  The moduli space of lattices of $\mathbb{C}$ is a Riemann surface of finite
hyperbolic area with the square lattice as an origin. We select a lattice from
the induced uniform distribution and calculate the statistics of the
Teichm\"uller distance to the origin. This in turn identifies distribution of
the distance in Teichm\"uller space to the central "square" punctured torus in
the moduli space of punctured tori. There are singularities in this p.d.f.
arising from the topology of the moduli space. We also consider the statistics
of the distance in Teichm\"uller space to the rectangular punctured tori and
the p.d.f and expected distortion of the extremal quasiconformal mappings.
