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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{n} = \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}}

(where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.