On Rounding Percentages

We assess the probability that a table of rounded percentages adds to 100 percent. This extends work of Mosteller, Youtz, and Zahn (1967) who found that the chance of rounding to 100 percent was about 3/4 with three categories, 2/3 with four categories, and (6/πc)1/2 with a large number c of categories. We give a mathematical treatment of this phenomenon when the table is drawn from a multinomial distribution or from a mixture of multinomial distributions. We discuss the very different small-sample behavior and treat Benford's leading digit data as an example.