Log-normal Distribution - Occurrence

Occurrence

• In biology, variables whose logarithms tend to have a normal distribution include:
  • Measures of size of living tissue (length, skin area, weight);
  • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth;
  • Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations)

Consequently, reference ranges for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.

• In hydrology, the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.
  • The image on the right illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of a cumulative frequency analysis.

• In economics, there is evidence that the income of 97%–99% of the population is distributed log-normally.

• In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as Benoît Mandelbrot have argued that log-Levy distributions which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. Indeed stock price distributions typically exhibit a fat tail.

• The distribution of city sizes is lognormal. This follows from Gibrat's law of proportionate (or scale-free) growth. Irrespective of their size, all cities follow the same stochastic growth process. As a result, the logarithm of city size is normally distributed. There is also evidence of lognormality in the firm size distribution and of Gibrat's law.

• In reliability analysis, the lognormal distribution is often used to model times to repair a maintainable system.

• In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution."

• It has been proposed that coefficients of friction and wear may be treated as having a lognormal distribution