Power Law Distributions

Charlottesville, Virginia

April 26, 2020

I think this post falls under the category of "if you want to feel smart." I had never heard of power law distributions prior to this week. I guess I fell into the trap of assuming more subjects qualify as normal distribution than is true. Bummer.

In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.

The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms, the sizes of power outages, criminal charges per convict, volcanic eruptions, human judgements of stimulus intensity and many other quantities. Few empirical distributions fit a power law for all their values, but rather follow a power law in the tail. Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature.

More examples [Lord knows I don't know what most of these are!]

More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income). Among them are:



  • number of charges per criminal offender



  • Kleiber's law relating animal metabolism to size, and allometric laws in general

  • The two-thirds power law, relating speed to curvature in the human motor system.

  • The Taylor's law relating mean population size and variance of populations sizes in ecology

  • Neuronal avalanches

  • The species richness (number of species) in clades of freshwater fishes

  • The Harlow Knapp effect, where a subset of the kinases found in the human body compose a majority of published research


  • The size of rain-shower cells, energy dissipation in cyclones, and the diameters of dust devils on Earth and Mars

General science



  • Population sizes of cities in a region or urban network, Zipf's law.

  • Distribution of artists by the average price of their artworks.

  • Distribution of income in a market economy.

  • Distribution of degrees in banking networks.


  • The mean absolute change of the logarithmic mid-prices

  • Number of tick counts over time

  • Size of the maximum price move

  • Time of directional change

  • Time of overshoot

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