I've given this some more thought, and I think that a "key" argument 
would make sense for a general selection function.

The general selection problem is: given a set of items A, and a number k 
between 1 and the number of items, return the k-th item. In Python 
terms, we would use a list, and 0 <= k < len(A) instead.

https://www.cs.rochester.edu/~gildea/csc282/slides/C09-median.pdf

I've had the idea of adding a select(A, k) function to statistics for a 
while now. Then the median_low would be equivalent to select(A, 
len(A)//2) and median_high would be select(A, len(A)//2 + 1). I'd leave 
the median_* functions as they are, and possibly include a key function 
in select.

I don't think it makes sense to add key arguments to mode, mean, 
variance etc. I'm having trouble thinking of what that would even mean 
(no pun intented): it's unlikely that the mean will actually a data 
value (except by accident, or by careful construction of the data). 
Variance has the wrong units (it is the units of your data, squared) and 
the stdev is conceptually a difference between data values, not a data 
value itself, so it doesn't even make sense to apply a key function and 
return one of the data points.

And mode counts objects, so it already applies to non-numeric data. 
It's even documented as applying to nominal data.