 Sample mean

For observations $X_1,\dots ,X_n$ with $n\ge 1$, the sample mean is $\overline{X}=n^{-1}{\sum }_i{X}_i$. 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 $\mathrm{Var}(\overline{X})={\sigma }^2/n$. 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.