Rank-size plots, Zipf’s law, and scaling

Rank-size plots, also called Zipf plots, have a role to play in representing statistical data. The method is somewhat peculiar, but throws light on one aspect of the notions of concentration. This chapter’s first goals are to define those plots and show that they are of two kinds. Some are simply an analytic restatement of standard tail distributions but other cases stand by themselves. For example, in the context of word frequencies in natural discourse, rank-size plots provide the most natural and most direct way of expressing scaling. Of greatest interest are the rank-size plots that are rectilinear in log-log coordinates. In most cases, this rectilinearity is shown to simply rephrase an underlying scaling distribution, by exchanging its coordinate axes. This rephrasing would hardly seem to deserve attention, but continually proves its attractiveness. Unfortunately, it is all too often misinterpreted and viewed as significant beyond the scaling distribution drawn in the usual axes. These are negative but strong reasons why rank-size plots deserve to be discussed in some detail. They throw fresh light on the meaning and the pitfalls of infinite expectation, and occasionally help understand upper and lower cutoffs to scaling.