Let $X = \{x_1, x_2, \dots, x_n\}$ be a dataset. A statistician defines a measure of overall pairwise variance as $D = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n (x_i - x_j)^2$. Which of the following is the correct relationship between $D$ and the standard population variance $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$?