIBUTIONS for a list of supported
distributions.
Dataplot allows you to specify the range of the shape parameter. Dataplot generates 50 probability plots in equally spaced
intervals from the smallest value of the shape parameter to the largest value of the shape parameter. For example, to generate
a Weibull PPCC plot for values of the shape parameter gamma from 2 to 4, enter the commands:
LET GAMMA1 = 2
LET GAMMA2 = 4
WEIBULL PPCC PLOT Y
The command LIST DISTRIBUTIONS gives the name of the shape parameter for the supported distributions. The "1" and
"2" suffixes imply the minimum and maximum value for the shape parameter, respectively.
Whenever Dataplot generates a PPCC plot, it saves the following internal parameters:
MAXPPCC - the maximum correlation coefficient from the PPCC plot.
SHAPE - the value of the shape parameter that generated the maximum correlation coefficient.
Dataplot Command for Proportion Defective Confidence Interval The Dataplot command for a confidence interval for the
proportion defective is
PROPORTION CONFIDENCE LIMITS Y
where Y is a response variable. Note that for large samples, Dataplot generates the interval based on the exact binomial
probability, not the normal approximation.
The following command sets the lower and upper bounds that define a success in the response variable:
ANOP LIMITS <lower bound> <upper bound>
Dataplot Command for Q-Q Plot The Dataplot command to generate a q-q plot is
QUANTILE-QUANTILE PLOT Y1 Y2
The CHARACTER and LINE commands can be used to control the appearance of the q-q plot. For example, to draw the
quantile points as circles and the reference line as a solid line, enter the commands
LINE BLANK SOLID
Dataplot Commands for the Generation of Random Walk Numbers To generate a random walk with 1,000 points requires the
following Dataplot commands:
LET Y = UNIFORM RANDOM NUMBERS FOR I = 1 1 1000
LET Y2 = Y - 0.5
LET RW = CUMULATIVE SUM Y2
Dataplot Commands for Rank Sum Test The Dataplot commands for a rank sum (Wilcoxon rank sum, Mann-Whitney) test are
RANK SUM TEST Y1 Y2
RANK SUM TEST Y1 Y2 A
where Y1 contains the data for sample 1, Y2 contains the data for sample 2, and A is a scalar value (either a number or a
parameter). Y1 and Y2 need not have the same number of observations.
The first syntax is used to test the hypothesis that two sample means are equal. The second syntax is used to test that the
difference between two means is equal to a specified constant.
Dataplot Commands for the Run Sequence Plot The Dataplot command to generate a run sequence plot is
RUN SEQUENCE PLOT Y
Equivalently, you can enter
PLOT Y

The appearance of the plot can be controlled with appropriate settings of the LINE, CHARACTER, SPIKE, and BAR
commands and their associated attribute-setting commands.
Dataplot Command for the Runs Test The Dataplot command for a runs test is
RUNS TEST Y
where Y is a response variable.
Dataplot Commands for Measures of Scale The various scale measures can be computed in Dataplot as follows:
LET A = VARIANCE Y
LET A = STANDARD DEVIATION Y
LET A = AVERAGE ABSOLUTE DEVIATION Y
LET A = MEDIAN ABSOLUTE DEVIATION Y
LET A = RANGE Y
LET A1 = LOWER QUARTILE Y
LET A2 = UPPER QUARTILE Y
LET IQRANGE = A2 - A1
Dataplot Commands for Scatter Plots The Dataplot command to generate a scatter plot is
PLOT Y X
The appearance of the plot can be controlled by appropriate settings of the CHARACTER and LINE commands and their
various attribute-setting commands.
Dataplot Commands for Scatterplot Matrix The Dataplot command to generate a scatterplot matrix is
SCATTER PLOT MATRIX X1 X2 ... XK
In addition, Dataplot provides a number of SET commands to control the appearance of the scatterplot matrix. The most
common commands are:
SET MATRIX PLOT LOWER DIAGONAL <ON/OFF>
This command controls whether or not the plots below the diagonal are plotted.
SET MATRIX PLOT TAG <ON/OFF>
If ON, the last variable on the SCATTER PLOT MATRIX command is not plotted directly. Instead, it is used as a
group-id variable. You can use the CHARACTER and LINE commands to set the plot attributes for each group.
SET MATRIX PLOT FRAME <DEFAULT/USER/CONNECTED>
If DEFAULT, the plot frames are connected (that is, it does a FRAME CORNER COORDINATES 0 0 100 100). The
axis tic marks and labels are controlled automatically. If CONNECTED, then it is similar to DEFAULT except the
current value of FRAME CORNER COORDINATES is used. This is useful for putting a small gap between the plots
(e.g., enter FRAME CORNER COORDINATES 3 3 97 97 before generating the scatterplot matrix). If USER, Dataplot
does not connect the plot frames. The tic marks and labels are as the user set them.
SET MATRIX PLOT FIT <NONE/LOWESS/LINEAR/QUADRATIC> This controls whether a lowess fit, a linear fit,
a quadratic fit line, or no fit is superimposed on the plot points. If lowess, a rather high value of the lowess fraction is
recommended (e.g., LOWESS FRACTION 0.6).
In Dataplot, enter HELP SCATTER PLOT MATRIX for additional options for this plot.
Dataplot Commands for Seasonal Subseries Plot The Dataplot commands to generate a seasonal subseries plot are
LET PERIOD = <value>
LET START = <value>
SEASONAL SUBSERIES PLOT Y
The value of PERIOD defines the length of the seasonal period (e.g., 12 for monthly data) and START identifies which group
the series starts with (e.g., if you have monthly data that starts in March, set START to 3).
Dataplot Commands for Sign Test The Dataplot commands for a sign test are
SIGN TEST Y1 A
SIGN TEST Y1 Y2
SIGN TEST Y1 Y2 A
parameter). Y1 and Y2 should have the same number of observations.
The first syntax is used to test the hypothesis that the mean for one sample equals a specified constant. The second syntax is
used to test the hypothesis that two sample means are equal. The third syntax is used to test that the difference between two
means is equal to a specified constant.

Dataplot Commands for Signed Rank Test The Dataplot commands for a signed rank (or Wilcoxon signed-rank) test are
SIGNED RANK TEST Y1 A
SIGNED RANK TEST Y1 Y2
SIGNED RANK TEST Y1 Y2 A
parameter). Y1 and Y2 should have the same number of observations.
The first syntax is used to test the hypothesis that the mean for one sample equals a specified constant. The second syntax is
used to test the hypothesis that two sample means are equal. The third syntax is used to test that the difference between two
means is equal to a specified constant.
Dataplot Commands for Skewness and Kurtosis The Dataplot commands for skewness and kurtosis are
LET A = SKEWNESS Y
LET A = KURTOSIS Y
where Y is the response variable. Dataplot can also generate plots of the skewness and kurtosis for grouped data or one-factor
data with the following commands:
SKEWNESS PLOT Y X
KURTOSIS PLOT Y X
Dataplot Command for the Spectral Plot The Dataplot command to generate a spectral plot is
SPECTRAL PLOT Y
Dataplot Command for the Standard Deviation Plot The Dataplot command to generate a standard deviation plot is
STANDARD DEVIATION PLOT Y X
where Y is a response variable and X is a group id variable.
Dataplot supports this command for a number of other common scale statistics. For example, AAD PLOT Y X and MAD
PLOT Y X compute the average absolute deviation and median absolute deviation, respectively, instead of the standard
deviation for each group.
Dataplot Command for the Star Plot The Dataplot command to generate a star plot is
STAR PLOT X1 TO XP FOR I = 10 1 10
where there are p response variables called X1, X2, ... , XP. Note that this syntax prints one star, specifically the tenth row of
the X1, X2, ..., XP variables.
Typically, multiple star plots will be displayed on the same page. For example, to plot the first 25 rows on the same page,
enter the following sequence of commands
MULTIPLOT CORNER COORDINATES 0 0 100 100
MULTIPLOT 5 5
LOOP FOR K = 1 1 25
STAR PLOT X1 TO XP FOR I = K 1 K
END OF LOOP
Dataplot Command to Generate a Table of Summary Statistics The Dataplot command to generate a table of summary
statistics is
SUMMARY Y
Dataplot Commands for the t Probability Functions Dataplot can compute the probability functions for the t distribution with
the following commands.
cdf LET Y = TCDF(X,NU,A,B)
pdf LET Y = TPDF(X,NU,A,B)
ppf LET Y = TPPF(X,NU,A,B)
random numbers LET NU = value
LET Y = T RANDOM NUMBERS FOR I = 1 1 1000
probability plot LET NU = value
T PROBABILITY PLOT Y
ppcc plot LET NU = value
T PPCC PLOT Y
In the above, X can be a number, a parameter, or a variable. NU is the shape parameter (= number of degrees of freedom).
NU can be a number, a parameter, or a variable. However, it is typically either a number or a parameter. A and B are the
location and scale parameters, respectively, and they are optional (a location of 0 and scale of 1 are used if they are omitted).
If given, A and B can be a number, a parameter, or a variable. However, they are typically either a number or a parameter.

PLOT TPDF(X) FOR X = -4 0.01 4
Dataplot Command for Tietjen-Moore Test The Dataplot command for the Tietjen-Moore test is
LET NOUTLIER = <value>
TIETJEN-MOORE <MINIMUM/MAXIMUM> TEST Y
where Y is the response variable and NOUTLIER specifies the number of outliers to test. The MINIMUM or MAXIMUM
keyword is optional. If it is omitted, outliers will be checked in both the minimum and the maximum direction.
Dataplot Command for Tolerance Intervals The Dataplot command for tolerance intervals is
TOLERANCE Y
where Y is the response variable. Both normal and nonparametric tolerance intervals are printed.
Dataplot Command for Two-Sample t-Test The Dataplot command to generate a two-sample t-test is
T TEST Y1 Y2
where Y1 contains the data for sample 1 and Y2 contains the data for sample 2. Y1 and Y2 do not need to have the same
number of observations.
Return to the Two-Sample t-Test Page
Dataplot Commands for the Tukey-Lambda Probability Functions Dataplot can compute the probability functions for the
Tukey-Lambda distribution with the following commands.
cdf LET Y = LAMCDF(X,LAMBDA,A,B)
pdf LET Y = LAMPDF(X,LAMBDA,A,B)
ppf LET Y = LAMPPF(X,LAMBDA,A,B)
random numbers LET LAMBDA = value
LET Y = TUKEY-LAMBDA RANDOM NUMBERS FOR I = 1 1 1000
probability plot LET LAMBDA = value
TUKEY-LAMBDA PROBABILITY PLOT Y
ppcc plot TUKEY-LAMBDA PPCC PLOT Y
In the above, X can be a number, a parameter, or a variable. LAMBDA is the shape parameter and is required. It can be a
number, a parameter, or a variable. It is typically a number or a parameter. A and B are the location and scale parameters,
respectively, and they are optional (a location of 0 and scale of 1 are used if they are omitted). If given, A and B can be a
number, a parameter, or a variable. However, they are typically either a number or a parameter.
PLOT LAMPDF(X,0.14) FOR X = -5 0.01 5
Dataplot Commands for the Uniform Probability Functions Dataplot can compute the probability functions for the uniform
cdf LET Y = UNICDF(X,A,B)
pdf LET Y = UNIPDF(X,A,B)
ppf LET Y = UNIPPF(X,A,B)
hazard LET Y = UNIHAZ(X,A,B)
cumulative hazard LET Y = UNICHAZ(X,A,B)
survival LET Y = 1 - UNICDF(X,A,B)
inverse survival LET Y = UNIPPF(1-X,A,B)
random numbers LET Y = UNIFORM RANDOM NUMBERS FOR I = 1 1 1000
probability plot UNIFORM PROBABILITY PLOT Y
parameter estimation The method of moment estimators can be computed with the commands
LET YMEAN = MEAN Y
LET YSD = STANDARD DEVIATION Y
LET A = YMEAN - SQRT(3)*YSD
LET B = YMEAN + SQRT(3)*YSD
The maximum likelihood estimators can be computed with the commands
LET YRANGE = RANGE Y
LET YMIDRANG = MID-RANGE Y
LET A = YMIDRANG - 0.5*YRANGE
LET B = YMIDRANG + 0.5*YRANGE
In the above, X can be a number, a parameter, or a variable. A and B are the lower and upper limits of the uniform distribution
and they are optional (A is 0 and B is 1 if they are omitted). The location parameter is A and the scale parameter is (B - A). If
given, A and B can be a number, a parameter, or a variable. However, they are typically either a number or a parameter.

PLOT UNIPDF(X) FOR X = 0 0.1 1
Dataplot Commands for the Generation of Uniform Random Numbers The Dataplot commands to generate 1,000 uniform
random numbers in the interval (-100,100) are
LET A = -100
LET B = 100
LET Y = UNIFORM RANDOM NUMBERS FOR I = 1 1 1000
LET Y = A + (B-A)*Y
A similar technique can be used for any package that can generate standard uniform random numbers. Simply multiply by the
scale value (equals upper limit minus lower limit) and add the location value.
Programs that automatically generate random numbers are typically controlled by a seed, which is usually an integer value.
The importance of the seed is that it allows the random numbers to be replicated. That is, giving the program the same seed
should generate the same sequence of random numbers. If the ability to replicate the set of random numbers is not important,
you can give any valid value for the seed.
In Dataplot, the seed is an odd integer with a minimum (and default) value of 305. Seeds less than 305 generate the same
sequence as 305 and even numbers generate the same sequence as the preceeding odd number. To change the seed value to
401 in Dataplot, enter the command:
SEED 401
Dataplot Commands for the Weibull Probability Functions Dataplot can compute the probability functions for the Weibull
cdf LET Y = WEICDF(X,GAMMA,A,B)
pdf LET Y = WEIPDF(X,GAMMA,A,B)
ppf LET Y = WEIPPF(X,GAMMA,A,B)
hazard LET Y = WEIHAZ(X,GAMMA,A,B)
cumulative hazard LET Y = WEICHAZ(X,GAMMA,A,B)
survival LET Y = 1 - WEICDF(X,GAMMA,A,B)
inverse survival LET Y = WEIPPF(1-X,GAMMA,A,B)
LET Y = WEIBULL RANDOM NUMBERS FOR I = 1 1 1000
WEIBULL PROBABILITY PLOT Y
WEIBULL PPCC PLOT Y
parameter estimation If your data are not censored, enter the commands
SET CENSORING TYPE NONE
WEIBULL MLE Y
If your data have type 1 censoring at fixed time t0, enter the commands
WEIBULL MLE Y X
If your data have type 2 censoring, enter the commands
WEIBULL MLE Y X
Y is the response variable and X is the censoring variable where a value of 1
indicates a failure time and a value of 0 indicates a censoring time. In addition to
the point estimates, confidence intervals for the parameters are generated.
In the above, X can be a number, a parameter, or a variable. GAMMA is the shape parameter and is required. It can be a
number, a parameter, or a variable. It is typically a number or a parameter. A and B are the location and scale parameters,
respectively, and they are optional (a location of 0 and scale of 1 are used if they are omitted). If given, A and B can be a
number, a parameter, or a variable. However, they are typically either a number or a parameter.
PLOT WEIPDF(X,2) FOR X = 0.01 0.01 5
Return to the Weibull Distribution Page
Dataplot Commands for the Weibull Plot The Dataplot commands to generate a Weibull plot are
WEIBULL PLOT Y
WEIBULL PLOT Y X
where Y is the response variable containing failure times and X is an optional censoring variable. A value of 1 indicates the

item failed by the failure mode of interest while a value of 0 indicates that the item failed by a failure mode that is not of
interest.
The appearance of the plot can be controlled with appropriate settings for the LINE and CHARACTER commands. For
example, to draw the raw data with the "X" character and the 2 reference lines as dashed lines, enter the commands
LINE BLANK DASH DASH
CHARACTER X BLANK BLANK
WEIBULL PLOT Y X
Dataplot saves the following internal parameters after the Weibull plot.
ETA - the estimated characterstic life
BETA - the estimated shape parameter
SDETA - the estimated standard deviation of ETA
SDBETA - the estimated standard deviation of BETA
BPT1 - the estimated 0.1% point of failure times
BPT5 - the estimated 0.5% point of failure times
B1 - the estimated 1% point of failure times
B995 - the estimated 99.5% point of failure times
B999 - the estimated 99.9% point of failure times
Dataplot Command for the Wilk-Shapiro Normality Test The Dataplot command for a Wilk-Shapiro normality test is
WILK SHAPIRO TEST Y
The significance value is only valid if there is less than 5,000 points.
Dataplot Commands for Yates Analysis The Dataplot command for a Yates analysis is
YATES Y
where Y is a response variable in Yates order.
Dataplot Commands for the Youden Plot The Dataplot command to generate a Youden plot is
YOUDEN PLOT Y1 Y2 LAB
where Y1 and Y2 are the response variables and LAB is a laboratory (or run number) identifier. The LINE and
CHARACTER commands can be used to control the appearance of the Youden plot. For example, if there are 5 labs, a typical
sequence would be
LINE BLANK ALL
CHARACTER 1 2 3 4 5
YOUDEN PLOT Y X LAB
Dataplot Commands for the 4-plot The Dataplot command to generate the 4-plot is
4-PLOT Y
Dataplot Commands for the 6-Plot The Dataplot commands to generate a 6-plot are
FIT Y X
6-PLOT Y X
where Y is the response variable and X is the independent variable.

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