Variance

$\displaystyle V(X)$	$\displaystyle =$	$\displaystyle E(X^2)-\left[E(X)\right]^2$
	$\displaystyle =$	$\displaystyle \sum_{k=1}^{n}p_{k}\ k^2 - \left(\frac{n+1}{2}\right)^2$
	$\displaystyle =$	$\displaystyle \sum_{k=1}^{n} \frac{1}{n} \ k^2 - \frac{(n+1)^2}{4}$
	$\displaystyle =$	$\displaystyle \left( \frac{1}{n} \ \frac{n(n+1)(2n+1)}{6} \right)-\frac{(n+1)^2}{4}$
	$\displaystyle =$	$\displaystyle \frac{n^2-1}{12}$