How big can the commutator of two matrices be and how big is it typically?

Numerical experiments show that the Frobenius norm of the commutator of two large matrices typically clusters sharply around a certain value, which, moreover, is much smaller than one would predict. The purpose of this paper is to give a rigorous foundation of this phenomenon. We also discuss the question how big the Frobenius norm of commutator of the two matrices can actually be.