Let $k\in\mathbb N^+$ be a positive integer.

Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$.

For $i\in \{1,2,\ldots,2k+1\}$, let $Y_i\triangleq|\{j\mid X_j=i\}|$ denote the number of variables with the value $i$. Notice that each $Y_i$ is distributed $Bin\left(n,p\right)$, for $p\triangleq\frac{1}{2k+1}$.

Finally, let $Z\triangleq \text{Median}(\{Y_1,Y_2,\ldots,Y_{2k+1}\})$ be the median of the $Y_i$ variables.

An answer to this question in math.se indicates that $\mathbb E(Z)=\frac{n}{2k+1}-O(1)\approx \mathbb E(Y_i)$.

To me, it'll be very interesting to understand other properties of $Z$, and mainly:

What is $Var(Z)$?is it $o(np(1-p))$? i.e., is the variance of the median asymptotically smaller than $Var(Y_i)=np(1-p)$? by how much?

The special case of $k=1$, where we have only 3 $Y_i$ variables is also interesting for me.