Zipf's Law

Origins and word frequencies

The French stenographer Estoup (1916) had already noticed regularities in word frequencies, but it was George Kingsley Zipf of Harvard who systematized them. Zipf (1935, 1949) showed that in English texts the most frequent word occurs about twice as often as the second and three times as often as the third, and tried to explain this through a principle of least effort shared by speakers and listeners. Mandelbrot (1953) derived the distribution from an information-theoretic model. Piantadosi (2014) concluded that the law is observed widely across languages but that there is still no consensus about its cause.

City sizes and economic explanations

The German physicist Auerbach (1913) first noted that city populations tend to be inversely proportional to rank, a pattern known in urban economics as the rank–size rule. Herbert Simon (1955) proposed a stochastic process in which new population attaches to cities in proportion to their existing size, generating such skewed distributions. Krugman (1996) called the regularity of the urban hierarchy a mystery, and Gabaix (1999) showed mathematically that Zipf's law emerges naturally if cities follow Gibrat's law, meaning that their growth rates have the same distribution regardless of size. Eeckhout (2004), by contrast, argued that when all cities, including the smallest, are included, the distribution is closer to lognormal.

Firm sizes and research on power laws

Axtell (2001) reported that the size distribution of US firms closely follows Zipf's law, and Gabaix (2009) reviewed the power laws that appear in economic and financial phenomena such as income, wealth and stock price movements. Newman (2005) explained the close mathematical link between Zipf's law and the Pareto distribution, along with the statistical care needed to establish a power law in data. Although it is a simple empirical rule, Zipf's law is treated as a leading example of how mechanisms such as random proportional growth, preferential attachment and path dependence can produce macroscopic order.

FAQ

What is Zipf's law in simple terms?

If you rank things by size, the first is about twice as large as the second and three times as large as the third, so size falls in inverse proportion to rank.

Why does Zipf's law occur?

There is no single accepted answer. Proposed explanations include random growth proportional to size, minimization of effort and information-theoretic optimization.

Sources reviewed: 2026-10-26

References

1. Zipf, G. K. (1935). The Psycho-Biology of Language. Boston: Houghton Mifflin.
2. Zipf, G. K. (1949). Human Behavior and the Principle of Least Effort. Cambridge, MA: Addison-Wesley.
3. Auerbach, F. (1913). Das Gesetz der Bevölkerungskonzentration. Petermanns Geographische Mitteilungen, 59, 74–76.
4. Estoup, J.-B. (1916). Gammes sténographiques. Paris: Institut Sténographique de France.
5. Simon, H. A. (1955). On a class of skew distribution functions. Biometrika, 42(3–4), 425–440.
6. Mandelbrot, B. (1953). An informational theory of the statistical structure of language. In W. Jackson (Ed.), Communication Theory (pp. 486–502). London: Butterworths.
7. Krugman, P. (1996). Confronting the mystery of urban hierarchy. Journal of the Japanese and International Economies, 10(4), 399–418.
8. Gabaix, X. (1999). Zipf's law for cities: An explanation. Quarterly Journal of Economics, 114(3), 739–767.
9. Axtell, R. L. (2001). Zipf distribution of U.S. firm sizes. Science, 293(5536), 1818–1820.
10. Eeckhout, J. (2004). Gibrat's law for (all) cities. American Economic Review, 94(5), 1429–1451.
11. Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46(5), 323–351.
12. Gabaix, X. (2009). Power laws in economics and finance. Annual Review of Economics, 1, 255–293.
13. Piantadosi, S. T. (2014). Zipf's word frequency law in natural language: A critical review and future directions. Psychonomic Bulletin & Review, 21(5), 1112–1130.