Why is unbiased variance divided by n-1

First, we introduce the population mean into the sample variance.

$$ \begin{eqnarray} V_s &=& \frac{1}{n} \sum_{i=1}^{n} (X_i - \overline{X})^2 \\ &=& \frac{1}{n} \sum_{i=1}^{n} ((X_i - \mu) - (\overline{X} - \mu)) \\ &=& \frac{1}{n} \sum_{i=1}^{n} ((X_i - \mu)^2 - 2(X_i - \mu)(\overline{X} - \mu) + (\overline{X} - \mu)^2) \\ &=& \frac{1}{n} \sum_{i=1}^{n} (X_i - \mu)^2 - \frac{2}{n} \sum_{i=1}^{n} (X_i - \mu)(\overline{X} - \mu) + \frac{n}{n} (\overline{X} - \mu)^2 \end{eqnarray} $$

The second term can be rewritten.

$$ \begin{eqnarray} - \frac{2}{n} \sum_{i=1}^{n} (X_i - \mu)(\overline{X} - \mu) &=& - \frac{2}{n} (\overline{X} - \mu) \sum_{i=1}^{n} (X_i - \mu) \\ &=& -2 (\overline{X} - \mu) \frac{1}{n} \sum_{i=1}^{n} (X_i - \mu) \\ &=& -2 (\overline{X} - \mu) (\overline{X} - \mu) \\ &=& -2 (\overline{X} - \mu)^2 \end{eqnarray} $$

Then, the expected value of the sample variance is expressed by the population variance \(\sigma^2\).

$$ \begin{eqnarray} V_s &=& \frac{1}{n} \sum_{i=1}^{n} (X_i - \mu)^2 - (\overline{X} - \mu)^2 \\ E[V_s] &=& E[\frac{1}{n} \sum_{i=1}^{n} (X_i - \mu)^2 - (\overline{X} - \mu)^2] \\ &=& \frac{1}{n} \sum_{i=1}^{n} E[(X_i - \mu)^2] - E[(\overline{X} - \mu)^2] \\ &=& \frac{1}{n} (n \sigma^2) - \frac{1}{n} \sigma^2 \\ &=& \sigma^2 - \frac{1}{n} \sigma^2 \\ &=& \frac{n-1}{n} \sigma^2 \end{eqnarray} $$

Thus, the unbiased variance (the estimated value of the population variance) is given by denominator \(n-1\).

$$ \begin{eqnarray} \sigma^2 &=& \frac{n}{n-1} E[V_s] \\ &=& \frac{n}{n-1} \cdot \frac{1}{n} \sum_{i=1}^{n} (X_i - \overline{X})^2 \\ &=& \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \overline{X})^2 \\ \end{eqnarray} $$

Reference: http://www.sist.ac.jp/~kanakubo/research/statistic/fuhenbunsan.html (no longer exists. 9.28.2025)

First version：9.28.2025