While computing the mean deviation we ignore the negative signs of the deviations of the items from the central tendency. This is because in dispersion we are interested only in knowing how much, on an average, items deviate from central tendency irrespective of the fact that items are less than or more than central tendency. This ignoring of signs which arise during calculations, introduces some limitations on the measure. A mathematical solution for ignoring signs is squaring. As the square of any negative item becomes positive, a new measure of dispersion is defined in which deviations are first squared (to ignore the signs) and then averaged out. The value so obtained gives the average of the squares of the deviations and not of deviations directly. So, finally a square root of this value is extracted. Thus the result obtained will give an indirect average of deviations arithmetic mean or median or mode. Out of these three values, in every data, root mean square deviation about arithmetic mean is the least. So it is called Standard Deviation.

Thus, the standard deviation is defined as the position square
root of the variance. This concept was introduced by Karl Pearson in 1893. It
is 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. As
this measure is calculated by finding square root of the mean of the squares of
the deviation of items from the arithmetic mean, it is also called Root mean Square
Deviation. Standard Deviation is usually denoted by the Greek letter ‘Ïƒ’ (read
as sigma). Now, let us study the meaning, method of computation, merits and
limitations of standard deviation.

Computation

There are two methods of calculating standard deviation for
ungrouped and grouped distributions. They are: 1) direct method and 2) short
cut method.

Let us study these two methods.

1. Direct method: under this method, standard deviation is
calculated by taking deviations of the items from the actual arithmetic mean of
the distributions.

2. Short-Cut method: Under this method, standard deviation is calculated
by taking deviations of the items from the assumed mean. Among the above two
methods, short cut method is convenient when the size of items and their
numbers are large or the arithmetic mean comes out in fractions. If mean is
with fractional value, it is a time consuming process to find the deviations
and its square deviations to compute the standard deviation.