= Sample mean
{title2=$\overline X$}
{wiki}
For observations $X_1,\ldots,X_n$ with $n\geq1$, the <sample mean> is $\overline X=n^{-1}\sum_iX_i$. It is a <statistic> summarizing location: values $(2,3,7)$ give $\overline X=4$. For an <IID sample> with mean $\mu$ and finite <variance> $\sigma^2$, <linearity of expectation> gives $\mathbb E\overline X=\mu$, while <variance additivity for independent random variables> gives $\operatorname{Var}(\overline X)=\sigma^2/n$. Dependence changes the <variance> through cross-<covariances>. A <sample mean> is sensitive to extreme observations, and a population such as the <Cauchy distribution> need not have the finite mean needed for ordinary averaging guarantees.
Back to article page
