Suppose a sample of $n$ random variables $X_1, X_2, \dots, X_n$ each has variance $\sigma^2$, and the correlation between any pair $X_i$ and $X_j$ ($i \ne j$) is a constant $\rho > 0$. What is the variance of the sample mean $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$?