Find $y$ to minimize $\sum (x_i - y)^2$ I have a finite set of numbers $X$. I want to minimize the following expression by finding the appropriate value for y:
$$\sum\limits_{i=1}^n (x_i - y)^2$$

This is one of those problems where you just turn the crank and out pops the answer. The basic optimization technique of "set the derivative equal to zero and solve" to find critical points works in its simplest form without issue here.
And as the others have mentioned, the special form of being quadratic allows you to apply the specialized techniques you've learned for dealing with quadratic equations.
