In this paper, we introduce a geometric statistic called the
sprawl of a group
with respect to a generating set, based on the average distance in the word
metric between pairs of words of equal length. The sprawl quantifies a certain
obstruction to hyperbolicity. Group presentations with maximum sprawl (ie
without this obstruction) are called
statistically hyperbolic. We first relate
sprawl to curvature and show that nonelementary hyperbolic groups are
statistically hyperbolic, then give some results for products and for certain solvable
groups. In free abelian groups, the word metrics are asymptotic to norms
induced by convex polytopes, causing several kinds of group invariants to
reduce to problems in convex geometry. We present some calculations and
conjectures concerning the values taken by the sprawl statistic for the group
as the generators vary,
by studying the space
with various norms.