Turning a Sum Into an Integral

Question

I came up with a measure of dispersion for a sequence of $n$ numbers $(x_i)$, namely $\sqrt{\sum^n_{i=1}x^2_i - \sum^n_{j=2}\sum^{j-1}_{i=1}\frac{2x_i x_j}{n-1}}$. Is there a way to generalize this sum by turning it into an integral for probability distribution functions?

Answer 1

Using Sangchul Lee’s reply, I was able to deduce that if I scale my original calculation by a factor of $\frac{\sqrt{n-1}}{n}$, then my scaled equation becomes identical to the standard deviation of the set, regardless of the $n$ chosen.