3 Descriptive Numerical Summary Measures.ppt

Descriptive Statistics:
Numerical Summary Measures
– Single numbers which quantify the
characteristics of a distribution of values
 Measures of central tendency (location)
 Measures of dispersion

• A frequency distribution is a general
picture of the distribution of a variable
• But, can’t indicate the average value and
the spread of the values

Measures of Central Tendency (MCT)
• On the scale of values of a variable there is
a certain stage at which the largest number
of items tend to cluster.
• Since this stage is usually in the centre of
distribution, the tendency of the statistical
data to get concentrated at a certain value
is called “central tendency”
• The various methods of determining the
point about which the observations tend to
concentrate are called MCT.

• The objective of calculating MCT is to
determine a single figure which may be
used to represent the whole data set.
• In that sense it is an even more compact
description of the statistical data than the
frequency distribution.
• Since a MCT represents the entire data, it
facilitates comparison within one group or
between groups of data.

0
5
10
15
20
0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99
Position

Characteristics of a good MCT
A MCT is good or satisfactory if it possesses
the following characteristics.
1. It should be based on all the observations
2. It should not be affected by the extreme values
3. It should be as close to the maximum number of
values as possible
4. It should have a definite value
5. It should not be subjected to complicated and tedious
calculations
6. It should be capable of further algebraic treatment
7. It should be stable with regard to sampling

• The most common measures of central
tendency include:
– Arithmetic Mean
– Median
– Mode
– Others

1. Arithmetic Mean
A. Ungrouped Data
• The arithmetic mean is the "average" of the data
set and by far the most widely used measure of
central location
• Is the sum of all the observations divided by the
total number of observations.

3 Descriptive Numerical Summary Measures.ppt

The heart rates for n=10 patients were as follows (beats per minute):
167, 120, 150, 125, 150, 140, 40, 136, 120, 150
What is the arithmetic mean for the heart rate of these patients?

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Example. Compute the mean age of 169 subjects from the
grouped data.
Mean = 5810.5/169 = 34.48 years
Class interval Mid-point (mi) Frequency (fi) mifi
10-19
20-29
30-39
40-49
50-59
60-69
14.5
24.5
34.5
44.5
54.5
64.5
4
66
47
36
12
4
58.0
1617.0
1621.5
1602.0
654.0
258.0
Total __ 169 5810.5

The mean can be thought of as a “balancing
point”, “center of gravity”

When the data are skewed, the mean is
“dragged” in the direction of the skewness
• It is possible in extreme cases for all but one of the sample points
to be on one side of the arithmetic mean & in this case, the mean is
a poor measure of central location or does not reflect the center of
the sample.

Properties of the Arithmetic Mean.
• For a given set of data there is one and only
one arithmetic mean (uniqueness).
• Easy to calculate and understand (simple).
• Influenced by each and every value in a data
set
• Greatly affected by the extreme values.
• In case of grouped data if any class interval
is open, arithmetic mean can not be
calculated.

2. Median
a) Ungrouped data
• The median is the value which divides the data set
into two equal parts.
• If the number of values is odd, the median will be the
middle value when all values are arranged in order of
magnitude.
• When the number of observations is even, there is no
single middle value but two middle observations.
• In this case the median is the mean of these two
middle observations, when all observations have
been arranged in the order of their magnitude.

• The median is a better description (than the mean) of
the majority when the distribution is skewed
• Example
– Data: 14, 89, 93, 95, 96
– Skewness is reflected in the outlying low value of 14
– The sample mean is 77.4
– The median is 93

b) Grouped data
• In calculating the median from grouped data, we
assume that the values within a class-interval
are evenly distributed through the interval.
• The first step is to locate the class interval in
which the median is located, using the following
procedure.
• Find n/2 and see a class interval with a
minimum cumulative frequency which contains
n/2.
• Then, use the following formal.

~
x = L
n
2
F
f
W
m
c
m












where,
Lm
= lower true class boundary of the interval containing the median
Fc
= cumulative frequency of the interval just above the median
class
interval
fm
= frequency of the interval containing the median
W= class interval width
n = total number of observations

Example. Compute the median age of 169
subjects from the grouped data.
n/2 = 169/2 = 84.5
Class interval Mid-point (mi) Frequency (fi) Cum. freq
10-19
20-29
30-39
40-49
50-59
60-69
14.5
24.5
34.5
44.5
54.5
64.5
4
66
47
36
12
4
4
70
117
153
165
169
Total 169

• n/2 = 84.5 = in the 3rd
class interval
• Lower limit = 29.5, Upper limit = 39.5
• Frequency of the class = 47
• (n/2 – fc) = 84.5-70 = 14.5
• Median = 29.5 + (14.5/47)10 = 32.58 ≈ 33

Properties of the median
• There is only one median for a given set of data
(uniqueness)
• The median is easy to calculate
• Median is a positional average and hence it is
insensitive to very large or very small values
• Median can be calculated even in the case of
open end intervals
• It is determined mainly by the middle points and
less sensitive to the remaining data points
(weakness).

Quartiles
• Just as the median is the value above and
below which lie half the set of data, one
can define measures (above or below)
which lie other fractional parts of the data.
• The median divides the data into two
equal parts
• If the data are divided into four equal
parts, we speak of quartiles.

a) The first quartile (Q1
): 25% of all the
ranked observations are less than Q1.
b) The second quartile (Q2
): 50% of all the
ranked observations are less than Q2
. The
second quartile is the median.
c) The third quartile (Q3
): 75% of all the
ranked observations are less than Q3.

Percentiles
• Simply divide the data into 100 pieces.
• Percentiles are less sensitive to outliers
and not greatly affected by the sample
size (n).

3. Mode
• The mode is the most frequently occurring
value among all the observations in a set
of data.
• It is not influenced by extreme values.
• It is possible to have more than one mode
or no mode.
• It is not a good summary of the majority of
the data.

Mode
0
2
4
6
8
10
12
14
16
18
20
N
Mode
T. Ancelle, D. Coulombie
Mode

• It is a value which occurs most
frequently in a set of values.
• If all the values are different there is no
mode, on the other hand, a set of
values may have more than one mode.
a) Ungrouped data

• Example
• Data are: 1, 2, 3, 4, 4, 4, 4, 5, 5, 6
• Mode is 4 “Unimodal”
• Example
• Data are: 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, 6, 8
• There are two modes – 2 & 5
• This distribution is said to be “bi-modal”
• Example
• Data are: 2.62, 2.75, 2.76, 2.86, 3.05, 3.12
• No mode, since all the values are different

b) Grouped data
• To find the mode of grouped data, we
usually refer to the modal class, where
the modal class is the class interval with
the highest frequency.
• If a single value for the mode of
grouped data must be specified, it is
taken as the mid-point of the modal
class interval.

Properties of mode
 It is not affected by extreme values
 It can be calculated for distributions with
open end classes
 Often its value is not unique
 The main drawback of mode is that
often it does not exist

4. Geometric mean (GM)
• Mainly used in many types of laboratory data,
specifically data in the form of concentrations of
one substance in another
• Example: the minimum inhibitory concentration of
penicillin in urine for N. gonorrhoeae in 71 patients
(µg/ml) Frequency (µg/ml) Frequency
0.03125
0.0625
0.1250
21
6
8
0.250
0.50
1.0
19
17
3

If x x ..., x are n positive observed values, then
GM = x
1 2 n
i
i=1
n
n
, ,

and
logGM =
logx
n
i
i=1
n

.
The geometric mean is generally used with data measured on a logarithmic scale, such
as titers of anti-neutrophil immunoglobulin G.

Example:
logGM = [21log(0.03125) + 6log(0.0625) +
8log(0.125) + 19log(0.25) + 17log(0.5)
+ 3log(1.0)]/74 = -0.846
The GM = the antilogarithm of -0.846 = 0.143

5. Harmonic mean (HM)
• Just as the geometric mean is based on
an arithmetic mean of logarithms, so is
the harmonic mean based on arithmetic
mean of the reciprocals.
• Pertains to rates and time
• We define it as the reciprocal of the
arithmetic mean of the reciprocal of the
given numbers.

If the given numbers are x x ..., x , then
HM =
1
1
n
1
x
1 2 n
i
i=1
n
, ,


6. Weighted mean (WM)
• In a weighted mean, separate outcomes
have separate influences.
• The influence attached to an outcome is
the weight.
• Familiar is the calculation of a course
grade as a weighted average of scores on
separate outcomes.

Which measure of central tendency is best with a
given set of data?
• Two factors are important in making this
decisions:
– The scale of measurement (type of data)
– The shape of the distribution of the
observations

• The mean can be used for discrete and
continuous data
• The median is appropriate for discrete and
continuous data as well, but can also be
used for ordinal data
• The mode can be used for all types of
data, but may be especially useful for
nominal and ordinal measurements
• For discrete or continuous data, the
“modal class” can be used

• The geometric mean is used primarily for
observations measured on a logarithmic
scale.
• Harmonic mean is a suitable MCT when
the data pertains to rates and time.
• Weighted mean is commonly used in the
calculation of mean for different
outcomes.

(a) Symmetric and unimodal distribution —
Mean, median, and mode should all be
approximately the same
Mean, Median & Mode

(b) Bimodal — Mean and median should be
about the same, but may take a value that
is unlikely to occur; two modes might be
best

(c) Skewed to the right (positively skewed) —
Mean is sensitive to extreme values, so
median might be more appropriate
Mode
Median
Mean

(d) Skewed to the left (negatively skewed)
— Same as (c)
Mode
Median
Mean

Measures of Dispersion
Consider the following two sets of data:
A: 177 193 195 209 226 Mean =
200
B: 192 197 200 202 209 Mean =
200
Two or more sets may have the same mean and/or median but they
may be quite different.

These two distributions have the same mean,
median, and mode

• MCT are not enough to give a clear
understanding about the distribution of
the data.
• We need to know something about the
variability or spread of the values —
whether they tend to be clustered close
together, or spread out over a broad
range

• Measures that quantify the variation or
dispersion of a set of data from its central
location
• Dispersion refers to the variety exhibited by
the values of the data.
• The amount may be small when the values are
close together.
• If all the values are the same, no dispersion

Other synonymous term:
– “Measure of Variation”
– “Measure of Spread”
– “Measures of Scatter”

• Measures of dispersion include:
– Range
– Inter-quartile range
– Variance
– Standard deviation
– Coefficient of variation
– Standard error
– Others

1. Range (R)
• The difference between the largest and
smallest observations in a sample.
• Range = Maximum value – Minimum value
• Example –
– Data values: 5, 9, 12, 16, 23, 34, 37, 42
– Range = 42-5 = 37
• Data set with higher range exhibit more
variability

Properties of range
 It is the simplest crude measure and can be
easily understood
 It takes into account only two values which
causes it to be a poor measure of dispersion
 Very sensitive to extreme observations
 The larger the sample size, the larger the
range

2. Interquartile range (IQR)
• Indicates the spread of the middle 50% of
the observations, and used with median
IQR = Q3 - Q1
• Example: Suppose the first and third quartile for
weights of girls 12 months of age are 8.8 Kg and
10.2 Kg, respectively.
IQR = 10.2 Kg – 8.8 Kg
i.e., 50% of the infant girls weigh between 8.8 and
10.2 Kg.

The two quartiles (Q3 &Q1) form the basis of the
Box-and-Whiskers Plots — Variables A, B, C
0
1
2
3
4
5
6
7
8
9
10
Variable A Variable B Variable C

Properties of IQR:
• It is a simple and versatile measure
• It encloses the central 50% of the observations
• It is not based on all observations but only on
two specific values
• It is important in selecting cut-off points in the
formulation of clinical standards
• Since it excludes the lowest and highest 25%
values, it is not affected by extreme values
• Less sensitive to the size of the sample

3. Quartile deviation (QD)
QD = Q Q
2
3 1


4. Coefficient of quartile deviation
(CQD)
• CQD =
• CQD is an absolute quantity (unitless)
and is useful to compare the variability
among the middle 50% observations.
Q Q
Q Q
3 1
3 1



5. Mean deviation (MD)
• Mean deviation is the average of the
absolute deviations taken from a central
value, generally the mean or median.
• Consider a set of n observations x1
,
x2
, ..., xn
. Then:
• ‘A’ is a central value (arithmetic mean or
median).
MD
1
n
x A
i
i 1
n
 



Properties of mean deviation:
 MD removes one main objection of the earlier
measures, that it involves each value
 It is not affected much by extreme values
 Its main drawback is that algebraic negative
signs of the deviations are ignored which is
mathematically unsound

6. Variance (2
, s2
)
• The main objection of mean deviation, that
the negative signs are ignored, is removed
by taking the square of the deviations from
the mean.
• The variance is the average of the squares
of the deviations taken from the mean.

• It is squared because the sum of the
deviations of the individual observations of
a sample about the sample mean is
always 0
0 = ( )
• The variance can be thought of as an
average of squared deviations
 - x
x i

• Variance is used to measure the
dispersion of values relative to the mean.
• When values are close to their mean
(narrow range) the dispersion is less than
when there is scattering over a wide
range.
– Population variance = σ2
– Sample variance = S2

a) Ungrouped data
 Let X1
, X2
, ..., XN
be the measurement on N
population units, then:



2
i
2
i 1
N
i
i=1
N
(X )
N
where
=
X
N
is the population mean.






A sample variance is calculated for a sample of
individual values (X1, X2, … Xn) and uses the sample
mean (e.g. ) rather than the population mean µ.

Degrees of freedom
• In computing the variance there are (n-1)
degrees of freedom because only (n-1) of the
deviations are independent from each other
• The last one can always be calculated from
the others automatically.
• This is because the sum of the deviations
from their mean (Xi-Mean) must add to zero.

b) Grouped data
where
mi
= the mid-point of the ith
class interval
fi
= the frequency of the ith
class interval
= the sample mean
k = the number of class intervals
S
(m x) f
f - 1
2
i
2
i
i=1
k
i
i=1
k




x

Properties of Variance:
 The main disadvantage of variance is
that its unit is the square of the unite of
the original measurement values
 The variance gives more weight to the
extreme values as compared to those
which are near to mean value, because
the difference is squared in variance.
• The drawbacks of variance are
overcome by the standard deviation.

7. Standard deviation (, s)
• It is the square root of the variance.
• This produces a measure having the
same scale as that of the individual
values.
 
 2
and S = S2

• Following are the survival times of n=11
patients after heart transplant surgery.
• The survival time for the “ith” patient is
represented as Xi for i= 1, …, 11.
• Calculate the sample variance and SD.

Example. Compute the variance and SD of the age of 169
subjects from the grouped data.
Mean = 5810.5/169 = 34.48 years
S2
= 20199.22/169-1 = 120.23
SD = √S2 = √120.23 = 10.96
Class
interval (mi) (fi) (mi-Mean) (mi-Mean)2 (mi-Mean)2
fi
10-19
20-29
30-39
40-49
50-59
60-69
14.5
24.5
34.5
44.5
54.5
64.5
4
66
47
36
12
4
-19.98
-9-98
0.02
10.02
20.02
30.02
399.20
99.60
0.0004
100.40
400.80
901.20
1596.80
6573.60
0.0188
3614.40
4809.60
3604.80
Total 169 1901.20 20199.22

Properties of SD
• The SD has the advantage of being expressed in
the same units of measurement as the mean
• SD is considered to be the best measure of
dispersion and is used widely because of the
properties of the theoretical normal curve.
• However, if the units of measurements of variables
of two data sets is not the same, then there
variability can’t be compared by comparing the
values of SD.

SD Vs Standard Error (SE)
• SD describes the variability among individual
values in a given data set
• SE is used to describe the variability among
separate sample means obtained from one
sample to another
• We interpret SE of the mean to mean that
another similarly conducted study may give a
mean that may lie between  SE.

Standard Error
• SD is about the variability of individuals
• SE is used to describe the variability in
the means of repeated samples taken
from the same population.
• For example, imagine 5,000 samples, each of the same size n=11.
This would produce 5,000 sample means. This new collection has
its own pattern of variability. We describe this new pattern of
variability using the SE, not the SD.

Example: The heart transplant surgery
n=11, SD=168.89, Mean=161 days
• What happens if we repeat the study? What will our next mean be? Will
it be close? How different will it be? Focus here is on the
generalizability of the study findings.
• The behavior of mean from one replication of the study to the next
replication is referred to as the sampling distribution of mean.
• We can also have sampling distribution of the median or the SD
• We interpret this to mean that a similarly conducted study might
produce an average survival time that is near 161 days, ±50.9 days.

8. Coefficient of variation (CV)
• When two data sets have different units
of measurements, or their means differ
sufficiently in size, the CV should be
used as a measure of dispersion.
• It is the best measure to compare the
variability of two series of sets of
observations.
• Data with less coefficient of variation is
considered more consistent.

CV
S
x
100
 
• “Cholesterol is more variable than systolic blood
pressure”
SD Mean CV (%)
SBP
Cholesterol
15mm
40mg/dl
130mm
200mg/dl
11.5
20.0
•CV is the ratio of the SD to the mean multiplied by 100.

NOTE:
• The range often appears with the median
as a numerical summary measure
• The IQR is used with the median as well
• The SD is used with the mean
• For nominal and ordinal data, a table or
graph is often more effective than any
numerical summary measure

3 Descriptive Numerical Summary Measures.ppt

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