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Zipf–Mandelbrot law

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Probability mass function
Cumulative distribution function
Parameters N \in \{1,2,3\ldots\} (integer)
q \in [0;\infty) (real)
s>0\, (real)
Support k \in \{1,2,\ldots,N\}
Template:Probability distribution/link mass \frac{1/(k+q)^s}{H_{N,q,s}}
cdf \frac{H_{k,q,s}}{H_{N,q,s}}
Mean \frac{H_{N,q,s-1}}{H_{N,q,s}}-q
Mode 1\,
Char. func.

In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto-Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoît Mandelbrot, who subsequently generalized it.

The probability mass function is given by:


where H_{N,q,s} is given by:

H_{N,q,s}=\sum_{i=1}^N \frac{1}{(i+q)^s}

which may be thought of as a generalization of a harmonic number. In the formula, k is the rank of the data, and q and s are parameters of the distribution. In the limit as N approaches infinity, this becomes the Hurwitz zeta function \zeta(q,s). For finite N and q=0 the Zipf–Mandelbrot law becomes Zipf's law. For infinite N and q=0 it becomes a Zeta distribution.


The distribution of words ranked by their frequency in a random text corpus is generally a power-law distribution, known as Zipf's law.

If one plots the frequency rank of words contained in a large corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Gelbukh & Sidorov, 2001).

In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf–Mandelbrot law.[1]

Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandlebrot distributions.[2]


  1. Mouillot, D, Lepretre, A (2000). Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity. Environmental Monitoring and Assessment 63 (2): 279–295.
  2. Manris, B, Vaughan, D, Wagner, CS, Romero, J, Davis, RB. Evolutionary Music and the Zipf-Mandelbrot Law: Developing Fitness Functions for Pleasant Music. Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003) 611.


  • Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics" B.B. Wolman and E. Nagel Scientific psychology, Basic Books. Reprinted as
    • Mandelbrot, Benoît [1965] (1968). "Information Theory and Psycholinguistics" R.C. Oldfield and J.C. Marchall Language, Penguin Books.
  • Zipf, George Kingsley (1932). Selected Studies of the Principle of Relative Frequency in Language, Cambridge, MA: Harvard University Press.

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