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 $X$). It is defined as the square root of the variance of these values, where variance is defined as

$\sigma ^{2}=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\operatorname {E} [X^{2}]-(\operatorname {E} [X])^{2}$

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

Thus the standard deviation is

$\sigma ={\sqrt {\operatorname {E} [(X-\operatorname {E} [X])^{2}]}}={\sqrt {\operatorname {E} [X^{2}]-(\operatorname {E} [X])^{2}}}$

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

$S_{n}={\sqrt {\sum _{n}(X_{n}-{\bar {X}})^{2} \over n-1}}$

(where ${\bar {X}}={\sum _{n}X_{n} \over N}$ 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.