The Pareto distribution
The Pareto distribution describes quantities where a few large values dominate: incomes and wealth, city sizes, file sizes, insurance losses, word frequencies. It is named after Vilfredo Pareto, who noticed around 1900 that about 20% of the population owned about 80% of the land in Italy.
It has two parameters: a scale xm, the smallest possible value, and a shape α, which controls how heavy the tail is. Smaller α means more extreme inequality.
Formulas
For x ≥ xm:
- density: f(x) = α × xm^α ÷ x^(α + 1);
- cumulative: P(X ≤ x) = 1 − (xm ÷ x)^α;
- mean: α × xm ÷ (α − 1), finite only if α > 1;
- median: xm × 2^(1/α);
- variance: xm² × α ÷ ((α − 1)² × (α − 2)), finite only if α > 2.
Example
xm = 1, α = 3, x = 2:
- P(X ≤ 2) = 1 − (1/2)³ = 0.875, so 12.5% of values exceed 2;
- density at 2: 3 ÷ 2⁴ = 0.1875;
- mean 1.5, median 1.26, variance 0.75;
- the top 20% hold 0.2^(1 − 1/3) ≈ 34.2% of the total.
The 80/20 rule
For a Pareto distribution with α > 1, the share of the total held by the top fraction p of the population is p^(1 − 1/α). The classic 80/20 split corresponds to α = log 5 ÷ log 4 ≈ 1.16. With that α, the top 1% hold about 53% of the total, which shows how quickly concentration rises in the extreme tail.
Infinite means and variances
When α ≤ 1, the mean is infinite: the sample average keeps growing as more data comes in, driven by rare huge values. When 1 < α ≤ 2, the mean exists but the variance does not, so averages converge slowly and standard errors are unreliable. Many real-world phenomena, such as city sizes and some financial losses, sit near α between 1 and 2.
Fitting α
Given data above a threshold xm, the maximum likelihood estimate is α = n ÷ Σ ln(xi ÷ xm). On a log-log plot of the survival function, Pareto data follow a straight line with slope −α.
Related distributions
The lognormal also produces skewed data but has lighter tails. The exponential distribution is what you get by taking the logarithm of Pareto data: ln(X ÷ xm) is exponential with rate α.