## Definition

$S_n^2=\sum_{i=1}^{n}\frac{(X_i-\overline{X}_n)^2}{n-1}$

## Bias

$bias(S_n^2) = E[S^2_n] - \sigma^2 = 0$

• Demonstration

## Standard error $$se$$ of $$\widehat{\theta}$$

$se(\widehat{\theta})=\sqrt{V(\widehat{\theta})}$

## Consistency

$$S_n^2$$ is consistent because it converges in probability to $$\sigma^2$$

• Demonstration …

### Sample standard deviation

$$S_n$$ is consistent because it converges in probability to $$\sigma$$.

### Sample uncorrected variance

Uncorrected $$S_n^2$$ is consistent because it converges in probability to $$\sigma^2$$.