# Rank-size distribution

﻿
Rank-size distribution

Rank-size distribution or the rank-size rule (or law) describes the remarkable regularity in many phenomena including the distribution of city sizes around the world, sizes of businesses, particle sizes (such as sand), lengths of rivers, frequencies of word usage, wealth among individuals, etc. All are real-world observations that follow power laws such as those called Zipf's law, the Yule distribution, or the Pareto distribution. If one ranks the population size of cities in a given country or in the entire world and calculates the natural logarithm of the rank and of the city population, the resulting graph will show a remarkable log-linear pattern. This is the rank-size distribution. [ [http://people.few.eur.nl/vanmarrewijk/geography/zipf/ Zipf's Law, or the Rank-Size Distribution] Steven Brakman, Harry Garretsen, and Charles van Marrewijk]

In the case of city populations, the resulting distribution in a country, region or the world will be characterized by a largest city, with other cities decreasing in size respective to it, initially at a rapid rate and then more slowly. This results in a few large cities, and a much larger number of cities orders of magnitude smaller. For example, a rank 3 city would have ⅓ the population of a country's largest city, a rank four city would have ¼ the population of the largest city, and so on.

Why should simple rank be able to predict so easily such complex distributions? In short, why does the rank size rule “work?” One study has shown why this is so. [ [http://www-personal.umich.edu/~copyrght/image/monog08/fulltext.pdf The Urban Rank-Size Hierarchy] James W. Fonseca]

The distributions mentioned above such as Zipf, Pareto, Yule, etc., also called power laws, are all also related to the distribution known as the Fibonacci sequence and to that of the equiangular spiral. In the Fibonacci sequence, each term is approximately 1.618 (the Golden ratio) times the preceding term. A special case of the Fibonacci sequence is the Lucas sequence consisting of these sequentially additive numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199 ,…

When any log-linear factor is ranked, the ranks follow the Lucas sequence as above and each of the terms in the sequence can also be approximated by the successive values of powers of 1.618. For example, the third term in the sequence above, 4, is approximately 1.6183 or 4.236 (which is approximately 4); the fourth term in the sequence, 7, is approximately 1.6184 or 6.854 (which is approximately 7); the eight term in the series, 47, is approximately 1.6188 or 46.979 (which is approximately 47). With higher and higher values, the figures converge.

Thus it is shown that the rank size rule “works” because it is a “shadow” or coincidental measure of the true phenomenon. The true value of rank size is thus not as an accurate mathematical measure (since other power-law formulas are more accurate, especially at ranks lower than 10) but rather as a handy measure or “rule of thumb” to spot power laws. When presented with a ranking of data, is the third-ranked variable approximately ⅓ the value of the highest-ranked one? Or, conversely, is the highest-ranked variable approximately ten times the value of the tenth-ranked one? If so, the rank size rule has possibly helped spot another power law relationship. A 2002 study found that, Zipf’s Law worked for 44 of 73 countries tested. [ [http://cep.lse.ac.uk/pubs/download/dp0641.pdf Kwok Tong Soo (2002)] ] The study also found that variations of the Pareto exponent are better explained by political variables than by economic geography variables like proxies for economies of scale or transportation costs. [ [http://www.oup.com/uk/orc/bin/9780199280988/01student/zipf/ Zipf's Law, or the Rank-Size Distribution] ]

References

*cite journal
author = Brakman, S.
coauthors = Garretsen, H.; Van Marrewijk, C.; Van Den Berg, M.
year = 1999
title = The Return of Zipf: Towards a Further Understanding of the Rank-Size Distribution
journal = Journal of Regional Science
volume = 39
issue = 1
pages = 183–213
doi = 10.1111/1467-9787.00129

*cite journal
author = Guérin-Pace, F.
year = 1995
title = Rank-Size Distribution and the Process of Urban Growth
journal = Urban Studies
volume = 32
issue = 3
pages = 551–562
doi = 10.1080/00420989550012960

*cite journal
author = Reed, W.J.
year = 2001
title = The Pareto, Zipf and other power laws
journal = Economics Letters
volume = 74
issue = 1
pages = 15–19
doi=10.1016/S0165-1765(01)00524-9

*Douglas R. White, Laurent Tambayong, and Nataša Kejžar. 2008. Oscillatory dynamics of city-size distributions in world historical systems. "Globalization as an Evolutionary Process: Modeling Global Change". Ed. by George Modelski, Tessaleno Devezas, and William R. Thompson. London: Routledge. ISBN 9780415773614

ee also

* Pareto distribution
* Pareto principle
* The Long Tail

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• rank-size rule — UK US noun [S] (also rank size law, rank size distribution) ECONOMICS ► the principle that many things all over the world, for example the sizes of cities or businesses, or how rich people are, follow the same pattern in relation to their rank on …   Financial and business terms

• Occupancy frequency distribution — In macroecology and community ecology, an occupancy frequency distribution (OFD) is the distribution of the numbers of species occupying different numbers of areas. It was first reported in 1918 by the Danish botanist Christen C. Raunkiær in… …   Wikipedia

• Sample size determination — is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample …   Wikipedia

• Effect size — In statistics, an effect size is a measure of the strength of the relationship between two variables in a statistical population, or a sample based estimate of that quantity. An effect size calculated from data is a descriptive statistic that… …   Wikipedia

• Multivariate normal distribution — MVN redirects here. For the airport with that IATA code, see Mount Vernon Airport. Probability density function Many samples from a multivariate (bivariate) Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the… …   Wikipedia

• Noncentral t-distribution — Noncentral Student s t Probability density function parameters: degrees of freedom noncentrality parameter support …   Wikipedia

• Frequency distribution — In statistics, a frequency distribution is an arrangement of the values that one or more variables take in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval, and… …   Wikipedia

• Student's t-distribution — Probability distribution name =Student s t type =density pdf cdf parameters = u > 0 degrees of freedom (real) support =x in ( infty; +infty)! pdf =frac{Gamma(frac{ u+1}{2})} {sqrt{ upi},Gamma(frac{ u}{2})} left(1+frac{x^2}{ u} ight)^{ (frac{… …   Wikipedia

• Wishart distribution — Probability distribution name =Wishart type =density pdf cdf parameters = n > 0! deg. of freedom (real) mathbf{V} > 0, scale matrix ( pos. def) support =mathbf{W}! is positive definite pdf =frac{left|mathbf{W} ight|^frac{n p 1}{2… …   Wikipedia

• Multinomial distribution — Multinomial parameters: n > 0 number of trials (integer) event probabilities (Σpi = 1) support: pmf …   Wikipedia

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.