Pareto Distribution Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The Pareto distribution models quantities where a minority of extreme values dominates the distribution. Its cumulative distribution function is F(x) = 1 − (xₘ/x)^α, where xₘ is the minimum possible value and α a shape parameter. For xₘ=2 and α=3, the probability that X exceeds 4 (twice xₘ) is exactly (1/2)³=12.5%.
Explanation
The Pareto distribution, named after the Italian economist Vilfredo Pareto who formulated it by observing the distribution of wealth, describes phenomena where a small proportion of extreme values concentrates a disproportionate share of the total — it is the statistical law behind the famous "80/20 principle" (80% of effects come from 20% of causes), even though that precise ratio corresponds only to a particular value of the shape parameter α, not to all Pareto distributions. The scale parameter xₘ sets the minimum possible value of the distribution (no observation can be below xₘ), while the shape parameter α controls the thickness of the extreme-value tail: an α close to 1 produces a very heavy-tailed distribution, where extremely high values stay relatively probable, while a high α produces a distribution closer to a concentration around xₘ. This distribution differs markedly from the log-normal distribution, another asymmetric positive-valued distribution: the Pareto distribution has an even heavier tail (a "power law"), which makes it better suited to phenomena where extreme events are significantly more frequent than a log-normal distribution would predict — the size of a country's largest cities, the popularity of the most viral content online, or the magnitude of the largest earthquakes. A remarkable and sometimes surprising property of this distribution is that its expected value becomes infinite as soon as α drops to 1 or below: the extreme-value tail is then so heavy that no finite mean can summarize it, a phenomenon observed in some real wealth distributions on a global scale.
Example: xₘ=2, α=3, x=4
Inputs
Scale value xₘ = 2. Shape parameter α = 3. Value x = 4 (twice xₘ).
Calculation
P(X > 4) = (2/4)³ = (0.5)³ = 0.125. P(X ≤ 4) = 1 − 0.125 = 0.875. Expected value = (3×2)/(3−1) = 6/2 = 3.
Result
The probability of exceeding twice the minimum value is only 12.5%, and the mean value of the distribution is 3 (1.5 times xₘ).
Frequently asked questions
Why can the expected value be infinite?
When α is less than or equal to 1, the extreme-value tail of the distribution is so heavy that the sum (integral) defining the mathematical mean no longer converges to a finite number: increasingly large values stay probable enough to prevent the mean from settling. It is a counterintuitive but real phenomenon in some economic or physical systems, which is why this calculator requires α strictly greater than 1 to display an expected value.
Does the "80/20" ratio apply to all Pareto distributions?
No, this precise ratio corresponds only to a specific value of the shape parameter (α≈1.161, obtained by solving the equation giving exactly 80% of the total mass to the top 20% of values). The general principle — a minority of values dominating the majority of the total — remains true for the whole family of Pareto distributions with a relatively low α, but the exact ratio varies with the precise value of α, it is never automatically 80/20.
What is the difference between the scale parameter xₘ and the distribution's expected value?
xₘ is the MINIMUM possible value of the distribution (no observation can be below it), fixed by the model's construction, while the expected value is the AVERAGE value of all observations, always strictly greater than xₘ. The ratio between the two, α/(α−1), shows that the expected value approaches xₘ when α is very high (very concentrated distribution) and moves far from it when α approaches 1 (heavy tail pulling the mean upward) — a sharp contrast with the exponential distribution, whose tail decreases much faster and whose expected value is always finite whatever its rate parameter.