I am stuck at the following problem: given $n$ numbers I uniformly sample $\sqrt{n}$ of them (at once) and take the sample median $X_m$. How does it correspond to the real median $n/2$? In more detail, what is $\mathbb{E}(X_m)$? Perhaps there is a nice way to show it is "near" the real median (I hope it is).
I can't really come up with some nice formula that I can solve besides some combinatorial expressions that define $\mathbb{P}(X_m = k)$ where I count the number of sets that would yield a median $k$.