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The positive square root of variance is called standard deviation. Symbolically,
σ² = ∑(xi-µ)²/N, for population sample
S²=∑ (xi−Ẍ)²/n, for sample data
The standard deviation expressed in the same units as the observations themselves and is a measure of the average spread around the mean, Karl Pearson (1857-1936), founder of the science of statistics”, is credited with the name standard deviation, the most useful measure of dispersion. The sample variance in some texts is defined as
Where n is replaced by n-1 on the basis of argument the knowledge of any n-1 deviations automatically determines the remaining deviation as the sum of n deviations must be zero. This is, in fact, an unbiased estimator of the population variance. Variance S² =∑(xi-ẍ)²/n, for small samples, underestimates the population variance .
When the data are grouped in to a frequency distribution having k classes with midpoints x1,x2,….xk and the corresponding frequencies f1,f2…fk(∑fi=n), the sample variance and standard deviation are given by
s² =∑fi(xi-ẍ)²/n ,and
It should be noted that for a frequency distribution, as the number of observations or the total frequency n is usually large, dividing the sum of square deviations by n-1 is practically equivalent to dividing it by n.
The standard deviation has definite mathematical meaning, utilizes all observed values and is amenable to mathematical treatment but is affected by extreme values. The standard deviation is an absolute measure of dispersion. Its relative measure called coefficient of standard deviation, is defined as
Coefficient of S.D = standard deviation/ mean
The quantity [∑(xi-a)²/n]½, where a is some arbitrary origin, is called the root mean square deviation which becomes the standard deviation when this arbitrary origin coincides with the mean.
The variance is the mean of squares minus the square of the mean. The corresponding formula for sample variance is
S² = ∑xi²/n- (∑xi/n) ²
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