Consider an odd number of random i.i.d. Gaussian real numbers $x_1,x_2,\dots x_n$. Let $M$ be their median. What can be said about the Hermite--Fourier expansion of $M=M(x_1,x_2,\dots,x_n)$?   

Of course, an actual formula will be of interest but I will be happy to see also some numerical data. 
The case $n=3$ is already interesting to me. 
In particular, what is the contribution (sum of squares) of the coefficients for degree-$k$ terms?