Pratt’s measure for some bibliometric distributions and its relation with the 80/20 - rule

Abstract

Pratt’s measure C on the class concentration of distributions is calculated and interpreted for the laws of Zipf, Mandelbrot, and Lotka, and for the geometric distribution. Comparisons between each are made. We show that phenomena agreeing with Zipf’s law are more concentrated than phenomena agreeing with Mandelbrot’s law. On the other hand, data following Lotka’s law are more concentrated than data following Zipf’s law. We also find that the geometric distribution is more concentrated than the Lotka distribution only for high values of the maximal production a source can have. An explicit mathematical formula (in case of the law of Lotka) between C and x(e), the fraction of the sources needed to obtain a fraction 8 of the items produced by these sources (see my earlier article on the 80/20 rule), is derived and tested, unifying these two theories on class concentration. So far, C and x(e) appeared separate in the literature.