Difference between revisions of "Standard deviation"
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| − | '''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 [[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> | ||
Revision as of 15:06, January 19, 2009
| This article/section deals with mathematical concepts appropriate for late high school or early college. |
Standard deviation is a measure in statistics of the dispersion of a set of values (represented as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ). It is defined as the square root of the variance of these values, where variance is defined as
![{\displaystyle \sigma ^{2}=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\operatorname {E} [X^{2}]-(\operatorname {E} [X])^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2c16e5cbc1f3fd32dfd976f5a7d93a8e1ded5e)
where the expected value of X is E(X).
Thus the standard deviation is
![{\displaystyle \sigma ={\sqrt {\operatorname {E} [(X-\operatorname {E} [X])^{2}]}}={\sqrt {\operatorname {E} [X^{2}]-(\operatorname {E} [X])^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6351cdec12c1d239008c4088badf1a861744e97)
The formula for standard deviation must not be confused with the formula
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{n} = \sqrt {\sum_n(x - \bar x)^2 \over n - 1}}
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.