Standard deviation

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This article/section deals with mathematical concepts appropriate for late high school or early college.

Standard deviation is a measure in statistics of the dispersion of a set of values (represented as <math>X</math>). It is defined as the square root of the variance of these values, where variance is defined as

<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the expected value of X is E(X).

Thus the standard deviation is

<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

<math>S_{n} = \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>

(where <math>\bar X = {\sum_n X_n \over N}</math> is the sample mean).

which is the formula for a point estimate of the true standard deviation from a sample size of n. As such this statistical estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

It may be important to consider the standard deviation is but a measure of dispersion, and not the only one. For instance, a Cauchy distribution has an undefined standard deviation (or mean), yet a glance at its pdf suggests that it is not "infinitely" dispersed.