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} $\square$