The following simple expression produces a sum operator as in (1). But I want it to be bigger like in (2), of course, without altering the size of inner braces. Any suggestion?
\documentclass[a4,12pt]{article}
\begin{document}
$$\sum_{i=1}^{N_P}\biggl(X^{(i)}-Y^{(i)}\biggr)^2$$
\end{document}


\biggin the first expression; if you use simple(and), the size of the summation symbol is just right. And there's no reason for increasing the size of\sumunder any circumstance, in my opinion. – egreg Nov 07 '14 at 17:07