  Even if quark and lepton masses are not uniquely predicted by the fundamental
theory, as may be the case in the string theory landscape, nevertheless their
pattern may reveal features of the underlying theory. We use statistical
techniques to show that the observed masses appear to be representative of a
scale invariant distribution, rho(m) ~ 1/m. If we extend this distribution to
include all the Yukawa couplings, we show that the resulting CKM matrix
elements typically show a hierarchical pattern similar to observations. The
Jarlskog invariant measuring the amount of CP violation is also well reproduced
in magnitude. We also apply this framework to neutrinos using the seesaw
mechanism. The neutrino results are ambiguous, with the observed pattern being
statistically allowed even though the framework does not provide a natural
explanation for the observed two large mixing angles. Our framework highly
favors a normal hierarchy of neutrino masses. We also are able to make
statistical predictions in the neutrino sector when we specialize to situations
consistent with the known mass differences and two large mixing angles. Within
our framework, we show that with 95% confidence the presently unmeasured MNS
mixing angle sin theta_{13} is larger than 0.04 and typically of order 0.1. The
leptonic Jarlskog invariant is found to be typically of order 10^{-2} and the
magnitude of the effective Majorana mass m_{ee} is typically of order 0.001 eV.
