  Let E be an elliptic curve defined over a number field k. In this paper, we
define the ``global discrepancy'' of a finite set Z of algebraic points on E
which in a precise sense measures how far the set is from being adelically
equidistributed. We then prove an upper bound for the global discrepancy of Z
in terms of the average canonical height of points in Z. We deduce from this
inequality a number of consequences. For example, we give a new and simple
proof of the Szpiro-Ullmo-Zhang equidistribution theorem for elliptic curves.
We also prove a non-archimedean version of the Szpiro-Ullmo-Zhang theorem which
takes place on the Berkovich analytic space associated to E. We then prove some
quantitative `non-equidistribution' theorems for totally real or totally p-adic
small points. The results for totally real points imply similar bounds for
points defined over the maximal cyclotomic extension of a totally real field.
