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 $$\begin{equation*} \langle I \rangle_M=\frac{1}{M}\sum_{l=1}^{M}\langle f\rangle_l. \end{equation*}$$