Changes

'''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>

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:Probability and Statistics]]

[[Category:Probability and Statistics]]

[[Category:Mathematics]]

[[Category:Mathematics]]