Zipf's law

Zipf's law () is an empirical law stating that when a set of measured values is sorted in decreasing order, the value of the n-th entry is often approximately inversely proportional to n. The best-known instance of Zipf's law applies to the frequency distribution of words in a text or corpus of natural language: w o r d f r e q u e n c y ∝ 1 w o r d r a n k . {\displaystyle \ {\mathsf {word\ frequency}}\ \propto \ {\frac {1}{\ {\mathsf {word\ rank}}\ }}~.} It is usually found that the most common word occurs approximately twice as often as the next common one, three times as often as the third most common, and so on. For example, in the Brown Corpus of American English text, the word "the" is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's law, the second-place word "of" accounts for slightly over 3.5% of words (36,411 occurrences), followed by "and" (28,852). It is often used in the following form, called the Zipf-Mandelbrot law: f r e q u e n c y ∝ 1 ( r a n k + b ) a {\displaystyle \ {\mathsf {frequency}}\ \propto \ {\frac {1}{\ \left(\ {\mathsf {rank}}+b\ \right)^{a}\ }}\ } where a {\displaystyle \ a\ } and b {\displaystyle \ b\ } are fitted parameters, with a ≈ 1 {\displaystyle \ a\approx 1} , and b ≈ 2.7 {\displaystyle \ b\approx 2.7~} . This law is named after the American linguist George Kingsley Zipf, and is still an important concept in quantitative linguistics. It has been found to apply to many other types of data studied in the physical and social sciences. In mathematical...