Variance:Probability and Statistics

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This article/section deals with mathematical concepts appropriate for late high school or early college.

Variance is a measure in statistics of the dispersion of a set of values (represented as $X$). It is defined as

$\sigma ^{2}=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\operatorname {E} [X^{2}]-(\operatorname {E} [X])^{2}$

where the expected value of X is E(X).

The formula for variance must not be confused with the formula

$S_{n}^{2}={\sum _{n}(X_{n}-{\bar {X}})^{2} \over n-1}$

(where ${\bar {X}}={\sum _{n}X_{n} \over N}$ is the sample mean).

which is the formula for a point estimate of the true variance from a sample of size n. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.