Rank-size distribution

Rank-size distribution is the distribution of size by rank, in decreasing order of size. For example, if a data set consists of items of sizes 5, 100, 5, and 8, the rank-size distribution is 100, 8, 5, 5 (ranks 1 through 4). This is also known as the rank-frequency distribution, when the source data are from a frequency distribution. These are particularly of interest when the data vary significantly in scale, such as city size or word frequency. These distributions frequently follow a power law distribution, or less well-known ones such as a stretched exponential function or parabolic fractal distribution, at least approximately for certain ranges of ranks; see below.

A rank-size distribution is not a probability distribution or cumulative distribution function. Rather, it is a discrete form of a quantile function (inverse cumulative distribution) in reverse order, giving the size of the element at a given rank.

In the case of city populations, the resulting distribution in a country, a region, or the world will be characterized by its 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 one-third the population of a country's largest city, a rank 4 city would have one-fourth the population of the largest city, and so on.

When any log-linear factor is ranked, the ranks follow the Lucas numbers, which consist of the sequentially additive numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, etc. Like the more famous Fibonacci sequence, each number is approximately 1.618 (the Golden ratio) times the preceding number. For example, the third term in the sequence above, 4, is approximately 1.618³, or 4.236; the fourth term, 7, is approximately 1.618⁴, or 6.854; the eighth term, 47, is approximately 1.618⁸, or 46.979. With higher values, the figures converge. An equiangular spiral is sometimes used to visualize such sequences.

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