Page 120 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes presented to the general public or to groups not familiar with statistical methods. This measure
is useful for small samples with no elaborate analysis required. Incidentally it may be mentioned
that the National Bureau of Economic Research has found in its work on forecasting business
cycles, that the average deviation is the most practical measure of dispersion to use for this
purpose.
• 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.
• 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.
• When the actual mean is in fractions, say, in the above case 123.674, it would be too cumbersome
to take deviations from it and then obtaining squares of these deviations. In such a case, either
the mean may be approximated or else the deviations be taken from an assumed mean and the
necessary adjustment be made in the value of standard deviation.
• The sum of the squares of the deviations of items in the series from their arithmetic mean is
minimum. In other words, the sum of the squares of the deviations of items of any series from
a value other than the arithmetic mean would always be greater. This is the reason why standard
deviation is always computed from the arithmetic mean.
• In a normal distribution there is a fixed relationship between the three most commonly used
measures of dispersion. The quartile deviation is smallest, the mean deviation next and the
standard deviation is largest, in the following proportion:
2 4
Q.D. = σ ; M.D. = σ
3 5
• These relationships can be easily memorised because of the sequence 2, 3, 4, 5. The same
proportions tend to hold true for many distributions that are quite normal. They are useful in
estimating one measure of dispersion when another is known, or in checking roughly the
accuracy of a calculated value. If the computed σ differs very widely from its value estimated
from Q.D. or M.D. either an error has been made or the distribution differs considerably from
normal.
• The Standard deviation discussed above is an absolute measure of dispersion. The corresponding
relative measure is known as the coefficient of variation. This measure developed by Karl Pearson
is the most commonly used measure of relative variation. It is used in such problems where we
want to compare the variability of two or more than two series. That series (or group) for which
the coefficient of variation is greater is said to be more variable or conversely less consistent,
less uniform, less stable or less homogeneous.
• The term variance was used to describe the square of the standard deviation by R.A. Fisher in 1918.
The concept of variance is highly important in advanced work where it is possible to split the total
into several parts, each attributable to one of the factors causing variation in the original series.

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