\[ \begin{align*} V &= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2 \\ &= \frac{1}{n} \sum_{i=1}^n \left \{(X_1-\bar{X})^2 + \cdots + (X_n-\bar{X})^2 \right \} \\ &= \frac{1}{n} \sum_{i=1}^n \left \{(X_1^2-2X_1\bar{X}+\bar{X}^2) + \cdots + (X_n^2-2X_n\bar{X}+\bar{X}^2) \right \} \\ &= \frac{1}{n} \left \{ (X_1^2 + \cdots +X_n^2)-2\bar{X}(X_1 + \cdots + X_n) +(\bar{X}^2 + \cdots +\bar{X}^2) \right \} \\ &= \frac{1}{n} \left \{ \sum_{i=1}^n X_i^2 - 2\bar{X}\sum_{i=1}^n X_i + n\bar{X}^2 \right \} \\ &= \frac{\displaystyle \sum_{i=1}^n X_i^2}{n} - \frac{\displaystyle 2\bar{X}\sum_{i=1}^n X_i}{n} + \frac{n\bar{X}^2}{n} \\ &= \frac{\displaystyle \sum_{i=1}^n X_i^2}{n} - 2\bar{X}\frac{\displaystyle \sum_{i=1}^n X_i}{n} + \bar{X}^2 \\ &= \frac{\displaystyle \sum_{i=1}^n X_i^2}{n} - 2\bar{X}^2 + \bar{X}^2 \\ &= \frac{\displaystyle \sum_{i=1}^n X_i^2}{n} - \bar{X}^2 \\ &= \frac{X_1^2 + X_2^2 + \cdots +X_n^2}{n} - \bar{X}^2 \\ \end{align*} \]