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 scales, 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. A rank-size (or rank–frequency) distribution is often segmented into ranges. This is frequently done somewhat arbitrarily or due to external factors, particularly for market segmentation, but can also be due to distinct behavior as rank varies. Most simply and commonly, a distribution may be split in two pieces, termed the head and tail. If a distribution is broken into three pieces, the third (middle) piece has several terms, generically middle, also belly, torso, and body. These frequently have some adjectives added, most significantly long tail, also fat belly, chunky middle, etc.