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{{short description|Probability distribution}}
[[File:Zipf's law on War and Peace.png|thumb|Zipf's Law on War and Peace.<ref name="piant2014" /> The lower plot shows the remainder when the Zipf law is divided away. It shows that there remains significant pattern not fitted by Zipf law.]]<!-- EDITORS! Please see Wikipedia:WikiProject Probability123#Standards for a discussion of standards used for probability distribution articles such as this one.-->

[[File:Zipf-engl-0 English - Culpeper herbal and War of the Worlds.svg|thumbnail|A plot of the frequency of each word as a function of its frequency rank for two English language texts: Culpeper's ''Complete Herbal'' (1652) and H. G. Wells's ''The War of the Worlds'' (1898) in a log-log scale.  The dotted line is the ideal law ''y'' [[Proportionality (mathematics)|∝]] 1/''x''.]]

'''Zipf's law''' ({{IPAc-en|z|ɪ|f}}, {{IPA-de|ts͡ɪpf|lang}}) is an empirical law that often holds, approximately, when a list of measured values is sorted in decreasing order.  It states that the value of the ''n''th entry is inversely proportional to ''n''.

The best known instance of Zipf's law applies to the frequency table of words in a text or [[Text corpus|corpus]] of [[Philosophy:Natural language|natural language]]:<math display="block">\text{word frequency} \propto \frac{1}{\text{word rank}}.</math> 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 [[Social:Brown Corpus|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).<ref name=fagan2010/> It is often used in the following form, called [[Zipf–Mandelbrot law|Zipf-Mandelbrot law]]:<math display="block">\text{frequency} \propto \frac{1}{(\text{rank} + b)^a}</math>where <math>a, b</math> are fitted parameters, with <math>a\approx 1</math>, and <math>b \approx 2.7</math>.<ref name="piant2014" />

This law is named after the American [[Social:Linguistics|linguist]] [[Biography:George Kingsley Zipf|George Kingsley Zipf]],<ref name=Powers1998/><ref name=zipf1935/><ref name=zipf1949/> and is still an important concept in [[Quantitative linguistics|quantitative linguistics]]. It has been found to apply to many other types of data studied in the physical and [[Social:Social science|social]] sciences.

In [[Mathematical statistics|mathematical statistics]], the concept has been formalized as the '''Zipfian distribution''': a family of related discrete [[Probability distribution|probability distribution]]s whose rank-frequency distribution is an inverse [[Power law|power law]] relation. They are related to [[Benford's law]] and the [[Pareto distribution]].

Some sets of time-dependent empirical data deviate somewhat from Zipf's law.  Such empirical distributions are said to be '''quasi-Zipfian'''.

==History==

In 1913, the German physicist [[Biography:Felix Auerbach|Felix Auerbach]] observed an inverse proportionality between the population sizes of cities, and their ranks when sorted by decreasing order of that variable.<ref name="Auerbach1913"/>

Zipf's law has been discovered before Zipf,{{efn|as Zipf acknowledged<ref name="zipf1949" />{{rp|546}}}} by the French stenographer Jean-Baptiste Estoup' ''Gammes Stenographiques'' (4th ed) in 1916,<ref name="mann1999" /> with G. Dewey in 1923,<ref>Dewey, Godfrey. ''Relativ frequency of English speech sounds''. Harvard University Press, 1923.</ref> and with E. Condon in 1928.<ref>Condon, EDWARD U. "Statistics of vocabulary." ''Science'' 67.1733 (1928): 300-300.</ref>

The same relation for frequencies of words in natural language texts was observed by George Zipf in 1932,<ref name=zipf1935/> but he never claimed to have originated it. In fact, Zipf did not like mathematics. In his 1932 publication,<ref name=zipf1932/> the author speaks with disdain about mathematical involvement in linguistics, a. o. ibidem, p.&nbsp;21: ''(…) let me say here for the sake of any mathematician who may plan to formulate the ensuing data more exactly, the ability of the highly intense positive to become the highly intense negative, in my opinion, introduces the devil into the formula in the form of √(-i)''. The only mathematical expression Zipf used looks like ''a''.''b''<sup>2</sup> = constant, which he "borrowed" from [[Biography:Alfred J. Lotka|Alfred J. Lotka]]'s 1926 publication.<ref name=king1942/>

The same relationship was found to occur in many other contexts, and for other variables besides frequency.<ref name=piant2014/> For example, when corporations are ranked by decreasing size, their sizes are found to be inversely proportional to the rank.<ref name=axte2001/>  The same relation is found for personal incomes (where it is called [[Pareto distribution#Occurrence and applications|Pareto principle]]<ref>{{Citation |last=Sandmo |first=Agnar |title=Chapter 1 - The Principal Problem in Political Economy: Income Distribution in the History of Economic Thought |date=2015-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780444594280000023 |volume=2 |pages=3–65 |editor-last=Atkinson |editor-first=Anthony B. |access-date=2023-07-11 |series=Handbook of Income Distribution |publisher=Elsevier |doi=10.1016/B978-0-444-59428-0.00002-3 |language=en |editor2-last=Bourguignon |editor2-first=François |archive-date=2023-10-29 |archive-url=https://web.archive.org/web/20231029222127/https://www.sciencedirect.com/science/article/abs/pii/B9780444594280000023 |url-status=live }}</ref>), number of people watching the same TV channel,<ref name=erik2014/> notes in music,<ref name=zann2004/> cells transcriptomes<ref name=lazz2021/><ref name=chen2011/> and more.

In 1992 bioinformatician [[Biography:Wentian Li|Wentian Li]] published a short paper<ref>{{Cite journal |last=Li |first=Wentian |date=1992 |title=Random texts exhibit Zipf's-law-like word frequency distribution |url=https://ieeexplore.ieee.org/document/165464 |journal=EEE Transactions on Information Theory |volume=38 |issue=6 |pages=1842-1845 |via=IEEE Xplore}}</ref> showing that Zipf's law emerges even in randomly generated texts. It included proof that the power law form of Zipf's law was a byproduct of ordering words by rank.

==Formal definition==
{{Probability distribution|
  name       =Zipf's law|
  type       =mass|
  pdf_image  =[[Image:Zipf distribution PMF.png|325px|Plot of the Zipf PMF for ''N'' = 10]]<br /><small>Zipf PMF for ''N'' = 10 on a log–log scale. The horizontal axis is the index ''k''&nbsp;. (The function is only defined at integer values of ''k''. The connecting lines do not indicate continuity.)</small>|
  cdf_image  =[[Image:Zipf distribution CMF.png|325px|Plot of the Zipf CDF for N=10]]<br /><small>Zipf CDF for ''N'' = 10. The horizontal axis is the index ''k''&nbsp;. (The function is only defined at integer values of ''k''. The connecting lines do not indicate continuity.)</small>|
  parameters =<math>s \geq 0\,</math> ([[Real number|real]])<br /><math>N \in \{1,2,3\ldots\}</math> ([[Integer|integer]])|
  support    =<math>k \in \{1,2,\ldots,N\}</math>|
  pdf        =<math>\frac{1/k^s}{H_{N,s}}</math> where ''H<sub>N,s</sub>'' is the ''N''th generalized [[Harmonic number|harmonic number]]|
  cdf        =<math>\frac{H_{k,s}}{H_{N,s}}</math>|
  mean       =<math>\frac{H_{N,s-1}}{H_{N,s}}</math>|
  median     =|
  mode       =<math>1\,</math>|
  variance   =<math>\frac{H_{N,s-2}}{H_{N,s}}-\frac{H^2_{N,s-1}}{H^2_{N,s}}</math>|
  skewness   =|
  kurtosis   =|
  entropy    =<math>\frac{s}{H_{N,s}}\sum\limits_{k=1}^N\frac{\ln(k)}{k^s}
+\ln(H_{N,s})</math>|
  mgf        =<math>\frac{1}{H_{N,s}}\sum\limits_{n=1}^N \frac{e^{nt}}{n^s}</math>|
  char       =<math>\frac{1}{H_{N,s}}\sum\limits_{n=1}^N \frac{e^{int}}{n^s}</math>
}}

Formally, the Zipf distribution on {{mvar|N}} elements assigns to the element of rank {{mvar|k}} (counting from 1) the probability

:<math> f(k;N) = \frac{1}{H_N}\,\frac{1}{k}</math>

where {{mvar|H}}<sub>{{mvar|N}}</sub> is a normalization constant, the {{mvar|N}}th [[Harmonic number|harmonic number]]:

:<math> H_N = \sum_{k=1}^N \frac{1}{k}\ .</math>

The distribution is sometimes generalized to an inverse power law with exponent {{mvar|s}} instead of 1.<ref name=adam2000/>  Namely,

:<math>f(k;N,s) = \frac{1}{H_{N,s}}\,\frac{1}{k^s}</math>

where {{mvar|H}}<sub>{{mvar|N}},{{mvar|s}}</sub> is a generalized harmonic number

:<math> H_{N,s} = \sum_{k=1}^N \frac{1}{k^s}\ .</math>

The generalized Zipf distribution can be extended to infinitely many items ({{mvar|N}} = ∞) only if the exponent {{mvar|s}}  exceeds 1.  In that case, the normalization constant {{mvar|H}}<sub>{{mvar|N}},{{mvar|s}}</sub> becomes [[Riemann zeta function|Riemann's zeta function]],
:<math>\zeta (s) = \sum_{k=1}^\infty \frac{1}{k^s} < \infty\ .</math>
If the exponent {{mvar|s}} is 1 or less, the normalization constant {{mvar|H}}<sub>{{mvar|N}},{{mvar|s}}</sub> diverges as {{mvar|N}} tends to infinity.

==Empirical testing==

Empirically, a data set can be tested to see whether Zipf's law applies by checking the [[Goodness of fit|goodness of fit]] of an empirical distribution to the hypothesized power law distribution with a [[Kolmogorov–Smirnov test]], and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential distribution or lognormal distribution.<ref name="Clausetetal2009">Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-Law Distributions in Empirical Data. SIAM Review, 51(4), 661–703. {{doi|10.1137/070710111}}</ref>

Zipf's law can be visuallized by [[Graph of a function|plotting]] the item frequency data on a log-log graph, with the axes being the [[Logarithm|logarithm]] of rank order, and logarithm of frequency. The data conform to Zipf's law with exponent {{mvar|s}} to the extent that the plot approximates a [[Linear equation|linear]] (more precisely, affine) function with slope −{{mvar|s}}.  For exponent {{mvar|s}} = 1, one can also plot the reciprocal of the frequency (mean interword interval) against rank, or the reciprocal of rank against frequency, and compare the result with the line through the origin with slope 1.<ref name=Powers1998/>

==Statistical explanations==

Although Zipf's Law holds for most natural languages, even some non-natural ones like [[Social:Esperanto|Esperanto]],<ref name=mana2006/> the reason is still not well understood.<ref name=bril1959/> Recent reviews of generative processes for Zipf's law include.<ref>{{Cite journal |last=Mitzenmacher |first=Michael |date=January 2004 |title=A Brief History of Generative Models for Power Law and Lognormal Distributions |url=http://www.internetmathematicsjournal.com/article/1385 |journal=Internet Mathematics |language=en |volume=1 |issue=2 |pages=226–251 |doi=10.1080/15427951.2004.10129088 |s2cid=1671059 |issn=1542-7951 |access-date=2023-07-25 |archive-date=2023-07-22 |archive-url=https://web.archive.org/web/20230722052227/https://www.internetmathematicsjournal.com/article/1385 |url-status=live |doi-access=free }}</ref><ref>{{Cite journal |last1=Simkin |first1=M. V. |last2=Roychowdhury |first2=V. P. |date=2011-05-01 |title=Re-inventing Willis |url=https://www.sciencedirect.com/science/article/pii/S0370157310003339 |journal=Physics Reports |language=en |volume=502 |issue=1 |pages=1–35 |doi=10.1016/j.physrep.2010.12.004 |issn=0370-1573 |arxiv=physics/0601192 |bibcode=2011PhR...502....1S |s2cid=88517297 |access-date=2023-07-25 |archive-date=2012-01-29 |archive-url=https://web.archive.org/web/20120129100824/http://www.sciencedirect.com/science/article/pii/S0370157310003339 |url-status=live }}</ref>

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" with different lengths follow the macro-trend of the Zipf's law (the more probable words are the shortest with equal probability).<ref name="liwe1992" /> In 1959, Vitold Belevitch observed that if any of a large class of well-behaved statistical distributions (not only the [[Normal distribution|normal distribution]]) is expressed in terms of rank and expanded into a [[Taylor series]], the first-order truncation of the series results in Zipf's law. Further, a second-order truncation of the Taylor series resulted in [[Zipf–Mandelbrot law|Mandelbrot's law]].<ref name="bele1959" /><ref name="neum2011" />

The [[Social:Principle of least effort|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.<ref name="zipf1949" /><ref name="ferr2003" />

A minimal explanation assumes that words are generated by [[Infinite monkey theorem|monkeys typing randomly]]. If language is generated by a single monkey typing randomly, with fixed and nonzero probability of hitting each letter key or white space, then the words (letter strings separated by white spaces) produced by the monkey follows Zipf's law.<ref>{{Cite journal |last1=Conrad |first1=B. |last2=Mitzenmacher |first2=M. |date=July 2004 |title=Power laws for monkeys typing randomly: the case of unequal probabilities |url=https://ieeexplore.ieee.org/document/1306541 |journal=IEEE Transactions on Information Theory |volume=50 |issue=7 |pages=1403–1414 |doi=10.1109/TIT.2004.830752 |s2cid=8913575 |issn=1557-9654 |access-date=2023-08-20 |archive-date=2022-10-17 |archive-url=https://web.archive.org/web/20221017103738/http://ieeexplore.ieee.org/document/1306541/ |url-status=live }}</ref>

Another possible cause for the Zipf distribution is a [[Preferential attachment|preferential attachment]] process, in which the value {{mvar|x}} of an item tends to grow at a rate proportional to {{mvar|x}} (intuitively, "the rich get richer" or "success breeds success").  Such a growth process results in the [[Yule–Simon distribution]], which has been shown to fit word frequency versus rank in language<ref name=linr2014/> and population versus city rank<ref name=vita2015/> better than Zipf's law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon.

A similar explanation is based on atlas models, systems of exchangeable positive-valued [[Diffusion process|diffusion process]]es with drift and variance parameters that depend only on the rank of the process.  It has been shown mathematically that Zipf's law holds for Atlas models that satisfy certain natural regularity conditions.<ref name=fern2020/><ref name=taot2012/> Quasi-Zipfian distributions can result from quasi-Atlas models.{{citation needed|date=May 2023}}

==Related laws==

A generalization of Zipf's law is the [[Zipf–Mandelbrot law]], proposed by [[Biography:Benoit Mandelbrot|Benoit Mandelbrot]], whose frequencies are:

:<math>f(k;N,q,s)=\frac{1}{C}\,\frac{1}{(k+q)^s}.\,</math>{{Clarify|date=September 2023|reason=denominator missing from equation}}

The constant {{mvar|C}} is the [[Hurwitz zeta function]] evaluated at ''s''.

Zipfian distributions can be obtained from [[Pareto distribution]]s by an exchange of variables.<ref name=adam2000/>

The Zipf distribution is sometimes called the '''discrete Pareto distribution'''<ref name=john1992/> because it is analogous to the continuous [[Pareto distribution]] in the same way that the discrete uniform distribution is analogous to the [[Uniform distribution (continuous)|continuous uniform distribution]].

The tail frequencies of the [[Yule–Simon distribution]] are approximately

:<math>f(k;\rho) \approx \frac{[\text{constant}]}{k^{\rho+1}}</math>

for any choice of ''ρ'' > 0.

In the [[Parabolic fractal distribution|parabolic fractal distribution]], the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.<ref name="Galien"/> Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.<ref name=efte2006/>

It has been argued that [[Benford's law]] is a special bounded case of Zipf's law,<ref name="Galien"/> with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena.<ref name=piet2001/> The ratios of probabilities in Benford's law are not constant. The leading digits of data satisfying Zipf's law with {{mvar|s}} = 1 satisfy Benford's law.
{| class="wikitable" style="text-align: center;"
|-
!<math>n</math>
!Benford's law: <math>P(n) = </math><br/><math>\log_{10}(n+1)-\log_{10}(n)</math>
!<math>\frac{\log(P(n)/P(n-1))}{\log(n/(n-1))}</math>
|-
| 1
| 0.30103000
|  
|-
| 2
| 0.17609126
| −0.7735840
|-
| 3
| 0.12493874
| −0.8463832
|-
| 4
| 0.09691001
| −0.8830605
|-
| 5
| 0.07918125
| −0.9054412
|-
| 6
| 0.06694679
| −0.9205788
|-
| 7
| 0.05799195
| −0.9315169
|-
| 8
| 0.05115252
| −0.9397966
|-
| 9
| 0.04575749
| −0.9462848
|}

==Occurrences==

===City sizes===

Following Auerbach's 1913 observation, there has been substantial examination of Zipf's law for city sizes.<ref name=gaba1999/> However, more recent empirical<ref name=arsh2018/><ref name=ganl2006/> and theoretical<ref name=verb2020/> studies have challenged the relevance of Zipf's law for cities.

===Word frequencies in natural languages===

[[File:Zipf 30wiki en labels.png|thumbnail|Zipf's law plot for the first 10 million words in 30 Wikipedias (as of October 2015) in a log-log scale.]]

In many texts in human languages, word frequencies approximately follow a Zipf distribution with exponent {{mvar|s}} close to 1: that is, the most common word occurs about ''n'' times the ''n''th most common one.

The actual rank-frequency plot of a natural language text deviates in some extent from the ideal Zipf distribution, especially at the two ends of the range.  The deviations may depend on the language, on the topic of the text, on the author, on whether the text was translated from another language, and on the spelling rules used.{{citation needed|date=May 2023}}  Some deviation is inevitable because of [[Sampling error|sampling error]].

At the low-frequency end, where the rank approaches {{mvar|N}}, the plot takes a staircase shape, because each word can occur only an integer number of times.

<gallery widths=280px heights=300px caption="Zipf's law plots for several languages">
Zipf-euro-4 German, Russian, French, Italian, Medieval English.svg|Texts in [[Social:German language|German]] (1669),  [[Social:Russian language|Russian]] (1972), [[Social:French language|French]] (1865), [[Social:Italian language|Italian]] (1840), and Medieval English (1460).
Zipf-euro-3 Spanish (Don Quixote) and Portuguese (Dom Casmurro).svg|Cervantes' Don Quixote Part I ([[Social:Spanish language|Spanish]], 1605) and Assis's Dom Casmurro ([[Social:Portuguese language|Portuguese]], 1899).
File:Zipf-semi-1 Arabic, Geez, Hebraic.svg|Ge'ez (14th century), [[Social:Arabic language|Arabic]] (~650), [[Social:Hebrew language|Hebrew]] (500-800), all with vowels.
File:Zipf-asia-1 Chinese, Tibetan, Vietnamese.svg|[[Social:Lhasa Tibetan|Lhasa Tibetan]], [[Social:Chinese language|Chinese]], [[Social:Vietnamese language|Vietnamese]], all with separated syllables.
Zipf-euro-2 Biblical texts - Latin Vulgate OT, Greek NT, Russian OT.svg|Biblical texts: Pentateuch from the Latin [[Religion:Vulgate|Vulgate]] and Russian Synodal Bible, the four [[Religion:Gospel|Gospel]]s from the Byzantine Greek [[Religion:Byzantine text-type|Majority version]] 
File:Zipf-span-1 Spanish - Don Quixote Parts 1 and 2.svg|Cervantes's Don Quixote, Part I (1605) and Part II (1615).
File:Zipf-heot-0 Hebrew - Books of the Torah.svg|First five books of the [[Religion:Old Testament|Old Testament]] (the Torah) in Hebrew, with vowels.
File:Zipf-laot-0 Vulgate Pentateuch books.svg|First five books of the [[Religion:Old Testament|Old Testament]] (the Pentateuch) in the Latin [[Religion:Vulgate|Vulgate]] version.
File:Zipf-lant-0 Vulgate Gospels.svg|First four books of the [[Religion:New Testament|New Testament]] (the Gospels) in the Latin [[Religion:Vulgate|Vulgate]] version.
</gallery>

[[Image:Wikipedia-n-zipf.png|thumb|A log-log plot of word frequency in Wikipedia (November 27, 2006). 'Most popular words are "the", "of" and "and", as expected. Zipf's law corresponds to the middle linear portion of the curve, roughly following the green (1/''x'')&nbsp; line, while the early part is closer to the magenta (1/''x''<sup>0.5</sup>) line while the later part is closer to the cyan (1/(''k''&nbsp;+&nbsp;''x'')<sup>2.0</sup>) line. These lines correspond to three distinct parameterizations of the Zipf–Mandelbrot distribution, overall a broken power law with three segments: a head, middle, and tail.]]

In some Romance languages, the frequencies of the dozen or so most frequent words deviate significantly from the ideal Zipf distribution, because of those words include articles inflected for grammatical gender and number.{{citation needed|date=May 2023}}

In many [[Place:East Asia|East Asia]]n languages, such as [[Social:Chinese language|Chinese]], [[Social:Lhasa Tibetan|Lhasa Tibetan]], and [[Social:Vietnamese language|Vietnamese]], each "word" consists of a single syllable; a word of English being often translated to a compound of two such syllables. The rank-frequency table for those "words" deviates significantly from the ideal Zipf law, at both ends of the range.{{citation needed|date=May 2023}}

Even in English, the deviations from the ideal Zipf's law become more apparent as one examines large collections of texts. Analysis of a corpus of 30,000 English texts showed that only about 15% of the texts in it have a good fit to Zipf's law.  Slight changes in the definition of Zipf's law can increase this percentage up to close to 50%.<ref name=more2016/>

In these cases, the observed frequency-rank relation can be modeled more accurately as by separate Zipf–Mandelbrot laws distributions for different subsets or subtypes of words. This is the case for the frequency-rank plot of the first 10 million words of the English Wikipedia. In particular, the frequencies of the closed class of function words in English is better described with ''s'' lower than 1, while open-ended vocabulary growth with document size and corpus size require ''s'' greater than 1 for convergence of the [[Harmonic series (mathematics)|Generalized Harmonic Series]].<ref name=Powers1998/>

[[File:Zipf-code-1 English plain, book-coded, Vigenere coded.svg|thumbnail|left|Well's ''War of the Worlds'' in plain text, in a book code, and in a Vigenère cipher.]]
When a text is encrypted in such a way that every occurrence of each distinct plaintext word is always mapped to the same encrypted word (as in the case of simple [[Substitution cipher|substitution cipher]]s, like the [[Caesar cipher]]s, or simple [[Social:Codebook|codebook]] ciphers), the frequency-rank distribution is not affected.  On the other hand, if separate occurrences of the same word may be mapped to two or more different words (as happens with the Vigenère cipher), the Zipf distribution will typically have a flat part at the high-frequency end.{{citation needed|date=May 2023}}

====Applications====

Zipf's law has been used for extraction of parallel fragments of texts out of comparable corpora.<ref name=moha2016/> [[Biography:Laurance Doyle|Laurance Doyle]] and others have suggested the application of Zipf's law for detection of alien language in the [[Astronomy:Search for extraterrestrial intelligence|search for extraterrestrial intelligence]].<ref name="doyle20162">{{Cite journal |last1=Doyle |first1=Laurance R. |date=2016-11-18 |title=Why Alien Language Would Stand Out Among All the Noise of the Universe |url=http://cosmos.nautil.us/feature/54/listening-for-extraterrestrial-blah-blah |url-status=dead |journal=Nautilus Quarterly |language=en |archive-url=https://web.archive.org/web/20200729120031/http://cosmos.nautil.us/feature/54/listening-for-extraterrestrial-blah-blah |archive-date=2020-07-29 |access-date=2020-08-30}}</ref><ref name="kersh20212">{{Cite book |last=Kershenbaum |first=Arik |title=The Zoologist's Guide to the Galaxy: What Animals on Earth Reveal About Aliens--and Ourselves |title-link=The Zoologist's Guide to the Galaxy |date=2021-03-16 |publisher=Penguin |isbn=978-1-9848-8197-7 |pages=251–256 |language=en |oclc=1242873084 |author-link=Arik Kershenbaum}}</ref>

The frequency-rank word distribution is often characteristic of the author and changes little over time. This feature has been used in the analysis of texts for authorship attribution.<ref name=droo2016/><ref name=droo2019/>

The word-like sign groups of the 15th-century codex Voynich Manuscript have been found to satisfy Zipf's law, suggesting that text is most likely not a hoax but rather written in an obscure language or cipher.<ref name=boyle2022/><ref name=mont2013/>

==See also==
* {{Annotated link|1% rule (Internet culture)}}
* {{Annotated link|Benford's law}}
* {{Annotated link|Bradford's law}}
* {{Annotated link|Brevity law}}
* {{Annotated link|Demographic gravitation}}
* {{Annotated link|Frequency list}}
* {{Annotated link|Gibrat's law}}
* {{Annotated link|Hapax legomenon}}
* {{Annotated link|Heaps' law}}
* {{Annotated link|King effect}}
* {{Annotated link|Long tail}}
* {{Annotated link|Lorenz curve}}
* {{Annotated link|Lotka's law}}
* {{Annotated link|Menzerath's law}}
* {{Annotated link|Pareto distribution}}
* {{Annotated link|Pareto principle}}, a.k.a. the "80–20 rule"
* {{Annotated link|Price's law}}
* {{Annotated link|Principle of least effort}}
* {{Annotated link|Rank-size distribution}}
* {{Annotated link|Stigler's law of eponymy}}
* [[Letter frequency]]
* Most common words in English

==Notes==
{{notelist}}

==References==
<references>
<ref name="Auerbach1913">Auerbach F. (1913) Das Gesetz der Bevölkerungskonzentration. Petermann’s Geographische Mitteilungen 59, 74–76</ref>
<ref name="zipf1935">George K. Zipf (1935): ''The Psychobiology of Language''. Houghton-Mifflin.</ref>
<ref name="zipf1932">George K. Zipf (1932): ''Selected Studies on the Principle of Relative Frequency in Language.'' Harvard, MA: Harvard University Press.</ref>
<ref name="king1942">{{Cite journal |last=Zipf |first=George Kingsley |date=1942 |title=The Unity of Nature, Least-Action, and Natural Social Science |url=https://www.jstor.org/stable/2784953 |journal=Sociometry |volume=5 |issue=1 |pages=48–62 |doi=10.2307/2784953 |jstor=2784953 |issn=0038-0431 |access-date=2022-11-20 |archive-date=2022-11-20 |archive-url=https://web.archive.org/web/20221120093519/https://www.jstor.org/stable/2784953 |url-status=live }}</ref>
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==Further reading==
* Alexander Gelbukh and Grigori Sidorov (2001) [http://www.gelbukh.com/CV/Publications/2001/CICLing-2001-Zipf.htm "Zipf and Heaps Laws’ Coefficients Depend on Language"]. Proc. CICLing-2001, ''Conference on Intelligent Text Processing and Computational Linguistics'', February 18–24, 2001, Mexico City. Lecture Notes in Computer Science N 2004, {{ISSN|0302-9743}}, {{isbn|3-540-41687-0}}, Springer-Verlag: 332–335.
* Kali R. (2003) "The city as a giant component: a random graph approach to Zipf's law," ''Applied Economics Letters 10'': 717–720(4)
* Shyklo A. (2017); [https://ssrn.com/abstract=2918642 Simple Explanation of Zipf's Mystery via New Rank-Share Distribution, Derived from Combinatorics of the Ranking Process], Available at SSRN: https://ssrn.com/abstract=2918642.
* Clara Moskowitz, Jen Christiansen, and Ni-Ka Ford, "Cells by Count and Size: The larger a cell type is, the rarer it is in the body – and vice versa", ''Scientific American'', vol. 330, no. 1 (January 2024), pp. 94–95. "'As you double the volume of a cell, the frequency of cells of that size is halved,'" ecologist Ian A. Hatton of McGill University and his fellow Zipf's-law researchers found, says Hatton. "Teensy, nonnucleated [[Biology:Red blood cell|red blood cell]]s are by far the most common cells in our bodies, whereas the comparatively gigantic muscle cells in our arms and legs are the scarcest. Being able to use a cell's size to estimate its frequency in the body could help doctors better understand certain body systems and hard-to-count cell types... The study suggests, for instance, that immune cells called [[Biology:Lymphocyte|lymphocyte]]s are far more common than biologists realized." (p. 94.)

==External links==
{{Library resources box}}
*{{Cite news | last = Strogatz | first = Steven |  title = Guest Column: Math and the City | date = 2009-05-29 | url = http://judson.blogs.nytimes.com/2009/05/19/math-and-the-city/ | access-date = 2009-05-29 | work = The New York Times | archive-date = 2015-09-27 | archive-url = https://web.archive.org/web/20150927204318/http://judson.blogs.nytimes.com/2009/05/19/math-and-the-city/ | url-status = dead }}—An article on Zipf's law applied to city populations
*[https://www.theatlantic.com/issues/2002/04/rauch.htm Seeing Around Corners (Artificial societies turn up Zipf's law)]
*[https://web.archive.org/web/20021018011011/http://planetmath.org/encyclopedia/ZipfsLaw.html PlanetMath article on Zipf's law]
*[http://www.hubbertpeak.com/laherrere/fractal.htm Distributions de type "fractal parabolique" dans la Nature (French, with English summary)] {{Webarchive|url=https://web.archive.org/web/20041024144850/http://www.hubbertpeak.com/laherrere/fractal.htm |date=2004-10-24 }}
*[https://www.newscientist.com/article.ns?id=mg18524904.300 An analysis of income distribution]
*[http://www.lexique.org/listes/liste_mots.txt Zipf List of French words] {{Webarchive|url=https://web.archive.org/web/20070623154627/http://www.lexique.org/listes/liste_mots.txt |date=2007-06-23 }}
*[http://1.1o1.in/en/webtools/semantic-depth Zipf list for English, French, Spanish, Italian, Swedish, Icelandic, Latin, Portuguese and Finnish from Gutenberg Project and online calculator to rank words in texts] {{Webarchive|url=https://web.archive.org/web/20110408115104/http://1.1o1.in/en/webtools/semantic-depth |date=2011-04-08 }}
*[https://arxiv.org/abs/physics/9901035 Citations and the Zipf–Mandelbrot's law]
*[http://www.geoffkirby.co.uk/ZIPFSLAW.pdf Zipf's Law examples and modelling (1985)]
*[http://www.nature.com/nature/journal/v474/n7350/full/474164a.html Complex systems: Unzipping Zipf's law (2011)]
*[http://terrytao.wordpress.com/2009/07/03/benfords-law-zipfs-law-and-the-pareto-distribution/ Benford’s law, Zipf’s law, and the Pareto distribution] by Terence Tao.
*{{springer|title=Zipf law|id=p/z130110}}

{{ProbDistributions|discrete-finite}}

[[Category:Discrete distributions]]
[[Category:Computational linguistics]]
[[Category:Power laws]]
[[Category:Statistical laws]]
[[Category:Tails of probability distributions]]
[[Category:Bibliometrics]]

{{Sourceattribution|Zipf's law}}