I am not sure that the following will be useful, however...
Similarly to the answer of Mark Sapir let us consider the "anti-commutativity" graph (an edge connects $a$ and $b$ if $ab\ne ba$). Of course, it is not connected, and the number of its connected components can be taken as some measure of non-abelian-ness. E.g., this measure equals 2 (minimal value) for $S_n (n\ge 5)$.