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'''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:Probability and Statistics|variance]] of these values, where variance is defined as

 

'''Standard deviation''' is a measure in [[Statistics|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>

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

where the [[expectation|expected value]] of ''X'' is E(''X'').

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is  

Thus the standard deviation is  

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

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

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

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

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.

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

[[category:statistics]]

[[Category:Probability and Statistics]]

[[Category:Mathematics]]