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
| <math>\frac{d}{dx} \sin x=?\,</math> | 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 <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>
where the expected value of X is E(X).
Thus the standard deviation is
- <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>S_{n} = \sqrt {\sum_n(x - \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.