For a group $G$, the commutator of two elements is defined as $[a,b]=aba^{-1}b^{-1}$, and is usually said to measure the extent to which the elements $a$ and $b$ fail to commute.

I'm having some trouble making sense of the last bit: I understand that if $a$ and $b$ commute, then $[a,b]=e$. But if $a$ and $b$ don't commute, in what sense is the commutator actually capturing the extent of their failure to commute, since there is no way to talk about how "far" an element $g\in G$ is from the identity?

Am I just interpreting the word "measure" too literally here, or is there actually a way to think about commutators that makes it clear in what sense they compare the way two pairs of elements fail to commute?