Sample mean

Definition

Let $X_1, \dots, X_n$ be real random variables, for example an i.i.d. sample. The sample mean is the statistic $$\bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i.$$ For observed values $x_1, \dots, x_n$, the sample mean is the number $\bar{x} = (x_1 + \dots + x_n)/n$.

Remarks

If the $X_i$ are i.i.d. with mean $\mu$, then $E[\bar{X}_n] = \mu$ by linearity of expectation. So $\bar{X}_n$ is an unbiased estimator of $\mu$. If also $\operatorname{Var}(X_i) = \sigma^2 < \infty$, then $\operatorname{Var}(\bar{X}_n) = \sigma^2 / n$. For i.i.d. $\mathcal{N}(\mu, \sigma^2)$ variables, $\bar{X}_n \sim \mathcal{N}(\mu, \sigma^2/n)$. By the law of large numbers, $\bar{X}_n \to \mu$ almost surely.

Explanations

L1 human

The sample mean is the ordinary average of the data. For the measurements $4$, $7$ and $10$, the sample mean is $21/3 = 7$.