The variance is nothing but a measure of the extent to which \( f \) deviates from its average over the region of integration. The standard deviation is defined as the square root of the variance. If we consider the above results for a fixed value of \( N \) as a measurement, we could recalculate the above average and variance for a series of different measurements. If each such measumerent produces a set of averages for the integral \( I \) denoted \( \langle f\rangle_l \), we have for \( M \) measurements that the integral is given by $$ \begin{equation*} \langle I \rangle_M=\frac{1}{M}\sum_{l=1}^{M}\langle f\rangle_l. \end{equation*} $$