For computational purposes one usually splits up the estimate of \( \mathrm{err}_X^2 \), given by Eq. (15), into two parts

$$ \mathrm{err}_X^2 = \frac{1}{n}\mathrm{var}(x) + \frac{1}{n}(\mathrm{cov}(x)-\mathrm{var}(x)), $$

which equals

$$ \begin{equation} \frac{1}{n^2}\sum_{k=1}^n (x_k - \bar x_n)^2 +\frac{2}{n^2}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n) \tag{17} \end{equation} $$

The first term is the same as the error in the uncorrelated case, Eq. (16). This means that the second term accounts for the error correction due to correlation between the measurements. For uncorrelated measurements this second term is zero.