Zipf's Law. Yes, it's real, and yes, it's really cool.

The exact statement of Zipf’s Law involves more formal math than we usually explain here, but it basically states that if you take a specific country, count the number of cities at a certain population level and then count the number of cities with twice that population, the former number will often be close to double the latter.

Although Zipf’s Law holds for most languages, even for non-natural languages like Esperanto,[10] the reason is still not well understood.[11] However, it may be partially explained by the statistical analysis of randomly generated texts. Wentian Li has shown that in a document in which each character has been chosen randomly from a uniform distribution of all letters (plus a space character), the "words" follow the general trend of Zipf's law (appearing approximately linear on log-log plot).[12] Vitold Belevitch in a paper, On the Statistical Laws of Linguistic Distribution offered a mathematical derivation. He took a large class of well-behaved statistical distributions (not only the normal distribution) and expressed them in terms of rank. He then expanded each expression into a Taylor series. In every case Belevitch obtained the remarkable result that a first-order truncation of the series resulted in Zipf's law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot's law.[13][14]



The principle of least effort is another possible explanation: Zipf himself proposed that neither speakers nor hearers using a given language want to work any harder than necessary to reach understanding, and the process that results in approximately equal distribution of effort leads to the observed Zipf distribution.[15][16]



Similarly, preferential attachment (intuitively, "the rich get richer" or "success breeds success") that results in the Yule-Simon distribution has been shown to fit word frequency versus rank in language[17] and population versus city rank[18] better than Zipf's law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon.

This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: {\displaystyle f(k;\rho )} f(k;\rho ) can be used to model, for example, the relative frequency of the {\displaystyle k} kth most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a (typically small) power of {\displaystyle k} k.