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Line 1: − {{math-h}}+ − '''Standard deviation''' is a measure in [[Statistics|statistics]] for how much a set of values ''varies''. It allows for one to find how likely it is for a specific value to be obtained by doing a [[Z-test]].+ − ==Standard Deviation==+ − The standard deviation of a set of values is a measure of how widely the values differ from each other. Specifically, standard deviation follows the equation:+ − + − This is the square root of the variance, which is:+ − + − Where:+ − *<math>\bar x</math> is the arithmetic mean of all values of x − *<math>\sum</math> is the [[summation]] function − *<math>n</math> is the number of <math>x</math> values + + + + − If the distribution of the values is [[normal distribution|normal]] then it follows the [[Empirical rule]], which states that: − :{{main|Empirical rule}} − − *68% of the values will fall within 1<math>\sigma</math> of the mean. − *95% of all values will fall within 2<math>\sigma</math> of the mean. − *99.7% of all values will fall within 3<math>\sigma</math> of the mean. − + +
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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
:<math>\sigma(x) = \sqrt {\sum(x - \bar x) \over n - 1}</math>
:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>
The formula for standard deviation must not be confused with the formula
:<math>Var(x) = {\sum (x - \bar x) \over n - 1}</math>
:<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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{{math-h}}
[[category:statistics]]
[[Category:Probability and Statistics]]
[[Category:Mathematics]]