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Zipf's law is an experimental law, not a theoretical one. The causes of Zipfian distributions in real life are a matter of some controversy. However, Zipfian distributions are commonly observed in many kinds of phenomena. Zipf's law is often demonstrated by scatterplotting the data, with the axes being log(rank order) and log(frequency). If the points are close to a single straight line, the distribution follows Zipf's law.
The classic case of Zipf's law is a "1/f function". Given a set of Zipfian distributed frequencies, sorted from most common to least common, the second most common frequency will occur 1/2 as often as the first. The third most common frequency will occur 1/3 as often as the first. The nth most common frequency will occur 1/n as often as the first.
Mathematically, it is impossible for the classic version of Zipf's law to hold exactly if there are infinitely many words in a language, since for any constant of proportionality c > 0, the sum of all relative frequencies is proportional to the harmonic series and must be
Empirical studies have found that in English, the frequencies of the approximately 1000 most-frequently-used words are approximately proportional to 1/ns where s is just slightly more than one.
As long as the exponent s exceeds 1, it is possible for such a law to hold with infinitely many words, since if s > 1 then
The value of this sum is ζ(s), where ζ is Riemann's zeta function.
The term Zipf's law has consequently come to be used to refer to frequency distributions of "rank data" in which the relative frequency of the nth-ranked item is given by the Zeta distribution, 1/(nsζ(s)), where s > 1 is a parameter indexing this family of probability distributions. Indeed, the term Zipf's law sometimes simply means the zeta distribution, since probability distributions are sometimes called "laws". This distribution is sometimes called the Zipfian distribution or Yule distribution.
A more general law proposed by Benoit Mandelbrot has frequencies
This is the Zipf-Mandelbrot law. The "constant" in this case is the reciprocal of the Hurwitz zeta function evaluated at s.
In the tail of the Yule-Simon distribution the frequencies are approximately
for any choice of ρ > 0.
The log-normal distribution is the distribution of a random variable whose logarithm is normally distributed, useful when small fluctuations multiply a quantity rather than add to it.
In the parabolic fractal distributionIn the parabolic fractal distribution the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship (see external link belo, 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 (see external link below).
It has been pointed out (see external link below) that Zipfian distributions can also be regarded as being Pareto distributionThe Pareto distribution named after the Italian economist Vilfredo Pareto is a power law distribution found in a large number of real-world situations. This distribution is also known, mostly outside economics, as the Bradford distribution''. If X is a ras with an exchange of variables.
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