A median of an AVL. How to take advantage of the AVL?
If you modify the AVL tree by storing the size of the subtree at each node rather than just its height, then you can find the median in time $O(\log n)$ using the fact that the tree is balanced. To accomplish this, you write a more general procedure Select which accepts a node $v$ and a number $k$, and finds the $k$th smallest node at the subtree rooted at $v$.

Suppose that the left subtree (if any) has $L$ nodes. If $k \leq L$ then we recurse to the left subtree. If $k = L+1$ then we return $v$. Otherwise we recurse to the right subtree, reducing $k$ by $L+1$.

The running time of this algorithm is linear in the height of the tree, which is $O(\log n)$.