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):
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="https://i.sstatic.net/KuJkg.png" width="300" alt="S3Table" />
<br />
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 $C_2$:
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="https://i.sstatic.net/EXCod.png" width="300" alt="S3RedC2" />
<br />
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!