Statistic

Definition

Let $X_1, \dots, X_n$ be random variables with values in a measurable space $(E, \mathcal{E})$. For example, they can be a random sample from a law with an unknown parameter $\theta$. A statistic is a random variable of the form $T = g(X_1, \dots, X_n)$, where $g \colon E^n \to \mathbb{R}^k$ is a measurable function that does not depend on $\theta$.

Remarks

Examples are the sample mean $\bar{X}_n = (X_1 + \dots + X_n)/n$, the sample variance, the sample maximum $\max_i X_i$ and the order statistics. A statistic is a random variable, so it has a law: the sampling distribution of the statistic. A statistic that one uses to estimate a parameter is an estimator. A statistic that one uses to decide a hypothesis test is a test statistic.

Explanations

L1 human

A statistic is a number that you can compute from your data alone, such as the average of the measurements. It must not contain the unknown quantity that you want to learn about.