Pairwise-difference estimator of the variance

Let $X_{1},X_{2},…,X_{n}$ be i.i.d. random variables having $N(\mu,\sigma^{2})$ distribution, where $\mu\in\mathbb{R}$ and $\sigma>0$. Define $$W=\frac{1}{2n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\left(X_{i}-X_{j}\right)^{2}$$ Then $W$, as an estimator of $\sigma^{2}$, is