Zipf's law

Zipf's law is the observation made by Harvard linguist George Kingsley Zipf that for many frequency distributions, the n-th largest frequency is proportional to a negative power of the rank order n. A distribution that is observed to obey Zipf's law, is sometimes referred to as Zipfian distribution. The phrase "Zipf's law" is also sometimes used to refer to the corresponding probability distribution, the zeta distribution. This probability distribution is an instance of a power law.

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.

For example, if f₁ is the frequency (in percent) of the most common English word, f₂ is the frequency of the second most common English word and so on, then there exist two positive numbers a and b such that for all n ≥ 1:

Note that the frequencies f_n have to add up to 100%, so if this relationship were strictly true for all n ≥ 1, and we had infinitely many words, then b would have to be greater than one and a would have to be equal to ζ(b), i.e., the value of the Riemann zeta function at b.

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

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.

See also: power law, Pareto distribution, Pareto principle, Benford's law, Mathematical economics, Bradford's law, law (principle), harmonic number of order

Further reading

• 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.

External links

• Comprehensive bibliography of Zipf's law
• Zipf, Power-laws, and Pareto - a ranking tutorial
• Seeing Around Corners (Artificial societies turn up Zipf's law)
• PlanetMath article on Zipf's law