  We discuss the problem of defining an estimate for the error in quasi-Monte
Carlo integration. The key issue is the definition of an ensemble of
quasi-random point sets that, on the one hand, includes a sufficiency of
equivalent point sets, and on the other hand uses information on the degree of
uniformity of the point set actually used, in the form of a discrepancy or
diaphony. A few examples of such discrepancies are given. We derive the
distribution of our error estimate in the limit of large number of points. In
many cases, Gaussian central limits are obtained. We also present numerical
results for the quadratic star-discrepancy for a number of quasi-random
sequences.
