Zipf's law

Originally the term Zipf's law meant the observation of Harvard linguist George Kingsley Zipf (SAMPA: [zIf]) that the frequency of use of the nth-most-frequently-used word in any natural language is approximately inversely proportional to n.

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.

Table of contents

1 Theoretical issues
2 Related laws
3 Examples of collections approximately obeying Zipf's law:
4 See also
5 Further reading
6 External links

Theoretical issues

Mathematically, it is impossible for Zipf's law to hold exactly if there are infinitely many words in a language, since for any constant of proportionality c > 0, the sum of all relative frequencies is proportional to the harmonic series and must be

Empirical studies have found that in English, the frequencies of the approximately 1000 most-frequently-used words are approximately proportional to 1/n^s where s is just slightly more than one.

As long as the exponent s exceeds 1, it is possible for such a law to hold with infinitely many words, since if s > 1 then

The value of this sum is ζ(s), where ζ is Riemann's zeta function.

Related laws

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/(n^sζ(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". This distribution is sometimes called the Zipfian distribution or Yule distribution.

A more general law proposed by Benoit Mandelbrot has frequencies

This is the Zipf-Mandelbrot law. The "constant" in this case is the reciprocal of the Hurwitz zeta function evaluated at s.

Examples of collections approximately obeying Zipf's law:

frequency of accesses to web pages

in particular the access counts on the Wikipedia page, with s approximately equal to 0.3
page access counts on Polish Wikipedia (data for late July 2003) approximately obey Zipf's law with s about 0.5

words in the English language
- for instance, in Shakespeare's play Hamlet, with s approximately 0.5, see Shakespeare word frequency lists
sizes of settlements
income distribution amongst individuals
size of earthquakes
notes in musical performances

It has been pointed out (see external link below) that Zipfian distributions can also be regarded as being Pareto distributions with an exchange of variables.

External links

Zipfin laki