We show that for many families of OPUC, one has $||φ'_n||_2/n -> 1$, a condition we call normal behavior. We prove that this implies $|α_n| -> 0$ and that it holds if the sequence $α_n$ is in $\ell^1$. We also prove it is true for many sparse sequences. On the other hand, it is often destroyed by the insertion of a mass point. ; Comment: 36 pages, no figures. Minor corrections, to appear in the Journal of Approximation Theory