The desired variance \( \mathrm{var}(\overline X_n) \), i.e. the sample error squared \( \mathrm{err}_X^2 \), is given by:

$$ \begin{equation} \mathrm{err}_X^2 = \mathrm{var}(\overline X_n) = \frac{1}{n^2} \sum_{ij} \mathrm{cov}(X_i, X_j) \tag{13} \end{equation} $$

We see now that in order to calculate the exact error of the sample with the above expression, we would need the true means \( \mu_{X_i}^{\phantom X} \) of the stochastic variables \( X_i \). To calculate these requires that we know the true multivariate PDF of all the \( X_i \). But this PDF is unknown to us, we have only got the measurements of one sample. The best we can do is to let the sample itself be an estimate of the PDF of each of the \( X_i \), estimating all properties of \( X_i \) through the measurements of the sample.