Distribution of the sample variance This is my first post to this great website :) It seems like an excellent place to learn. I have a question however that is bothering me as I cannot figure it out through my textbook.
The sample variance $$ S^2 = \frac{1}{n-1}\sum\limits_{i=1}^{n}\left(X_i - \bar{X}\right)^2 $$
Of course, we need to assume that $X_i$ is normal but here is where I am very confused. The next step in my notes shows this:
$$X_i - \bar{X} = \sigma [(X_i - \mu) / \sigma - (\bar{X} - \mu) / \sigma]$$
Where did $\sigma$ come from? And can anyone help me with the proof of this? It appears that the distribution of the sample variance is that of a Chi-Squared R.V. with $n-1$ DoF. Thanks so much!

Your second display equation has nothing to do with a normality or any other assumption.  As long as $0<\sigma<\infty$ it holds.  All you're doing is to multiply by $\sigma$ and to divide by $\sigma$ and to add and subtract $\mu$.
You do need the normality assumption, however, to establish the $\chi_{n-1}^2$ distribution. Use the fact that the sum of the squares of $n-1$ independent normals has a $\chi_{n-1}^2$ distribution.
