The accuracy of Monte Carlo Glauber model descriptions of minimum-bias multiplicity frequency distributions is evaluated using data from the Relativistic Heavy Ion Collider (RHIC) within the context of a sensitive, power-law representation introduced previously by Trainor and Prindle (TP). Uncertainties in the Glauber model input and in the midrapidity multiplicity frequency distribution data are reviewed and estimated using the TP centrality methodology. The resulting errors in model-dependent geometrical quantities used to characterize heavy-ion collisions (i.e. impact parameter, number of nucleon participants N_part, number of binary interactions N_bin and average number of binary collisions per incident participant nucleon ν) are presented for minimum-bias Au–Au collisions at = 20, 62, 130 and 200 GeV and Cu–Cu collisions at = 62 and 200 GeV. Considerable improvement in the accuracy of collision geometry quantities is obtained compared to previous Monte Carlo Glauber model studies, confirming the TP conclusions. The present analysis provides a comprehensive list of the sources of uncertainty and the resulting errors in the above geometrical collision quantities as functions of centrality. The capability of energy deposition data from trigger detectors to enable further improvements in the accuracy of collision geometry quantities is also discussed.