Chapter 6

Chapter 6
Basic Statistics Concepts
Every physical measurement is subject to a degree of uncertainty. A result that is not
particularly accurate may be of great use if the limits of the errors affecting it can be set
with a high degree of certainly. Statistical methods are used to evaluate the probable
errors in analytical measurements and interpretation of environmental data. This module
discusses some basic concepts of statistics, which a chemist should be conversant with
in order to evaluate the accuracy of the laboratory data and the estimation of the
environmental characteristics based on the results of limited samples.
1 Frequency distribution
Results of 44 replicate analyses for hardness of a sample of water are given in Table 1.
Table 1. Results of 44 replicate analyses for hardness, mg/l as CaC03
Class Value No. of
Values
in the
Class
Frequency
(1) (2) (3) (4)
51-52 51.2 1 0.0227
52-53 52.0,52.8,52.8 3 0.0682
53-54 53.1,53.1,53.3,53.4,53.5,53.6,53.6,53.9, 8 0.1818
54-55 54.0,54.1,54.3,54.3,54.4,54.5,54.5,54.6,54.7,54.7,54.9 11 0.2500
55-56 55.1,55.1,55.3,55.3,55.4,55.4,55.6,55.7,55.7,55.8,55.9,55.9 12 0.2727
56-57 56.2,56.3,56.7,56.8,56.9,56.9 6 0.1364
57-58 57.3,57.5 2 0.0454
58-59 - 0
59-60 59.1 1 0.2227
The data are classified in 9 classes, column (1) and arranged in each class according to
their magnitude, column (2). By convention an observation equal to the upper value of
the class is counted in the next higher class. For example the value 52.0 is not placed in
class 51-52 but in 52-53. The number of values in each class are given in column (3).
The ratio of number of values is a class to the total number of observation is called
frequency and is given in column (4).
A plot of data of column (1) vs. column (4) is called frequency distribution diagram and
is shown in Figure 1. For clarity of presentation, the frequency value is multiplied by 100
and shown as %.
Note that if the number of observations is increased and the class interval is reduced,
the histogram can be replaced by a continuous curve.

Figurel: Example of a frequency distribution diagram
<m m m. ss -sr BB ®o
Central Tendency
Arithmetic mean: The arithmetic mean, X, of a set of data is calculated by adding all the
observed values of variable (results of analyses) and dividing it by the total number of
observations:
X = ( X 1 + X 2 + Xn)/n (1)
where Xi, X2,... Xn are the observed values and n is the total number of observations.
The arithmetic mean is the most common measure of the central tendency. The mean
value of the data of Table 1 was calculated as 54.9
Geometric mean: When there are a few very high values or very low, such as in the
cases of bacteriological analysis, the arithmetic mean is not necessarily representative
of the central tendency. In such cases the geometric mean, g, is used:
g=Xi x X 2 x X 3 •Xn)
1/n
(2)
Median: The median is the middle value of the set of data. If the sample size n is an
odd number, one-half of the values exceed the median and one-half are less. When n is
even, it is average of the two middle terms. For the data of Table 1, it is 54.8, which is
the average of 54.7 and 54.9, the 22nd and the 23rd terms, when the data are arranged
in ascending order.
Mode: The mode is the most commonly occurring value. It is not widely employed as it
forms a poor basis for any further arithmetic calculations.
Standard deviation
The data of Table 1 show that 52% of observations lie in the range of 54 - 56 and 84%
in the range of 53-57. This tendency of observations to cluster (or not to cluster) around
the mean value, 54.9, is measured by standard deviation, s. Standard deviation is
calculated as:

s=V{X1 -X)2
+ (X2 - X)2
+ (Xn - X)2
}/(n - 1 )s=
or s = VsXi2
- (EXi)2
/N (3)
N
A small value of s signifies that most of the observations are close to the mean value. A
large value indicates that the observed values are spread over a larger range. For
example, if the data of Table 1 has more observations in 52-53 and 57-58 classes and
correspondingly lesser number in the central classes 54-55 and 55-56, the frequency
distribution diagram will be flatter and spread wider at the base and the value of s will be
larger.
The standard deviation has the same units as the quantity measured. The standard
deviation for the data of Table 1 was calculated as 1.545 mg/L.
The square of the standard deviation is called the variance. If the random distribution of
data is due to r different reasons, the total variance is the sum of individual variance.
Normal Distribution
Most frequency distributions for random observations, when the total number of
observations is very large tending to be the same as the population, N, conform to
normal distribution. The distribution is give by the theoretical equation:
where y = frequency of observations /1 = the mean, and cr = standard deviation. Note
the distinction made in the notations for number of observations, mean and standard
deviation for a sample of a limited set of data used earlier.
The standard deviation is given by:
N
For a normal distribution, 68.3 % of the total observations lie in the range jj, +g, 95.5% in
the range i ±2c and 99.7% in the range ju +3a. This is illustrated in Figure 2.
The number of observations between any two limits of the variable can be equated to
the area bounded by the frequency distribution curve, the ordinates at these limits and
the x axis. .
When the data set is small, more often than not, sample mean, X will differ somewhat
from the population mean, Any error in X causes a corresponding error in the
standard deviation calculated with Equation 5. There is a tendency for the standard
deviation value calculated to be small as the number of measurements becomes
smaller. It can be shown that this bias can be largely eliminated by substituting the
degree of freedom as (n-1) in the calculation, Equation (3), which defines the standard
(xi -ji)21 2o2
y = (4)
cW2n
(5)

deviation for a limited set of measurements.
The rationale for using (n-1) is as follows. When jj, is not known, we nnust extract two
quantities, namely, X and s from the set of data. The need to establish X from the data
removes one degree of freedom: that is if their signs are retained, the sum of individual
deviations must total zero; once (n-1) deviations have been established, the final
deviation is necessarily known. Thus only (n-1) deviations provide independent
measures for the standard deviation of the set.
2 Precision and Accuracy of Experimental Data
Classes of errors
The phenomena that are responsible for uncertainties in an analytical measurement can
be divided in two broad categories: determinate or system errors and indeterminate or
random errors. The total error is the sum of the two types of errors.
Determinate errors have assignable causes and are unidirectional. It may be possible to
eliminate some of these, for example, errors because of improper calibration of the
equipment, or presence of interfering substances in the sample. Errors inherent in the
method, such as addition of a reagent in excess of theoretical requirement to cause an
indicator to undergo colour change in a titration, may be difficult to eliminate.
Indeterminate errors are encountered whenever a measuring system is extended to its
maximum sensitivity. The results fluctuate in a random manner about a mean value.
The sources of these fluctuations can never be identified because they are made up of
a myriad of instrumental, personal and method uncertainties that are individually so
small that they can never be detected. For example, in he case of addition of an exact
volume of reagent in an analysis through a pipette the uncertainties could be the time
allowed for draining the pipette, the angle at which the pipette is held during delivery,
temperature of the reagent, visual judgement of water level with respect to the
graduation mark, etc. What is observed in the final result is then a summation of a very
large number of minute unobservable uncertainties. The cumulative effect is likewise
variable. Ordinarily they tend to cancel one other and thus exert a minimal effect.
Occasionally, however, they act in concert to produce a relatively large positive or
negative error.
Precision
The term precision is employed to describe the reproducibility of results. It can be
defined as the agreement between the numerical values of two or more measurements
that have been made in an identical fashion.
Absolute methods for expressing precision: The absolute average deviation from
the mean is a common method for describing precision. The spread or range of a set of
data is also a measure of precision and is simply the numerical difference of the highest
and the lowest result. Standard deviation, described earlier, is a more significant
measure of precision.

Example 1
From the data of replicate chloride analysis of a water sample given below,
calculate the precision in terms of the average deviation from the mean:
Analysis Chloride, mg/L Abs. dev. from the Mean, mg/L
1 24.39 0.077
2 24.19 0.123
3 24.36 0.047
Total 72.94 0.247
Mean = 24.313 ave. dev. = 0.247/3 = 0.082
Relative methods for expressing precision: It is frequently more informative to
indicate the relative precision. The relative parameters are dimensionless and
therefore can be used to compare two ore more sets of data. Thus, for example, the
relative deviation of analysis 1, in the above example, is (0.077 X 100J/24.313 =
0.32%.
Relative standard deviation or coefficient of variation is defined as:
CV = 100s (6)
x
Example 2
Monitoring results at three sampling locations are listed below. Compare the
sampling records in terms of variability.
A B C
40.0 19.9 37.0
29.2 24.1 33.4
18.6 22.1 36.1
29.3 19.8 40.2
X 29.28 21.48 36.68
s 8.74 2.05 2.80
CV 0.30 0.10 0.07
In terms of the coefficient of variation the concentration at A varies the most,
followed by B and then C.
Accuracy
The term accuracy is used to describe the total error of the observation, which is the
sum of the systematic and random errors. It denotes the nearness of the
measurement to its accepted value.
In Example 1, if the accepted value for the chloride concentration is 24.35 mg/L, the
absolute error of the mean is 24.31 - 24.35 = 0.04 mg/L.

3. Propagation of errors
The indeterminate error or uncertainty is most commonly expresses as standard
deviation. When the final result is computed from two or more data, each of which
has an indeterminate error associated with it, the error of the result will depend on
the arithmetic computations. The following examples illustrate the procedure to be
followed:
Example 3:
For the sum
+ 0.50 (+0.02)
+ 4.10 (+ 0.03)
- 1.97 (+ 0.05)
+ 2.63 (+? ?)
where the numbers in parantheses are the absolute standard
Statistically the most probable standard deviation would be given by the
of the sum of individual variances:
s = V(+0,02)2
+(+0.03)2
+ (+0.05)2
deviations,
square root
= +0.078
Therefore the result could be reported as:
2.63 (+0.078)
Example 4:
The case when products and quotients are involved is illustrated in the following
calculations:
4.10( + 0.02)x 0.0050(+0.0001)
= 0.0104(+?)
1.97(+0.04)
In this case it is necessary to calculate the relative standard deviations:
(Sa)r = + 0.02 = +0.0049
4.10
(Sa)r = + 0.0001 =+ 0.020
0.005
(Sr.)r = + 0.04 = + 0.020
1.97
(Sy) r =V (+0.0049)2
+ (+ 0.02)2
+(+0.02)2
=+ 0.029

The absolute standard deviation of the result will bel
Sy = 0.0104 x ( + 0.029) = + (0.0003)
Therefore the uncertainty of the result can be written as:
0.0104 (+ 0.0003)
For calculations that involve both sums and differences as well as products and
quotients, the uncertainties associated with the former are evaluated first and then
the latter following procedures illustrated in the two examples.

Chapter 6

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Chapter 6