Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3, \ldots$ is an D3357: Identically distributed random collection
(ii)	$X_1, X_2, X_3, \ldots$ is an D3842: Uncorrelated random collection
(iii)	\begin{equation} \mathbb{E} |X_1|^2 < \infty \end{equation}
(iv)	\begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}
(v)	$N \in \{ 1, 2, 3, \ldots \}$ is a D5094: Positive integer
(vi)	\begin{equation} \overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n \end{equation}

Using results

(i)	R5459: Sample mean of identically distributed random real numbers is an unbiased estimator for expectation
(ii)	R2355: Variance is homogeneous to degree two
(iii)	R2259: Variance of a finite sum of random real numbers

we have \begin{equation} \begin{split} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N (X_n - \overline{X}_N)^2 \right) \\ & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X^2_n - 2 X_n \overline{X}_N + \overline{X}_N^2 \right) \\ & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X^2_n - 2 \overline{X}_N \frac{1}{N} \sum_{n = 1}^N + \overline{X}_N^2 \right) \\ & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X^2_n - 2 \overline{X}_N^2 + \overline{X}_N^2 \right) \\ & = \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X^2_n - \overline{X}_N^2 \right) \\ & = \mathbb{E} X^2_1 + \mathbb{E} \overline{X}_N^2 \\ & = \sigma^2 + (\mathbb{E} X_1)^2 - \text{Var} \overline{X}_N - (\mathbb{E} \overline{X}_N)^2 \\ & = \sigma^2 + (\mathbb{E} X_1)^2 - \frac{1}{N^2} (N \sigma^2) - (\mathbb{E} X_1)^2 \\ & = \sigma^2 - \frac{1}{N} \sigma^2 \\ & = \frac{N - 1}{N} \sigma^2 \end{split} \end{equation} $\square$