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)	$N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer
(vi)	\begin{equation} \overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n \end{equation}

Result R3626: Sample variance is an unbiased estimator for the variance of uncorrelated identically distributed random real numbers shows that \begin{equation} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) = \frac{N - 1}{N} \sigma^2 \end{equation} Multiplying both sides by the rational number $N / (N - 1)$ and using the linearity of expectation, one has \begin{equation} \begin{split} \mathbb{E} \left( \frac{1}{N - 1} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) & = \frac{N}{N - 1} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) \\ & = \frac{N}{N - 1} \frac{N - 1}{N} \sigma^2 \\ & = \sigma^2 \end{split} \end{equation} $\square$