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

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 \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

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 \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

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

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