Standard deviation

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\frac{d}{dx} \sin x=?\, This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

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.

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