Sample Variance

Empirical Sample Variance

sn2=1n−1∑i=1n(Xi−Xˉ(n))2\begin{equation} s_{n}^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}_{(n)}\right)^{2} \end{equation} V(t)=1n∑k=1n(tk−M(t))2V(t)=\frac{1}{n} \sum_{k=1}^n\left(t_k-M(t)\right)^2

Properties
If we have E(X14)<∞\mathbb{E}\left(X_1^4\right)<\infty , then for the Sample Variance S(n)2S^2_{(n)} we have:

E(S(n)2)=σ2,V(S(n)2)=1n(V((X1−μ)2)+2n−1σ4)\mathbb{E}\left(S_{(n)}^2\right)=\sigma^2, \quad \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.