Sample mean ($\overline{X}$)

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.

For independent samples of sizes $n,m$ from populations with means ${\mu }_X,{\mu }_Y$ and finite variances ${\sigma }_X^2,{\sigma }_Y^2$, the difference of their sample means has expected value ${\mu }_X-{\mu }_Y$ and variance ${\sigma }_X^2/n+{\sigma }_Y^2/m$. This uses independence between and within the samples. With dependent samples, a covariance term is required instead.

Partition a sample into two disjoint groups $A,B$ with fixed positive sizes $n_A,n_B$, and let $w=n_A/(n_A+n_B)$. The overall sample mean is the weighted meanSubtracting gives the displayed identity. The subgroup and overall sample means share observations, so they are not independent. If the two disjoint groups are sampled independently, thenThus treating the subgroup and the combined sample as two independent samples uses the wrong standard error. Equality of the subgroup and overall population means is equivalent to equality of subgroup and complement means when both groups have positive weight, but the sampling distribution must respect the shared observations.

For independent and identically distributed observations with finite positive variance, the central limit theorem, consistency of the sample variance and Slutsky's theorem give $T_n\stackrel{d}{\to }N(0,1)$. Under an independent normal distribution sample it instead has the exact finite-sample Student t-distribution with $n-1$ degrees of freedom, as described by the studentized normal sample mean. The asymptotic statement does not require normal individual observations.