= Mathematical statistics
<Mathematical statistics> studies the mathematical properties of procedures for learning from data: <estimation>, <statistical hypothesis tests> and quantification of uncertainty under explicit <statistical models>. It asks what a method estimates, how its error behaves and which assumptions justify that behavior.
If $X_1,\ldots,X_n$ are <independent and identically distributed random variables> with <expected value> $\mu$ and finite <variance> $\sigma^2$, the <sample mean> $\overline X$ satisfies $\mathbb E\overline X=\mu$ and $\operatorname{Var}(\overline X)=\sigma^2/n$. Independence removes cross-covariances in the second calculation. These identities explain why averaging can improve precision, while also exposing a limitation: strongly dependent observations may provide much less information. Theorems about <estimators> are therefore conditional statements about a sampling model, rather than universal guarantees for every dataset.
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