Considering an unknown custom data structure (Balanced AVL?) that has functions insert(), delete(), mean(a,b) where it returns an arithmetic mean of all numbers between a and b, and std(a,b) where it returns standard deviation for numbers between a and b.

What is an example of a custom data structure that supports these functions in worst-case time O(logn) and please explain in details why it does so for each function (probably pseudocode will do)

I am thinking that a simple balanced AVL tree should do the job, but I am not sure, nor do I know how to properly explain the reasoning why all four functions will have a worst-case time of O(logn).

Please note that "heap sort" is not a data structure.