Page 104 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes 7.2 The Standard Deviation

The standard deviation concept was introduced by Karl Pearson in 1893. It is by far the most important
and widely used measure of studying dispersion. Its significance lies in the fact that it is free from
those defects from which the earlier methods suffer and satisfies most of the properties of a good
measure of dispersion. Standard deviation is also known as root-mean square deviation for the reason
that it is the square root of the means of the squared deviations from the arithmetic mean. Standard
deviation is denoted by the small Greek letter σ (read as sigma).
The standard deviation measures the absolute dispersion or variability of a distribution; the greater
the amount of dispersion or variability, the greater the standard deviation, the greater will be the
magnitude of the deviations of the values from their mean. A small standard deviation means a high
degree of uniformity of the observations as well as homogeneity of a series; a large standard deviation
means just the opposite. Thus if we have two or more comparable series with identical or nearly
identical means, it is the distribution with the smallest standard deviation that has the most
representative mean. Hence standard deviation is extremely useful in judging the representativeness
of the mean.
Difference between Average Deviation and Standard Deviation
Both these measures of dispersion are based on each and every item of the distribution. But they
differ in the following respects:
(i) Algebraic signs are ignored while calculating mean deviation whereas in the calculation of
standard deviations, signs are taken into account.
(ii) Mean deviation can be computed either from median or mean. The standard deviation, on the
other hand, is always computed from the arithmetic mean because the sum of the squares of
the deviations of items from arithmetic mean is the least.
Calculation of Standard Deviation—Individual Observations

In case of individual observations, standard deviation may be computed by applying any of the
following two methods:
1. By taking deviations of the items from the actual mean.
2. By taking deviations of the items from an assumed mean.
1. Deviations taken from Actual Mean: When deviations are taken from actual mean the following
formula is applied:

∑x 2
σ =
N
where x = ( XX and N = number of observations.
) −

Steps : (i) Calculate the actual mean of the series, i.e., X .
) −
(ii) Take the deviations of the items from the mean, i.e., find ( XX . Denote these
deviation by x.
2
(iii) Square these deviations and obtain the total ∑x .
2
(iv) Divide ∑x by the total number of observations, i.e., N, and extract the square-
root. This gives us the value of standard deviation.
Example 6: Calculate standard deviation from the following observations of marks of 5 students
of a tutorial group:

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