Say I have a rectangular function as follows $$f(x) = \begin{cases} 1 & \text{$|x|$$\leq$$x_0$} \\[2ex] 0 & \text{otherwise} \end{cases}$$
I want to match its width (which in this case is $2x_0$) to that of a Gaussian defined as, say, $exp(-2x^2/\sigma^2)$ (i.e., $\sigma^2$. I want to write some relation between $\sigma^2$ and $x_0$). Is there a standard or conventional way to do this?