Zipf

Zipf's law is an empirical principle describing a relationship between the frequency of items in a dataset and their rank when the items are sorted by frequency. In its simplest form the frequency f of the item with rank r is proportional to 1/r^s, with s close to 1 in many natural languages. When s equals 1, the product r·f(r) remains approximately constant across many ranks. The law was formulated by American linguist George Kingsley Zipf in the 1930s based on studies of word frequencies, and it has since been observed in a variety of domains beyond language.

Zipf's law is most famous for language, where a small set of words account for a large fraction of usage, while there is a long tail of rarely used words. It has also been reported in city-size distributions (the rank-size rule), firm sizes, income distributions, and other phenomena. Several refinements exist, such as the Zipf–Mandelbrot law, which introduces a second parameter to better fit low-rank behavior: f(r) ∝ 1/(r+q)^s. The law is not exact; deviations occur at very high or very low ranks, and empirical tests often yield mixed results. Competing models such as lognormal, Pareto, or other heavy-tailed distributions can describe similar data, and the underlying mechanisms remain debated. Explanations range from cognitive efficiency and communicative constraints to preferential attachment and random growth processes.