Properties of the sample variance

Summary

• Given an IID random sample \(x_1,x_2,...,x_n\) where \(E(x_i)=μ\) , \(Var(x_i)=σ^2\) for \(i=1…n\) , the sample variance

\[s^2= \frac{1}{n-1}\sum_{i=1}^{n}{ ‍ }{\left( x_i- \bar{x} \right)}^2\]

• has the following property:

\[E(s^2)=σ^2\]

• Given an IID random sample of size \(n\) from \(N(μ,σ^2)\) we have

\[ \frac{(n-1)s^2}{σ^2} ~ χ_{n-1}^2\]