Originally the term ** Zipf's law** meant the observation of Harvard linguist George Kingsley Zipf that the frequency of use of the the

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 *n*th-ranked item is given by the Zeta distribution, 1/(*n*^{s}ζ(*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:

- frequency of accesses to web pages
- in particular the access counts on the Wikipedia most popular page, with
*b*approximately equal to 0.3 - page access counts on Polish Wikipedia (data for late July 2003) approximately obey Zipf's law with
*b*about 0.5

- in particular the access counts on the Wikipedia most popular page, with
- words in the English language
- for instance, in Shakespeare's play Hamlet, with
*b*approximately 0.5, see Shakespeare word frequency lists

- for instance, in Shakespeare's play Hamlet, with
- sizes of settlements
- income distribution amongst individuals
- size of earthquakes

Table of contents |

2 Further reading 3 External links |

- Benford's law,
- Bradford's law,
- harmonic number of order
- law (principle),
- Mathematical economics,
- Pareto distribution,
- Pareto principle,
- power law,
- Zipf-Mandelbrot law

- Zipf, George K.;
*Human Behaviour and the Principle of Least-Effort*, Addison-Wesley, Cambridge MA, 1949 - W. Li, "Random texts exhibit Zipf's-law-like word frequency distribution",
*IEEE Transactions on Information Theory*,**38**(6), pp.1842-1845, 1992.