  We define a class of insulators with gapless surface states protected from
localization due to the statistical properties of a disordered ensemble, namely
due to the ensemble's invariance under a certain symmetry. We show that these
insulators are topological, and are protected by a $\mathbb{Z}_2$ invariant.
Finally, we prove that every topological insulator gives rise to an infinite
number of classes of statistical topological insulators in higher dimensions.
Our conclusions are confirmed by numerical simulations.
