Definition

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

Bias

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

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 \(\sigma^2\)

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 \(\sigma^2\).