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

$$\frac{(n-1)s^2}{σ^2} ~ χ_{n-1}^2$$