There are many kinds of averages, like the arithmetic-geometric mean and the logarithmic mean, which are defined for pairs of (positive) real numbers rather than on arbitrary multisets of (positive) reals. Is there a canonical process for converting these to multiset functions?
I'm not sure what is the best definition to use here for an average. At a minimum it would require that the function is symmetric in all its arguments, homogeneous in the sense that multiplying all elements by $k$ should increase the average by a factor of $k$, increasing in all of its arguments, and yields a value between the minimum and the maximum of its arguments. I also expect* that if the average of $A$ equals the average of $B$ then they also equal the average of $A\uplus B$ (where $A\uplus B$ is the union of $A$ and $B$ with multiplicity, usually called the multiset sum I think). I'm open to a narrower/less-inclusive definition if it makes the answer work.
* In general it's hard to ensure that the whole is the sum of the parts, so to speak; compare the consistency criterion in social choice theory. If you have a different idea of how to express that the new method is 'essentially' the same as the old, go ahead and use it.