Zipf's Law

Zipf's Law is a power law function that has found many applications in the physical, biological, and behavioral sciences. Popularized by George Zipf, it is also known as the Rank-Size Rule,the Pareto-Zipf Law, the Pareto-Estoup-Zipf Law and Zipf's Curve.

Zipf worked with language and text. He would take a book and count the number of times each word was used. Then he would rank these words by most common to least common. In analyzing the results he formulated Zipf's Law:

r f = C

where r is the word rank
f is the frequency (or how many times it occurred)
C is a constant that depends on the text being analyzed.

In English text C tends to be about N/10, where N is the number of words in the text.

So, for a text with 200,000 words we would expect to see the most common word about 20,000 times; the second ranked word 10,000 times, the third 6,667 times ... (20,000 / 1, 20,000 / 2, 20,000 / 3 ... ) and the 50th most common word 400 times (20,000 / 50).

A generalization from this can be made: a few things happen a lot, a bunch of things happen fairly often, and a lot of things rarely happen at all.

Zipf's Law, though would seem to apply to only a few phenomena, has actually been discovered to apply to a multitude of events. The first pattern that George Zipf discovered was the English language which the occurrence of words used in common speech follows the distribution pattern. "The" is used approximately twice as often as the second most common word, and three times as much as the third, and so on. However, city sizes in the United States have also met this pattern throughout history, even though the cities in the ranking have changed. New York the largest city, was both four times as large as Los Angles in 1950, when it was the fourth largest city, and four times as large as the fourth city in 1998, Houston. This pattern also holds true for natural occurrences such as earthquakes. The applications of this simplistic relationship seem to be never ending.