ThmDex – An index of mathematical definitions, results, and conjectures.
Result R3633 on D3363: Estimator bias
Bessel-corrected sample variance is an unbiased estimator for the variance of uncorrelated identically distributed random real numbers

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) $$\mathbb{E} |X_1|^2 < \infty$$ (iv) $$\sigma^2 : = \text{Var} X_1$$ (v) $N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer (vi) $$\overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n$$
Then $$\mathbb{E} \left( \frac{1}{N - 1} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) = \sigma^2$$
Proofs
Proof 0
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) $$\mathbb{E} |X_1|^2 < \infty$$ (iv) $$\sigma^2 : = \text{Var} X_1$$ (v) $N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer (vi) $$\overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n$$
Result R3626: Sample variance is an unbiased estimator for the variance of uncorrelated identically distributed random real numbers shows that $$\mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) = \frac{N - 1}{N} \sigma^2$$ Multiplying both sides by the rational number $N / (N - 1)$ and using the linearity of expectation, one has $$\begin{split} \mathbb{E} \left( \frac{1}{N - 1} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) & = \frac{N}{N - 1} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) \\ & = \frac{N}{N - 1} \frac{N - 1}{N} \sigma^2 \\ & = \sigma^2 \end{split}$$ $\square$