$$\begin{align} \sigma^2 =& \frac{\sum_{i=1}^n(x_i-\overline{X})^2 }{n} \\ =&\frac{\sum_{i=1}^nx_i^2}{n} + \frac{\sum_{i=1}^n\overline{X}^2}{n} - \frac{\sum_{i=1}^n2x_i\overline{X}}{n}\\=& \frac{\sum_{i=1}^nx_i^2}{n} + \overline{X}^2 - \frac{2\overline{X}\sum_{i=1}^nx_i}{n} \\=&\frac{\sum_{i=1}^nx_i^2}{n} - \overline{X}^2 \end{align}$$
