Power-law tails, and which α your “80/20” actually requires.
Which α your 80/20 actually needs
The only α for which the top 20% hold exactly 80%. The variance is infinite here — nothing mean-squared exists.
Pareto(α = 1.160964, xₘ = 1.000000)
The top 20.000% hold 80.0000% of the total
For the top 20% to hold exactly 80%, α would have to be 1.160964. Yours is 1.160964.
Top 20.000% share
80.0000%
of the total
α for exactly 80/20
1.160964
log 5 ÷ log 4
Mean
7.212569
αxₘ/(α−1)
Variance
does not exist
infinite at or below α = 2
Median
1.816741
xₘ · 2^(1/α)
P(X ≤ 4.0000)
0.800000
1 − (xₘ/x)^α
Gini coefficient
0.75647
1/(2α−1)
Tail exponent
2.1610
density falls as x^−(α+1)
Share held by the top slice, at α = 1.1610
The highlighted point is your chosen slice. A single α produces a whole family of “X/Y rules” — at your α the top 1% hold 52.8088% and the top 50% hold 90.8371%. The 80/20 version is one point on this curve, not a law.
The mean exists at α = 1.1610 but the variance does not. A standard deviation computed from a sample will not converge as n grows — it will keep jumping upward each time a large observation arrives. Confidence intervals built on it are not valid, and neither is anything that assumes finite variance. This is the range most fitted income and city-size distributions land in.
The 80/20 rule is one specific α, not a property of the distribution. Solving p^((α−1)/α) = 0.8 at p = 0.2 gives α = log 5 ÷ log 4 = 1.160964. At α = 2 the top 20% hold 44.72%; at α = 3, 34.20%. Quoting “80/20” about a dataset is a claim about its α, and it is usually wrong.
Scale invariance is the defining property. The share held by the top 1% of the top 1% is the same as the share held by the top 1% overall. That self-similarity is what makes a Pareto a power law, and it is why plotting the survival function on log-log axes gives a straight line with slope −α — the standard way to check whether your data is Pareto at all.
What this tool shows
The 80/20 rule is one specific α: log 5 ÷ log 4 = 1.160964. At α = 2 the top 20% hold 44.72%, and at α = 3, 34.20%. The tool computes the share for your own α and any top slice, and reports what does not exist — the mean below α = 1, the variance below α = 2.
PDF, CDF, median and the Gini coefficient for any α and minimum
The share held by the top slice you choose, plotted across the whole range
The exact α that produces the 80/20 rule, and how far yours is from it
Which moments exist: the mean above α = 1, the variance above α = 2
A warning when a sample mean or standard deviation will never converge
The tail exponent, for checking a power-law fit on log-log axes
Top-slice share Missing moments Exact 80/20 α Gini included
Updated 12 September 2026 · Works in any browser, no installation
A Pareto distribution has a minimum value and a tail that falls as a power of x rather than exponentially. The shape parameter α controls how heavy that tail is, and it decides everything: what share the top slice holds, whether the mean exists, and whether a standard deviation computed from a sample will ever settle down.
At a glance
Formula shown
The density is f(x) = αxₘ^α / x^(α+1) for x ≥ xₘ, and the CDF is F(x) = 1 − (xₘ/x)^α. The mean is αxₘ/(α−1), which exists only for α > 1; the variance is xₘ²α / ((α−1)²(α−2)), which exists only for α > 2. The share of the total held by the top fraction p is p^((α−1)/α), so setting p = 0.2 and the share to 0.8 gives α = log 5 / log 4 = 1.160964. The Gini coefficient is 1/(2α−1).
Scenario support
Income and wealth tails, city sizes, file sizes and web traffic, insurance claim severity, word frequencies, inventory concentration, and any claim of the form “the top X% account for Y%”.
Educational estimate
Planning support from the values you enter — not professional advice.
The 80/20 rule is one value of α, not a law
It is quoted as though it were a property of skewed data in general. It is a property of exactly one Pareto distribution.
The share held by the top fraction p is p^((α−1)/α). Set p = 0.2 and the share to 0.8, solve, and you get α = log 5 ÷ log 4 = 1.160964. No other α produces that pairing.
Move α a little and the rule changes a lot. At α = 1.5 the top 20% hold 58.48%. At α = 2, 44.72%. At α = 3, 34.20%. “80/20” is not a rough description of all of these; it is one point among them.
And a single α generates a whole family of rules. At the 80/20 α the top 1% hold 52.81% and the top 50% hold 90.84%. The tool plots the full curve so you can read off whichever slice matters rather than repeating the one that is famous.
So “our data follows 80/20” is a testable claim about α, and the way to test it is to fit the tail and read the exponent — not to check whether the top fifth happens to be near four fifths in one particular sample.
The mean and variance often do not exist
This is the part that breaks analyses, and it breaks them silently, because a computer will always return a number.
The mean exists only for α > 1. Below that the integral diverges and there is no value for a sample average to converge to. Collect more data and the average grows rather than settling.
The variance exists only for α > 2. Between 1 and 2 the mean is fine and the variance is not, so a standard deviation computed from a sample will ratchet upward every time a large observation arrives and never come back down.
That interval covers most fitted values. Income tails are commonly fitted around α = 1.5 to 2.5, and city sizes near 1. Analyses in that range that quote a standard deviation, a confidence interval built from one, or anything relying on the central limit theorem are computing with a quantity that is not there.
The tool says which moments exist rather than printing all of them. Where the variance is undefined it prints “does not exist”, because an infinity is more honest than a finite number that happens to be what this sample produced.
Scale invariance is the defining property
Power laws are the distributions that look the same at every magnification, and that is not a metaphor.
The share held by the top 1% of the top 1% equals the share held by the top 1% overall. Zoom into the tail and you find the same distribution again, with the same α. No other family does this.
Which is why the log-log plot is the standard diagnostic. Plot the survival function 1 − F(x) against x on log axes and a Pareto gives a straight line with slope −α. Curvature means it is not a power law, whatever the histogram suggests.
The minimum xₘ only sets the units. It does not affect the share held by any top slice, the Gini coefficient, or which moments exist — all of those depend on α alone. Change from dollars to rupees and only xₘ moves.
And the tail exponent, not the fit, is what transfers. Two datasets with the same α have the same concentration structure however different their units or medians, which is what makes α worth reporting on its own.
The Gini coefficient falls out of α directly
For a Pareto population the whole Lorenz curve is determined by one number, so the standard inequality measure is a formula rather than an integral over data.
Gini = 1/(2α − 1). At α = 2 it is exactly ⅓; at the 80/20 α of 1.161 it is 0.7576; at α = 3, 0.2.
So a Gini and a top-share claim are the same claim. Given either, the other follows, provided the population really is Pareto. Quoting both as independent evidence double-counts.
The caveat is the assumption. Real income distributions are Pareto in the upper tail and something else below it, so a Gini computed from the whole population will not match the one this formula gives from a tail-fitted α. A mismatch is informative rather than an error.
And a Gini near 1 does not need α near 0. It needs α near 0.5, where the mean has long since ceased to exist — which is a useful reminder that extreme inequality measures live in the region where the arithmetic stops behaving.
Fitting one, and the mistakes that come with it
Estimating α is easy and estimating it correctly is less so.
The maximum-likelihood estimate is n divided by the sum of log(x/xₘ). Simple, and highly sensitive to xₘ — which is usually chosen by eye and rarely reported.
Choosing xₘ too low is the commonest error. It pulls non-power-law body data into the fit and biases α upward, which makes the tail look lighter than it is. Choosing it too high throws away data and inflates the standard error.
Fitting a straight line to a log-log histogram is worse. The bins in the tail hold few observations, so ordinary least squares weights noise heavily and the slope it returns is biased. Maximum likelihood on the raw values is the better route.
And a straight-ish log-log plot is not proof. A lognormal over a limited range looks much the same, which is why claims that something “is a power law” usually need a comparison against that alternative rather than a single plot.
Reporting a concentration claim
Four things that turn an impressive sentence into a checkable one.
Report α and xₘ together. Neither means much alone, and xₘ is the input most often omitted despite doing most of the work in the fit.
Give the top-share figure for the slice you care about. Not the famous one. If your decision turns on the top 5%, quote the top 5%.
Say whether the mean and variance exist. If α is below 2, any standard deviation in the same report is describing your sample rather than the population, and should be labelled as such.
And show the log-log survival plot. It is the one piece of evidence that distinguishes a power law from a merely skewed distribution, and it costs one chart.
Sources and methodology
References for power laws and the Pareto distribution.
Method. The moment conditions are enforced rather than described: the mean returns NaN at or below α = 1 and the variance at or below α = 2, and the suite checks all three regimes explicitly. The top-share formula p^((α−1)/α) is verified to give exactly 0.8 at the 80/20 α and the published figures of 44.72% and 34.20% at α = 2 and 3, and the share is confirmed to fall monotonically as α rises across 61 points. The density is checked to integrate to its own CDF range by Simpson’s rule and the CDF to be monotone. That engine is verified on every change against 75 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Lognormal DistributionTakes log-scale parameters or your data's own mean and SD and converts between them — because μ describes ln x, not x, and entering the wrong one is wrong by orders of magnitude with nothing to flag it.
Cauchy DistributionRuns the averaging experiment rather than describing it: the interquartile range of a Cauchy sample mean is 1.85 at n = 1 and 2.07 at n = 1,000, while a normal's falls from 1.365 to 0.0436.
Exponential DistributionTime-until-event probabilities from a rate or a mean, always showing both — with a panel proving elapsed time changes nothing, which is a constant hazard rate and false for anything that wears out.
Weibull DistributionReliability probabilities with the hazard curve drawn beside the density — because the shape parameter is the hazard story: below 1 the risk falls, at 1 it is exactly an exponential, above 1 it rises.
Standard DeviationSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
Trimmed MeanTrimmed and Winsorized means at any fraction, plotted across the full sweep — because a 0% trim is the arithmetic mean and a 50% trim is exactly the median, so the trim fraction is a dial between them.
An educational tool. Fitting a power law is a strong claim that needs testing against alternatives such as the lognormal — a straight-looking log-log plot over a limited range is consistent with several distributions, and the choice changes what the model says about extreme values.
Published the Pareto distribution with the fact its popularity obscures: the 80/20 rule is one specific alpha, log 5 / log 4 = 1.160964, and nothing else. At alpha = 2 the top 20% hold 44.72%; at alpha = 3, 34.20%.
The top-share curve is plotted across the whole range, because a single alpha generates a family of rules - at the 80/20 alpha the top 1% hold 52.81% and the top 50% hold 90.84%.
Moment existence is enforced rather than described: the mean returns NaN at or below alpha = 1 and the variance at or below alpha = 2, with a warning that a sample standard deviation in that range will never converge. Most fitted income and city-size alphas land there.
The Gini coefficient is reported as 1/(2a-1), which makes a Gini claim and a top-share claim the same claim for a Pareto population rather than two pieces of evidence.
Verified that the top-share figure falls monotonically as alpha rises across 61 points, and pinned the published 44.72% and 34.20% figures exactly.
Add this calculator to your site
Responsive embed — and private: nothing your visitors type leaves their browser.