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We argue that recent observations of afterglows and some theoreticalconsiderations indicate that the gamma-ray fluence may be a better measure ofstrength of a burst than its peak flux. We discuss how the distribution of thefluence, or any physical quantity related to the peak counts (or flux), can beobtained with proper correction for threshhold effects. Using this method, wecompare the cumulative and differential distributions of flux and fluence. Wefind some differences between these distributions, but more remarkable are thesimilarities between the distributions. The other striking feature is howdifferent these distributions are from those of other extragalactic objects.Using the fluence distributions, we derive the expected comoving density rateevolution for different assumed total luminosity; we compare this with the GRBrate expected from the recently determined star formation rate.Comment: 5 pages, 4 postscript figures. To appear in the proceedings of the  4th Huntsville Gamma Ray Burst Symposium, C.A.Meegan, P.Cushman, ed

Distribution of fluences versus peak fluxes and the cosmological density evolution of the rate of GRBs