Log-Normal Distribution

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The log-normal distribution is a skewed distribution, well suited to fit skewed data with low mean and large variance. This occurs frequently in real life, such as latent periods of infectious disease and distribution of minerals on the Earth’s crust [1]. The random variables that make up the log-normal distribution have normally distributed logarithms. As Y = ln(x) only exists for positive values of x, log-normal distributions have all-positive values.

The general log-normal distribution has a probability density function (PDF) defined by

The log-normal distribution’s shape is defined by three parameters:

  • σ: the shape parameter, which changes the overall shape. This parameter is also the standard deviation for the log-normal distribution. Usually, this parameter is known from historical data.
  • m: the scale parameter (the median). This shrinks or stretches the graph.
  • θ (or sometimes μ): the location parameter, which tells you where on the x-axis the graph is located.

The standard lognormal distribution has θ = 0 and m = 1. If θ = 0, the distribution is a 2-parameter lognormal distribution [2].

log-normal distribution
Log-normal distribution for varying values of σ. The standard log-normal is shown in blue with σ = 1.

Data that fits a log-normal Distribution

The following phenomenon can all be modeled with a lognormal distribution [1].

  • Cow milk production.
  • Lifetimes of industrial units with fatigue-stress failure.
  • Amount of rainfall and size of droplets.
  • Volume of gas in a petroleum reserve.
  • Length of latent periods of infectious disease or species abundance.

Other Names for the Log-normal

  • The Galton distribution (after Francis Galton).
  • Galton-McAlister distribution (after Galton and Donald McAlister who published a description of the distribution with Galton).
  • Gibrat distribution, after the 20th century French economist Robert Gibrat.
  • Cobb-Douglas distribution (after 20th century economists Charles Cobb and Paul Douglas).

References

[1] Limpert, E. at al. (2001). Log-normal Distributions across the Sciences: Keys and Clues. BioScience. May  / Vol. 51 No. Online: https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

[2] NIST. Lognormal distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm

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