The Distribution of Sums of Rounded Percentages

Summary

When percentages are computed for counts in several categories or for several positive measurements each taken as a fraction of their sum, the rounded percentages often fail to add to 100 percent. We investigate how frequently this failure occurs and what the distributions of sums of rounded percentages are for (1) an empirical set of data, (2) the multinomial distribution in small samples, (3) spacings between points dropped on an interval—the broken-stick model—; and (4) for simulation for several categories. The several methods produce similar distributions.

We find that the probability that the sum of rounded percentages adds to exactly 100 percent is certain for two categories, about three-fourths for three categories, about two-thirds for four categories, and about $\sqrt {6/c\pi }$ for larger numbers of categories, c, on the average when categories are not improbable.

This work was facilitated by grants from the National Science Foundation.