Pareto Distribution
Heavy-tailed probability distribution following a power law, originally formulated to describe wealth inequality and now used across economics, linguistics, computing, and ecology.
The Pareto distribution is a power law probability distribution in which the probability of observing a value decays as an inverse power of that value. It is named after Italian economist Vilfredo Pareto (1848–1923), who in the 1890s used it to model wealth concentration in the Kingdom of Italy, where he estimated roughly 80% of the land was held by 20% of the population — the observation that later crystallised into the popular 80/20 rule or Pareto principle. Formally, a Pareto random variable has a survival function of the form P(X > x) = (x_m / x)^α for x ≥ x_m, where x_m is the minimum value and α is the shape parameter controlling tail heaviness. Lower α means a heavier tail and more extreme inequality. The distribution is closely related to Zipf's law and is one of several canonical heavy-tailed distributions including the log-normal and stretched exponential. Mark Newman's widely cited 2005 survey *Power laws, Pareto distributions and Zipf's law* catalogued the surprising range of phenomena that follow such distributions: city sizes, earthquake magnitudes, solar flares, moon craters, citation counts, web link counts, war fatalities, and personal income. In software and online communities, contribution activity per user often approximates a Pareto distribution — a small fraction of users produces the majority of content. This shows up in open source software commit histories, Wikipedia edits, OpenStreetMap mapping, and user-generated content platforms generally. Care is required because true power laws are statistically difficult to confirm — Aaron Clauset and colleagues' 2009 methodology paper showed many empirical 'power laws' are better fit by alternative heavy-tailed distributions. The Pareto family remains useful nonetheless as a working model of extreme inequality in production, wealth, and influence.