The probability density of obtaining a sample mean \( \bar x_n \) is the product of probabilities of obtaining arbitrary values \( x_1, x_2,\dots,x_n \) with the constraint that the mean of the set \( \{x_i\} \) is \( \bar x_n \):

$$ p_{\overline X_n}(x) = \int p_X^{\phantom X}(x_1)\cdots \int p_X^{\phantom X}(x_n)\ \delta\!\left(x - \frac{x_1+x_2+\dots+x_n}{n}\right)dx_n \cdots dx_1 $$

And in particular we are interested in its variance \( \mathrm{var}(\overline X_n) \).