Let us first take a look at what happens to the sample error as the size of the sample grows. In a sample, each of the measurements $x_i$ can be associated with its own stochastic variable $X_i$. The stochastic variable $\overline X_n$ for the sample mean $\bar{x}_n$ is then just a linear combination, already familiar to us:

$$\overline X_n = \frac{1}{n}\sum_{i=1}^n X_i$$

All the coefficients are just equal $1/n$. The PDF of $\overline X_n$, denoted by $p_{\overline X_n}(x)$ is the desired PDF of the sample means.