Sample Variance

Empirical Sample Variance

$$\begin{array}{c}{s}_n^2=\frac{1}{n-1}\sum _{i=1}^n{\left(X_i-{\bar{X}}_{(n)}\right)}^2\end{array}$$

$V(t)=\frac{1}{n}{\sum }_{k=1}^n{\left(t_k-M(t)\right)}^2$

Properties If we have $\mathbb{E}\left(X_1^4\right)<\infty$ , then for the Sample Variance $S_{(n)}^2$ we have: $\mathbb{E}\left(S_{(n)}^2\right)={\sigma }^2,\mathbb{V}\left(S_{(n)}^2\right)=\frac{1}{n}\left(\mathbb{V}\left({\left(X_1-\mu \right)}^2\right)+\frac{2}{n-1}{\sigma }^4\right)$ The Expectation is exactly the value we want to estimate. We can use the Variance to further describe the behaviour of the estimator.