  We study an ordinal rank on the class of Banach spaces with bases that
quantifies the distortion of the norm of a given Banach space. The rank
$AD(\cdot)$, introduced by P. Dodos, uses the transfinite Schreier familes and
has the property that $AD(X) < \omega_1$ if and only if $X$ is arbitrarily
distortable. We prove several properties of this rank as well as some new
results concerning higher order $\ell_1$ spreading models. We also compute this
rank for for several Banach spaces. In particular, it is shown that class of
Banach spaces $\mathfrak{X}^{\omega^\xi}_{0,1}$ , which each admit $\ell_1$ and
$c_0$ spreading models hereditarily, and were introduced by S.A. Argyros, the
first and third author, satisfy $AD(\mathfrak{X}^{\omega^\xi}_{0,1}) =
\omega^\xi + 1$. This answers some questions of Dodos.
