Let $X = \{x_1, x_2, \dots, x_n\}$ be a set of positive real numbers. If the arithmetic mean $A = \frac{1}{n} \sum_{i=1}^n x_i$ and the geometric mean $G = \sqrt[n]{\prod_{i=1}^n x_i}$, which of the following statements about the variance $\sigma^2$ are true when the numbers are not all equal?