By the same procedure we can use the sample variance as an estimate of the variance of any of the stochastic variables \( X_i \)

$$ \mathrm{var}(X_i)=\langle x_i - \langle x_i\rangle\rangle \approx \langle x_i - \bar x_n\rangle\nonumber, $$

which is approximated as

$$ \begin{equation} \mathrm{var}(X_i)\approx \frac{1}{n}\sum_{k=1}^n (x_k - \bar x_n)=\mathrm{var}(x) \tag{14} \end{equation} $$

Now we can calculate an estimate of the error \( \mathrm{err}_X^{\phantom X} \) of the sample mean \( \bar x_n \):

$$ \begin{align} \mathrm{err}_X^2 &=\frac{1}{n^2}\sum_{ij} \mathrm{cov}(X_i, X_j) \nonumber \\ &\approx&\frac{1}{n^2}\sum_{ij}\frac{1}{n}\mathrm{cov}(x) =\frac{1}{n^2}n^2\frac{1}{n}\mathrm{cov}(x)\nonumber\\ &=\frac{1}{n}\mathrm{cov}(x) \tag{15} \end{align} $$

which is nothing but the sample covariance divided by the number of measurements in the sample.