$\begin{array}{l}{\sigma }^2=\frac{1}{20}\left[1^2+2^2+\dots +{20}^2\right]-{\left[\frac{1}{20}(1+2+\dots +20)\right]}^2\\ =\frac{1}{20}\frac{20\times 21\times (2\times 20+1)}{6}-{\left[\frac{1}{20}\frac{20\times 21}{2}\right]}^2\\ =\frac{7\times 41}{2}-\frac{441}{4}=\frac{133}{4}.\end{array}$

In fact, the variance of first $n$-natural numbers is $\frac{n^2-1}{12}$.