I understand that

$\displaystyle \frac{(n-1)S^2}{\sigma ^2} = \frac{(n-1)\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}{\sigma ^2}$ ~ $\displaystyle \chi ^2(n-1)$

I do not understand why

$\displaystyle \frac{n\hat{\sigma}^2}{\sigma ^2} = \frac{n\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2}{\sigma ^2}$ ~ $\displaystyle \chi ^2(n-2)$