Originally the term Zipf's law meant the observation of Harvard linguist George Kingsley Zipf that the frequency of use of the the nth-most-frequently-used word in any natural language is inversely proportional to n.
Mathematically, that is impossible if there are infinitely many words in a language, since (letting c > 0 denote the constant of proportionality that would make the sum of all relative frequencies equal to 1) we have
[A scholarly reference to support this assertion about word frequencies should be added here.]
As long as the exponent 1 + ε exceeds 1, it is possible for such a law to hold with infinitely many words, since if s > 1 then
The term Zipf's law has consequently come to be used to refer to frequency distributions of "rank data" in which the relative frequency of the nth-ranked item is given by the Zeta distribution, 1/(nsζ(s)), where s > 1 is a parameter indexing this family of probability distributions. Indeed, the term Zipf's law sometimes simply means the zeta distribution, since probability distributions are sometimes called "laws".
A more general law proposed by Benoit Mandelbrot has frequencies
Zipf's law is an experimental law, not a theoretical one. The causes of Zipfian distributions in real life are a matter of some controversy. However, Zipfian distributions are commonly observed in many kinds of phenomena.
Zipf's law is often demonstrated by scatterplotting the data, with the axes being log(rank order) and log(frequency). If the points are close to a single straight line, the distribution follows Zipf's law.
Examples of collections approximately obeying Zipf's law:
|Table of contents|
2 Further reading
3 External links