Power Law Distribution Calculator
Calculate Pareto and power law probabilities, mean, and percentiles.
Model heavy-tailed distributions like wealth, city populations, and word frequencies.
A power law distribution has a probability density that decays as a power of x:
f(x) = (alpha / x_min) * (x_min / x)^(alpha + 1) for x >= x_min
The complementary CDF (survival function) is:
P(X > x) = (x_min / x)^alpha
This is the Pareto distribution, named after Vilfredo Pareto who noticed that 80% of Italy’s land was owned by 20% of the population.
The exponent alpha. Small alpha (close to 1) means extremely heavy tails — a large fraction of the total is held by a tiny fraction of observations. As alpha increases, the distribution becomes less extreme. For alpha > 1, the mean exists: E[X] = alpha * x_min / (alpha - 1). For alpha > 2, the variance also exists.
The 80-20 rule is not fixed. The exact split depends on alpha. The Pareto principle (80% of outcomes from 20% of causes) corresponds to alpha = log(5) / log(4) = 1.161. For income distributions in wealthy countries, alpha is often 1.5-2.5, giving softer inequalities like 60-40.
Where power laws appear. City populations (Zipf’s law), word frequencies, earthquake magnitudes (Gutenberg-Richter), solar flare intensities, stock return tails, internet traffic, the number of citations per paper, and the size of forest fires all follow approximate power laws.
Fitting caution. Many distributions look power-law-like on a log-log plot but are actually lognormal or stretched exponential. Proper fitting requires maximum likelihood estimation and goodness-of-fit tests, not just visual inspection of a log-log line.