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Line 16: − + − It may be important to consider the standard deviation is but ''a'' measure of dispersion, and not the only one. For instance, a [[Cauchy distribution]] has an undefined standard deviation (or mean), yet a glance at its [[Probability density function | pdf]] suggests that it is not "infinitely" dispersed.+ − + − +
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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: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>
:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>
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.
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:Probability and Statistics]]
[[Category:Mathematics]]
[[Category:Mathematics]]