According to page 2 of this paper, if $X$ is an arbitrary measurement with mean $\mu$ and variance $\sigma^2$, and $\overline{x}$ is the sample mean from random samples of size n, then:
$$\displaystyle\sigma_{\overline{x}}^2 = \frac{\sigma^2}{n}.$$
This is not something obvious, and I wonder if it has a name or is stated formally somewhere. Thanks.
On
Most detailed and easy to understand explanation I can found is in this link:
$\textrm{var}(\bar{X})=\textrm{var}\left(\dfrac{X_1+X_2+\cdots+X_n}{n}\right)$
$=\textrm{var}\left(\dfrac{1}{n}X_1+\dfrac{1}{n}X_2+\cdots+\dfrac{1}{n}X_n\right)$
$=\dfrac{1}{n^2}\textrm{var}(X_1)+\dfrac{1}{n^2}\textrm{var}(X_2)+\cdots+\dfrac{1}{n^2}\textrm{var}(X_n)$
$=\dfrac{1}{n^2}[\sigma^2+\sigma^2+\cdots+\sigma^2]$
$=\dfrac{1}{n^2}[n\sigma^2]=\dfrac{\sigma^2}{n}$
Thus:
$$\textrm{var}(\bar{X})=\dfrac{\sigma^2}{n}$$
It doesn't have a name but it is an immediate consequence of two facts:
$Var (cY)=c^{2} var (Y)$ if $c$ is a constant.
Variance of a sum of independent random variables is the sum of the variances.
The variance of $\overline {x}$ is therefore $\frac 1 {n^{2}} {n \, var (X)}=\frac {\sigma^{2}} n$.