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Line 1: − The '''standard deviation''' of a set of values is a measure of how widely the values differ from each other. Specifically, the '''standard deviation''' is the square root of the average of the squares of the differences between the data values and their [[mean]]. If the distribution of the values is [[normal distribution|normal]] then about 68% of the values will fall within one standard deviation of the mean.+ − Examples help illustrate this concept. Learning the median height of basketball players tell us that half are above that height and half are below. Learning the '''standard deviation''' of their heights tells us how varied their heights are.+ − Mathematically, the standard deviation of a [[random variable]] ''X'' is:+ − :<math>\sigma = \sqrt{\operatorname{E}((X-\operatorname{E}(X))^2)} </math> <math>= \sqrt{\operatorname{E}(X^2) - (\operatorname{E}(X))^2}</math>+ − + − + + + + + + + + + + + + + + +
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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
:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>
where the [[expectation (math)|expected value]] of ''X'' is E(''X'').
Thus the standard deviation is
where the [[expected value]] of ''X'' is E(''X'').
:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>
[[category:statistics]]
The formula for standard deviation must not be confused with the formula
:<math>S_{n} = \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>
(where <math>\bar X = {\sum_n X_n \over N}</math> 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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[[Category:Probability and Statistics]]
[[Category:Mathematics]]