Lognormal Distribution

A lognormal distribution is a probability distribution in which the logarithm of the variable is normally distributed. If X is lognormally distributed, then ln(X) follows a normal distribution. The key property: a lognormal arises naturally from multiplicative processes — when a quantity is the product of many independent random factors, the central limit theorem applied to the logarithm produces a lognormal outcome.

Lognormal distributions are frequently confused with power laws in empirical data analysis, particularly because both produce heavy tails on linear scales and roughly straight lines on log-log plots. The distinction matters: a power law has no characteristic scale, while a lognormal has a characteristic scale at its mode. Clauset, Shalizi, and Newman's rigorous statistical work demonstrated that many distributions claimed as power laws are statistically indistinguishable from lognormals under proper testing. This distinction is not pedantic — different generating mechanisms (multiplicative random growth vs. criticality) have entirely different theoretical implications.

Lognormal distributions appear in firm size distributions (Gibrat's law predicts this), income distributions, biological organ sizes, reaction times, and many physical measurements. The Galton-Watson branching processes underlying population genetics also tend toward lognormal outcomes. The empiricist takeaway: before invoking scale-free network arguments or critical phenomena to explain a heavy-tailed distribution, first verify that the lognormal alternative can be ruled out.