Flip an unweighted coin $N$ times, where we denote $X_i = 1$ if flip $i$ resulted in heads, and $X_i = 0$ if flip $i$ resulted in tails. Now take the average result $\bar X = \frac{\sum_{i=1}^N X_i}{N}$.
Conduct the same experiment again, this time denoting the $N$ flips $Y_i$. What is the expected value of the square of the difference between the two averages? In other words, $E \left[ (\bar X - \bar Y)^2 \right] = E \left[ \left( \frac{\sum_{i=1}^N X_i - \sum_{i=1}^N Y_i}{N} \right)^2 \right] = ?$
I'm interested in all of the steps leading up to the answer, which I assume should end up being some function of $N$...