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Zipf's Law

Zipf’s Law is a statistical distribution in certain data sets, such as words in a linguistic corpus, in which the frequencies of certain words are inversely proportional to their ranks. Named for linguist George Kingsley Zipf, who around 1935 was the first to draw attention to this phenomenon, the law examines the frequency of words in natural language and how the most common word occurs twice as often as the second most frequent word, three times as often as the subsequent word and so on until the least frequent word. The word in the position n appears 1/n times as often as the most frequent one.

When words are ranked according to their frequencies in a large enough collection of texts and then the frequency is plotted against the rank, the result is a logarithmic curve. (Or if you graph on a log scale, the result is a straight line.)

The most common word in English is “the,” which appears about one-tenth of the time in a typical text; the next most common word (rank 2) is of,” which appears about one-twentieth of the time. In this type of distribution, frequency declines sharply as the rank number increases, so a small number of items appear very often, and a large number rarely occur.

A Zipfian distribution of words is universal in natural language: It can be found in the speech of children less than 32 months old as well as in the specialized vocabulary of university textbooks. Studies show that this phenomenon also applies in nearly every language.

Individually, neither syntax nor semantics is sufficient to induce a Zipfian distribution on its own. However, syntax and semantics work together for a Zipfian distribution.

Only recently has Zipf’s Law been tested rigorously on databases large enough to ensure statistical validity. Researchers at the Centre de Recerca Matematica, part of the Government of Catalonia's CERCA network, who are attached to the Universitat Autonoma de Barcelona Department of Mathematics, analyzed the full collection of English-language texts in the Project Gutenberg, a free database with more than 30,000 works. When the rarest words were left out, Zipf’s Law applied to more than half of the words.

The law can be applied to fields other than literature. Zipfian distributions have been found in the population ranks of cities in various countries, corporation sizes, income rankings and ranks of the number of people watching the same TV channel.

This was last updated in January 2018

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