In addition to the average value $\langle f \rangle$ the other important quantity in a Monte-Carlo calculation is the variance $\sigma^2$ and the standard deviation $\sigma$. We define first the variance of the integral with $f$ for a uniform distribution in the interval $x \in [0,1]$ to be $$\begin{equation*} \sigma^2_f=\sum_{i=1}^N(f(x_i)-\langle f\rangle)^2p(x_i), \end{equation*}$$ and inserting the uniform distribution this yields $$\begin{equation*} \sigma^2_f=\frac{1}{N}\sum_{i=1}^Nf(x_i)^2- \left(\frac{1}{N}\sum_{i=1}^Nf(x_i)\right)^2, \end{equation*}$$ or $$\begin{equation*} \sigma^2_f=\left(\langle f^2\rangle - \langle f \rangle^2\right). \end{equation*}$$