Let $G$ be a finite non-abelian group of $n$ elements. I would like a measure that intuitively captures the extent to which $G$ is non-commutative. One easy measure is a count of the non-commutative products. For example, for $S_3$, 9 products are non-commutative, or, 18 of the 36 entries in the multiplication table indicate non-commutivity (in the table, $r$=rotation; $f$=flip):
          
So one might say $S_3$ is 50% non-abelian.

Another idea is to determine the fewest element identifications needed to make the group abelian. If one identifies the elements $r$ and $r^2$ above, and calls the resulting merged element $a$, then I believe $S_3$ is reduced to the abelian $C_2$:
          
So one might say $S_3$ is one element identification away from being abelian.

My question is:

Is there some standard, accepted measure of how far a group is from being abelian?

Ideally such a measure would not be restricted to finite groups. Thanks for pointers!