Interactive module guide

Pareto Distribution Calculator & Guide

This Pareto distribution calculator lets you calculate power-law tail probabilities and see which moments exist at each tail index.

Open calculator

Free · No sign-up · Calculations stay in your browser

The Pareto distribution is the canonical power law: above a minimum xₘ, the probability of exceeding x falls off as (xₘ/x)^α, so a small share of cases carries most of the total—the 80/20 pattern of wealth, city sizes, insurance losses, and file sizes.

Everything about the family is governed by the tail index α: the k-th moment exists only when α > k, so the mean requires α > 1, the variance α > 2, and so on.

This page computes exact tail probabilities and quantiles at every α, and its summary panel shows dashes for whichever moments the chosen tail index cannot support.

Is this the right module for my question?

Use it when

Use it for size phenomena with genuine multiplicative or preferential-attachment mechanisms: incomes above a threshold, insurance claims above a retention, market shares, or traffic bursts, where straight-line log-log survival plots are the empirical fingerprint.
Use it for tail-risk reasoning: with a power-law tail, “ten times larger” events are only a constant factor rarer, and this page quantifies exactly how much probability lives beyond any multiple of the minimum.
Use it to teach moment existence: sliding α across 1, 2, 3, and 4 switches the mean, variance, skewness, and kurtosis on and off, making the abstract condition α > k concrete on the summary panel.

Choose another method when

Avoid it for data whose tail decays exponentially—waiting times, light-tailed measurements—where a gamma, Weibull, or log-normal model fits the whole range instead of only the tail.
Avoid fitting it below its minimum: the model starts at xₘ by construction, and applying it to full datasets including small values misstates both parameters.
Avoid trusting moment-based summaries near the existence boundary: at α slightly above 2 the variance exists but is enormous and sample variances converge miserably slowly; quantiles are the stable currency here.

Interactive tool

The calculator loads as you approach this section so the guide remains fast on mobile connections.

Example preview

Power-law wealth threshold

Inputs: X ~ Pareto(xₘ=1, α=2.5); evaluate exceeding 4 times the minimum.
Representative result: P(X>4) = 4^(−2.5) ≈ 0.0313, a tail far heavier than any exponential model.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

3:05 min

The player loads only after you press play. You can also watch on YouTube.

Read the complete transcript

The Pareto Distribution. Why can a tiny fraction of observations dominate totals while ordinary-looking averages become dangerously unstable? The Type One Pareto distribution starts at positive minimum x m and uses tail index alpha. Its density equals alpha times x m to alpha, divided by x to alpha plus one. More directly, survival beyond x is x m over x, raised exactly to alpha. Inverting survival gives quantile x m times one minus p, raised to negative one over alpha. On log-log axes, survival becomes a straight line whose slope is precisely negative alpha.

The mode is x m, while the median is x m times two to one over alpha. The kth positive raw moment exists exactly when alpha is strictly greater than k. Therefore the mean exists only above alpha one and equals alpha x m over alpha minus one. Variance requires alpha above two, with another denominator factor of alpha minus two. Skewness exists only above alpha three, and excess kurtosis only above alpha four. Multiplying any threshold by c multiplies exceedance probability by c to negative alpha. An idealized eighty-twenty split corresponds to alpha about one point one six, not every Pareto law.

Consider insurance claims above one retention unit, modeled with x m one and alpha two point five. A claim exceeds four units with probability one over thirty-two, exactly point zero three one two five. The ninety-ninth percentile is six point three zero nine six retention units. At ninety-nine point nine percent, the quantile jumps to fifteen point eight four eight nine. Its mean is one point six six seven, while variance is two point two two two two. With alpha point eight, quantiles still exist, but both mean and variance are undefined.

Pareto tails can model thresholded claims, wealth, city sizes, firm sizes, and large files. Diagnose the model using log survival against log x, focusing only on observations above the threshold. A few large observations or a straight-looking segment never proves a power law. Values below x m are impossible under Type One Pareto and must not enter the fit. Near moment boundaries, sample averages and variances converge painfully slowly despite technically existing. Far-tail predictions extend the fitted power law beyond data and require an explicit modeling warning. Do not confuse Type One Pareto with Lomax or the generalized Pareto distribution. For reproducibility, report x m, alpha, thresholded population, event endpoints, and every extrapolation range. Explore Pareto probabilities, quantiles, and moment thresholds free at Distri Scope dot com.

How to read the result

Read the survival function multiplicatively: doubling the threshold multiplies the exceedance probability by 2^{−α}, independent of the starting point—the scale-free property that defines power laws.

At α = 1.16 the family reproduces the classic 80/20 rule (top 20% of the population holds 80% of the mass); smaller α is more extreme concentration.

The gap between median and mean is diagnostic: at α = 2.5, the median is about 1.32·xₘ while the mean is 1.67·xₘ, pulled up by the tail the median ignores.

Inputs and parameter meaning

Two parameters: the minimum xₘ (support starts there; it is also the mode) and the tail index α from 0.5 to 10, deliberately including values below 1 where even the mean fails to exist.

The plotted range runs from xₘ to the 95th percentile (capped at 50·xₘ) because the tail extends far beyond any finite window.

The summary panel shows dashes for moments the current α cannot support—an honest display, not a bug.

Formula and available parameters

Density or probability mass

f(x) = \frac{\alpha x_m^{\alpha}}{x^{\alpha+1}}, \quad x \ge x_m

Cumulative distribution

F(x) = 1 - \left(\frac{x_m}{x}\right)^{\alpha}

Mean

\mu = \frac{\alpha x_m}{\alpha - 1} \quad (\alpha > 1)

Variance

\sigma^2 = \frac{x_m^2 \alpha}{(\alpha-1)^2(\alpha-2)} \quad (\alpha > 2)

Minimum (xₘ): interactive range 0.1 to 5; default 1.
Tail index (α): interactive range 0.5 to 10; default 2.5.

What the method is doing

Density, CDF, and quantile function are closed-form; the survival function is exactly (xₘ/x)^α, which is a straight line of slope −α on log-log axes.

Moments use the standard formulas with explicit existence conditions: mean αxₘ/(α−1) for α > 1, variance xₘ²α/((α−1)²(α−2)) for α > 2, with skewness and kurtosis gated at α > 3 and α > 4.

The median is xₘ·2^{1/α} for every α—quantiles exist at every tail index, which is why heavy-tail practice reasons in quantiles.

Worked example: claims above a retention

Insurance claims above a retention of xₘ = 1 (in units of the retention) are modeled as Pareto with tail index α = 2.5. Find the probability that a claim exceeds 4 retentions, the 99th-percentile claim, the mean claim, and state which summaries would fail at a heavier tail index of α = 0.8.

1Tail probability: P(X > 4) = 4^{−2.5} ≈ 0.0313—about one claim in 32 exceeds four retentions.
2Extreme quantile: Q(0.99) = (0.01)^{−1/2.5} ≈ 6.31 retentions, and Q(0.999) ≈ 15.85—each extra “nine” multiplies the size by 10^{0.4} ≈ 2.51.
3Mean: with α = 2.5 > 1 the mean exists: 2.5/(2.5−1) ≈ 1.67 retentions; the variance also exists (α > 2) at ≈ 0.74.
4Heavier tail: at α = 0.8 the summary panel shows dashes everywhere—no mean, no variance—and only quantile statements remain meaningful.

Interpretation

At α = 2.5 the book is heavy-tailed but summarizable: means and variances exist, though the 99.9th-percentile claim is nearly ten times the mean. The α = 0.8 contrast is the practical warning: for sufficiently heavy tails, expected-value reasoning collapses entirely, and pricing or capital decisions must be made in quantiles.

Common mistakes

Do not verify a power law by eye on a linear histogram; the log-log survival plot with a straight line of slope −α is the diagnostic, and even that requires many tail observations.
Do not report sample means near the existence boundary as stable estimates; just above α = 1 the sample mean converges so slowly it is practically useless.
Do not extrapolate the fitted tail far beyond the observed range without saying so; the power-law form is an assumption exactly where data run out.

Limits and independent validation

This page implements the Type I (two-parameter) Pareto; shifted (Lomax) and generalized Pareto variants used in extreme-value theory are out of scope.

Fitting α from data—typically by maximum likelihood on tail observations—is a data-fitting task this theoretical page does not perform.

The plotted window truncates the tail by necessity; numerical quantiles remain exact beyond it.

Before using the result

Check the straight-line signature: compute the survival function at a few thresholds and confirm log-survival falls linearly in log-x with slope −α.
Recompute the median by hand as xₘ·2^{1/α} and compare with the summary panel.
Vary α across 1, 2, 3, and 4 and confirm each moment switches on exactly at its threshold—dashes on one side, numbers on the other.

log-normal distribution →Compare with the rival heavy-tail model whose moments all exist.exponential distribution →Contrast power-law tails with the light exponential benchmark.

← Cauchy Distribution Laplace Distribution →

See every option in the Probability Distribution Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.