Mean Standard Deviation

Both the mean and standard deviation are measures of dispersion

. They are both based on all observations, they are readily comprehensible, they are easy to calculate, they are affected very little by fluctuations in sampling, and are amenable to algebraic treatment.

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Finding the Standard deviation:

The standard deviation is the square root of the arithmetic mean of the squares of all deviations, deviations being measured from the arithmetic mean of the observations. If the standard deviation be denoted by and a deviation from the arithmetic mean (M) by x, then the standard deviation is given by the equation: ^2 = 1/n.

To square all the deviations, when finding standard deviation, may seem at first sight an artificial procedure, but it may be remembered that it would be useless to take the mere sum of the deviations, in order to find standard deviation, since this sum is necessarily zero if deviations be taken from the mean. In order to obtain some quantity that shall vary with the dispersion, it is necessary to average the deviations by a process that treats them as if they were all of the same sign, and squaring is the simplest process for eliminating signs which leads to results of algebraical convenience.

Properties of standard deviation:

The standard deviation is the measure of dispersion which it is most easy to treat by algebraical methods, resembling in this respect the arithmetic mean amongst measures of position. The standard deviation possesses majority of the properties required in a measure of dispersion. It is rigidly defined; it is based on all the observations made; it is calculated with reasonable ease; it lends itself readily to algebraic treatment; and the measure is least affected by fluctuations of sampling.

Mean deviation:

We have already remarked that it would be useless to take the sum of deviations from the mean as the measure of dispersion because such sum is identically zero. We therefore remove the signs of the deviation by squaring to reach the standard deviation.

It is also possible to overcome this difficulty by adding the sum of deviations taken regardless of signs. The arithmetic mean of these absolute deviations is called the mean deviation. If we write ||to denote the deviation from an arbitrary values A taken as positive whatever its actual sign, the mean deviation is thus defined as:

m.d. = 1/n .

Empirical relation between mean deviation and standard deviation:

For symmetric or moderately skew distributions the mean deviation is about four fifths of the standard deviation.

by: Omkar Nayak

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