Support of Variance of Random IID Sample (Bounded) 
Let $X_i$, $i\in\{1, 2, ..., n\}$ be independent and identically distributed random variables with bounded support $[\alpha, \beta]$, with $\alpha,\beta \in \mathbb{R}$.
What is the support of the random variable that corresponds to the variance of this sample?  That is, if $\mu:=\frac{\sum{x_i}}{n}$ and $\sigma^2:=\sum\frac{(x_i-\mu)^2}{n}$, what are the smallest and largest values that $\sigma^2$ can take?

The smallest value $\sigma^2$ can take is clearly $0$ if all $x_i$ are identical, but what's the largest, and how can you prove it's the largest?  To be clear, I'm looking for this value to be written in terms of $\alpha, \beta,$ and $n$.

Largest value is obtained when all $X_i$ have the distribution $P(X_i=\alpha)=P(X_i=\beta)=\frac{1}{2}$.  The mean for one varialbe  is $\frac{\alpha+\beta}{2}$.  The variance for one variable is $(\frac{\beta-\alpha}{2})^2$ so the variance for $n$ independent variables is $\frac{(\beta-\alpha)^2}{4n}$.  The proof may be messy, but it is obvious.
