Power Laws and Pareto Distributions: What a Power Law Distribution Really Is

Q: What is a power law distribution?

A power law distribution is one whose probability density falls off as a fixed negative power of the variable, so that p(x) is proportional to x to the minus alpha above some lower cutoff. Its defining property is scale invariance: multiplying x by a constant multiplies the density by a constant but leaves its shape unchanged, so the distribution has no characteristic size. This is the opposite of the normal distribution, where the standard deviation fixes a scale and observations far beyond it are effectively impossible.

Q: What is the 80-20 rule?

The 80-20 rule, or Pareto principle, is the observation that a small share of cases accounts for most of the total. For a Pareto distribution the top q fraction of observations holds a share equal to q raised to the power one minus one over gamma, where gamma is the tail exponent. Setting that share to 0.8 at q equal to 0.2 gives gamma about 1.161. The 80-20 split is therefore one particular exponent rather than a law of nature, and Pareto himself typically fitted exponents near 1.5, which gives roughly a 58-20 split instead.

Q: What is the difference between a power law and a lognormal distribution?

A power law has a density that is exactly straight on log-log axes, while a lognormal has a density whose log-log plot is a downward parabola with curvature equal to one over the variance of the logarithm. When that variance is large the curvature is tiny, so a lognormal looks straight over two or three decades. The generating mechanisms differ too: multiplicative random growth alone produces a lognormal, whereas multiplicative growth with a lower barrier produces a power law.

Q: Why do power laws have infinite variance?

The k-th moment of a power law is an integral of x to the power k minus alpha, which converges at infinity only when k is less than alpha minus one. With the density exponent alpha between 2 and 3 the mean exists but the second moment does not, so the variance is infinite. The practical consequence is that a sample standard deviation computed on such data is not estimating anything: it grows without bound as the sample size increases rather than settling down.

Q: How do you test whether data follows a power law?

Fit the exponent by maximum likelihood using the Hill estimator rather than by regression on a log-log histogram, choose the lower cutoff by minimising the Kolmogorov-Smirnov distance between the fitted and empirical distributions, compute a goodness-of-fit p-value by comparing the observed KS distance against synthetic data drawn from the fitted model, and then run likelihood-ratio tests against the lognormal, the exponential and the power law with an exponential cutoff. This is the procedure of Clauset, Shalizi and Newman, published in SIAM Review in 2009.

In 1999 three brothers named Faloutsos presented a paper at SIGCOMM arguing that the connectivity of the Internet followed a power law distribution. From three snapshots of the inter-domain topology spanning November 1997 to December 1998, several graph properties — node degree, the rank-ordered degree sequence, the number of node pairs within a given number of hops — fell on straight lines on doubly logarithmic axes, with correlation coefficients of 96 percent or better. Within a few years “scale-free” had become a standard descriptor for almost every measured network.

Twenty years later the careful re-examinations were more guarded. Clauset, Shalizi and Newman, fitting Internet degree data by maximum likelihood, found a power law with an exponential cutoff fitted clearly better than a pure power law. Broido and Clauset, examining nearly a thousand networks, found strongly scale-free structure empirically rare. The straight lines were real; what they implied was not what had been claimed.

So: what is a power law distribution? It is a distribution whose density falls off as a fixed negative power of the variable, $p(x) \propto x^{-\alpha}$ , above some lower cutoff. Its defining feature is not heavy tails, though it has them, nor straightness on log-log paper, though it shows that. It is scale invariance: multiply $x$ by any constant $c$ and the density is multiplied by a constant, with its shape untouched. A power law therefore has no characteristic size. Ask how big a typical observation is and the question has no answer, because the distribution looks the same whether you measure in metres or kilometres, in dollars or in millions of dollars. That is exactly the opposite of the normal distribution, where the standard deviation sets a scale and observations ten of them out are effectively impossible. Everything that follows — infinite variances, the eighty-twenty arithmetic, the difficulty of testing — is a consequence of that one property.

The Pareto form, and the exponent convention that confuses everyone

The canonical continuous power law is the Pareto distribution, with density

$p(x) = \frac{\alpha-1}{x_{\min}}\left(\frac{x}{x_{\min}}\right)^{-\alpha}, \qquad x \ge x_{\min},$

and complementary cumulative distribution function

$P(X > x) = \left(\frac{x}{x_{\min}}\right)^{-(\alpha-1)}, \qquad x \ge x_{\min}.$

The lower cutoff is forced, not convenient. The function $x^{-\alpha}$ is not integrable on $(0,\infty)$ for any $\alpha$ : near zero $\int_0 x^{-\alpha}dx$ diverges whenever $\alpha \ge 1$ , and at infinity it diverges whenever $\alpha \le 1$ . A power law can only hold above some positive $x_{\min}$ . Every real power-law claim is therefore a claim about a tail — and the exponent is estimated from the part of the data you have least of.

Now the convention trap, responsible for a great deal of confused reading. The density exponent and the exceedance exponent differ by exactly one:

$p(x) \propto x^{-\alpha}, \qquad P(X>x) \propto x^{-\gamma}, \qquad \gamma = \alpha-1.$

This article uses $\alpha$ for the density exponent throughout, the convention of Clauset, Shalizi and Newman and of most network science. But economics and insurance, following Pareto himself, usually report the tail exponent and call it $\alpha$ . So “Pareto’s alpha is about 1.5” and “the fitted alpha is about 2.5” can describe the same data. Whenever you meet a reported exponent, ask which function it belongs to; if it is below 1 it is almost certainly a tail exponent, since a density exponent that small cannot be normalised.

Zipf’s law adds a third quantity again: if the $r$ -th commonest word has frequency $f(r)\propto r^{-s}$ , the number of types with frequency above $f$ scales as $f^{-1/s}$ , so $\gamma = 1/s$ and $\alpha = 1+1/s$ . Zipf’s $s \approx 1$ means $\alpha \approx 2$ . (The discrete analogue of the Pareto law is the zeta distribution, $p(k) = k^{-\alpha}/\zeta(\alpha)$ .)

Scale invariance, derived

Rescale the density:

$p(cx) = C(cx)^{-\alpha} = c^{-\alpha}\cdot Cx^{-\alpha} = c^{-\alpha}p(x).$

The rescaled density is the original times a constant depending on $c$ but not on $x$ — on log-log axes a vertical shift of a straight line, leaving the slope, which carries all the information, unchanged.

Nothing else does this. Suppose $p(cx) = g(c)\,p(x)$ for all $c, x > 0$ . Put $x=1$ : then $p(c) = g(c)p(1)$ , so $g(c) = p(c)/p(1)$ , and substituting back gives $p(cx)p(1) = p(c)p(x)$ . Writing $f(u) = p(u)/p(1)$ , this is the multiplicative Cauchy equation $f(cx) = f(c)f(x)$ , whose only solutions under any mild regularity condition one would want of a density are the power functions $f(u) = u^{-\alpha}$ . Hence $p(x) = p(1)x^{-\alpha}$ : the power laws are not merely an example of a scale-invariant family but the entire scale-invariant family.

This matters for what it rules out. Any argument of the form “this mechanism has no preferred scale” is, if correct, an argument for a power law specifically and not for heavy tails in general — and it makes scale invariance a strong, testable hypothesis rather than a vague one, by the same functional-equation reasoning that pins down the scale-invariant digit distribution in Benford’s law.

Moments, and the standard deviation that never settles

The $k$ -th moment is one integral:

$\mathbb{E}[X^k] = \int_{x_{\min}}^{\infty}\! x^k\,\frac{\alpha-1}{x_{\min}}\left(\frac{x}{x_{\min}}\right)^{-\alpha}dx = (\alpha-1)x_{\min}^{\alpha-1}\!\int_{x_{\min}}^{\infty}\! x^{k-\alpha}dx.$

The integrand is a pure power, so convergence is decided by its exponent: $\int^\infty x^{k-\alpha}dx$ converges if and only if $k-\alpha < -1$ , that is

$k < \alpha - 1 = \gamma,$

in which case $\mathbb{E}[X^k] = \frac{\alpha-1}{\alpha-1-k}x_{\min}^k$ . Existence of moments is governed entirely by one number:

Density exponent $\alpha$	Tail exponent $\gamma$	Mean	Variance
$\alpha \le 2$	$\gamma \le 1$	infinite	infinite
$2 < \alpha \le 3$	$1 < \gamma \le 2$	finite	infinite
$\alpha > 3$	$\gamma > 2$	finite	finite

The middle row is where a great many fitted exponents land — city sizes, firm sizes, upper-tail wealth, many network degree distributions. There the distribution has a well-defined mean and no variance at all. Operationally: take $x_{\min}=1$ and $\alpha = 2.5$ , so $\gamma = 1.5$ and the mean is $(\alpha-1)/(\alpha-2) = 1.5/0.5 = 3$ . Median sample statistics over repeated simulated samples:

Sample size $n$	Median sample mean	Median sample s.d.
100	2.70	3.58
1,000	2.82	5.70
10,000	2.95	9.82
100,000	2.97	14.93
1,000,000	2.98	19.66

The sample mean converges to 3, as the law of large numbers promises. The sample standard deviation does the opposite, growing by a factor of 5.5 while the sample grows four orders of magnitude. That is not noise: for $1<\gamma<2$ it scales as $n^{1/\gamma-1/2}$ , here $n^{1/6}$ , predicting $10^{1/6}\approx 1.47$ per decade and $1.47^4 \approx 4.6$ over four decades.

The blunt version: a sample standard deviation computed on infinite-variance data has not estimated anything. It is a function of how much data you collected, and more data makes it bigger. Every quantity built on it — coefficient of variation, standard error, two-sigma interval, Sharpe ratio — inherits that emptiness. The parallel warning about the mean is in mean vs median; the behaviour of the maximum that actually drives these samples belongs to extreme value theory.

The eighty-twenty rule, computed properly

Work with $\gamma$ and $x_{\min}=1$ , so $P(X>x) = x^{-\gamma}$ . Let $q$ be the fraction of observations in the top group. Its threshold satisfies $x_q^{-\gamma} = q$ , so $x_q = q^{-1/\gamma}$ . The mass above it is

$\int_{x_q}^{\infty}\! x\,p(x)\,dx = \gamma\!\int_{x_q}^{\infty}\! x^{-\gamma}dx = \frac{\gamma}{\gamma-1}x_q^{1-\gamma}$

for $\gamma>1$ , while the total mass is the mean $\gamma/(\gamma-1)$ . Dividing, the share held by the top $q$ fraction is

$S(q) = x_q^{1-\gamma} = q^{\,1-1/\gamma}.$

Set $S(0.2)=0.8$ :

$1 - \frac{1}{\gamma} = \frac{\ln 0.8}{\ln 0.2} = \frac{-0.22314}{-1.60944} = 0.13865 \ \Longrightarrow \ \gamma = 1.1610, \quad \alpha = 2.1610.$

Eighty-twenty is one specific exponent, not a general property. With it, the rest is forced:

Top fraction $q$	Share $S(q) = q^{0.13865}$
20%	80.0%
4%	64.0%
1%	52.8%
0.1%	38.4%
0.01%	27.9%

Two things. First, the self-similarity: the top four percent hold exactly $0.8\times 0.8 = 64$ percent, because the top fifth of the top fifth is the top twenty-fifth. That recursion is scale invariance made visible, and it is the only genuinely remarkable feature of the eighty-twenty framing. Second, $\gamma = 1.161$ sits well below 2: a distribution that really obeys the eighty-twenty rule has infinite variance. Change the exponent and the rule dissolves — at $\gamma=1.5$ , roughly what Pareto fitted, the top fifth hold $0.2^{1/3} = 58.5$ percent; at $\gamma = 2$ , $0.2^{1/2} = 44.7$ percent.

The history needs care, because the usual telling is partly invented. Vilfredo Pareto’s Cours d’économie politique (1896–97) contains the regularity that named the distribution: plotting the number of taxpayers with income above a level against that level gave an approximately straight log-log relationship, fitted across several countries rather than Italy alone. His subject was income and wealth, not land. The widely repeated claim that he found eighty percent of Italian land held by twenty percent of the population appears to be a later embellishment, as does the story about his garden peas.

The ratio and the name came from Joseph Juran, who had noticed in the 1920s that a few defect types produced most defects in quality control, generalised it to the vital few and the trivial many, and attached Pareto’s name in his 1951 Quality Control Handbook. He later retracted the attribution — “The Non-Pareto Principle: Mea Culpa”, Quality Progress, mid-1970s — conceding he had put the wrong name on the principle and noting that it had stuck anyway.

Mechanisms: where power laws come from

A fitted exponent is a description; a mechanism is an explanation. Three families are reasonably well understood.

Preferential attachment. Udny Yule, in “A Mathematical Theory of Evolution” (Philosophical Transactions of the Royal Society B, 1925), modelled species counts per biological genus by letting new species attach to genera in proportion to their current size. Herbert Simon generalised the argument in 1955 to word frequencies, city sizes and incomes. Barabási and Albert applied it to growing networks in 1999: add one node at a time with $m$ edges, attaching each to an existing node with probability proportional to its current degree. The asymptotic degree distribution is a power law with density exponent

$\alpha = 3,$

independent of $m$ . In outline: a node added at time $t_i$ has expected degree $k_i(t)\approx m(t/t_i)^{1/2}$ , so the fraction of nodes with degree above $k$ is the fraction with $t_i < m^2t/k^2$ , giving $\gamma = 2$ and $\alpha = 3$ . The result is asymptotic and does not hold for arbitrary finite networks generated along the way — which matters when the measured networks are small. Note too that $\alpha = 3$ sits exactly on the boundary of finite variance.

Criticality. At a critical point the correlation length diverges, and with no finite correlation length there is no characteristic size for the structures that form; by the functional-equation argument above, the only compatible distributions are power laws. Hence cluster sizes at the percolation threshold are power-law distributed, and “critical” and “scale-free” are near-synonyms in statistical mechanics. Percolation theory shows the mechanism with exponents derived rather than fitted. Its weakness as an account of natural data is that the system must be at the critical point, and most are not.

Optimisation. Power laws also fall out of constrained maximisation: minimising average cost in a branching transport network, minimising expected search time over a vocabulary with a transmission cost, maximising entropy under a constraint on the expected logarithm. Sound, and empirically underdetermined — they give the functional form without pinning the exponent.

Now the alternative that should be the default. If a quantity grows multiplicatively, $X_{t+1}=\eta_tX_t$ with independent random factors, then $\ln X_t$ is a sum of independent increments, so by the central limit theorem it tends to normality and $X_t$ tends to lognormal. This is Gibrat’s law, a reasonable first model for incomes, firm sizes and city populations, as the mechanics of log scales make clear.

Add one ingredient: a lower barrier below which the quantity cannot fall. Champernowne showed in 1953 (Economic Journal 63, 318–351) that multiplicative income dynamics with a floor at a minimum income has a Pareto stationary distribution, and Gabaix later showed that geometric Brownian motion with a lower reflecting boundary gives city sizes an upper-tail exponent slightly above one. Unbounded multiplicative drift spreads the log of the variable out forever and never settles; a floor supplies a reflecting condition, the stationary density becomes a power of the variable, and the exponent is set by the ratio of drift to variance.

So: multiplicative growth with a lower barrier gives a power law; multiplicative growth without one gives a lognormal. Two nearly identical stories, differing in a detail usually invisible in the data — and producing two distributions that are notoriously hard to separate.

Lognormal versus power law: the identification problem

Write the lognormal density as a function of $u = \ln x$ :

$\ln p(x) = -\frac{(u-\mu)^2}{2\sigma^2} - u - \ln\!\left(\sigma\sqrt{2\pi}\right).$

On log-log axes a lognormal is a downward parabola in $u$ with curvature

$\frac{d^2\ln p}{du^2} = -\frac{1}{\sigma^2},$

so the apparent exponent, $-1-(u-\mu)/\sigma^2$ , drifts by $D\ln(10)/\sigma^2$ across a window of $D$ decades. With $\sigma = 3$ , plausible for a highly dispersed economic quantity, three decades give a drift of $3\times 2.303/9 = 0.77$ ; with $\sigma = 2$ , 1.73.

A drift of 0.77 in the fitted slope over three decades, against the sampling noise of a sparse tail, is invisible to the eye — and two or three decades is about what most datasets span. Hence the central methodological fact: a lognormal with large log-variance is straight on log-log axes, to the precision of any eyeball judgement, over the range of data you are likely to have. Separating the two needs either many more decades or a formal likelihood comparison. Seeing a line on a log-log plot is not evidence of a power law; it is evidence of a right-skewed distribution, which was never in doubt. Clauset, Shalizi and Newman put it about as strongly as a paper can: over realistic ranges of $x$ the two are very closely equal, and they judged it unlikely that any test could separate them without an extremely large dataset.

How to test for a power law, and why most claims fail

What not to do. The traditional method bins the data, takes logs of both axes, and runs least squares; the slope is the exponent and $R^2$ is offered as evidence. It dates back to Pareto’s own work and is wrong in four independent ways. The exponent estimate is biased, substantially, in a direction set by the binning: tail bins hold few counts with large relative fluctuations, and empty bins are usually discarded, which removes the low side of those fluctuations and flattens the fit. Least squares assumes independent, homoscedastic, Gaussian errors on log-counts, whereas bin counts are roughly Poisson, so their log-scale variance grows as the counts fall and none of the standard-error machinery applies. The reported $R^2$ is not a test: high log-log correlation is equally consistent with a lognormal, a stretched exponential and a power law with a cutoff. And the procedure silently fixes $x_{\min}$ at the first bin edge, though the exponent is extremely sensitive to it.

What to do instead is the procedure of Aaron Clauset, Cosma Shalizi and Mark Newman, “Power-law distributions in empirical data”, SIAM Review 51(4), 661–703 (2009), in four steps.

Estimate $\alpha$ by maximum likelihood. Above a known $x_{\min}$ the Pareto log-likelihood is maximised at the Hill estimator

$\hat{\alpha} = 1 + n\left[\sum_{i=1}^{n}\ln\frac{x_i}{x_{\min}}\right]^{-1},$

with asymptotic standard error $(\hat{\alpha}-1)/\sqrt{n}$ . Worked example, ten observations with $x_{\min}=1$ : 1.2, 1.5, 2.0, 2.6, 3.1, 4.4, 6.0, 9.5, 18.0, 40.0. Their logarithms are 0.1823, 0.4055, 0.6931, 0.9555, 1.1314, 1.4816, 1.7918, 2.2513, 2.8904, 3.6889, summing to 15.4718, so $\hat{\alpha} = 1 + 10/15.4718 = 1.646$ with standard error $0.646/\sqrt{10} = 0.204$ . Ten observations buy an exponent known to about plus or minus 0.2 — the width of the gap between finite and infinite variance.

Choose $x_{\min}$ by minimising the Kolmogorov–Smirnov distance. For each candidate cutoff, fit $\hat\alpha$ above it and compute $D = \max_x \lvert S(x) - \hat{P}(x)\rvert$ between empirical and fitted CDFs; take the minimiser. This trades bias against variance automatically.

Compute a goodness-of-fit p-value. Generate many synthetic datasets from the fitted model, refit each from scratch, and record the fraction whose KS distance exceeds the observed one. Here a large value is the favourable one: the power law is the hypothesis under test, not the null. Clauset and colleagues use a threshold of 0.1 and warn that with few data points every model passes.

Compare against alternatives by likelihood ratio. A power law can pass its own goodness-of-fit test and still be the worse model. Vuong’s test gives the log-likelihood ratio against a competitor plus a p-value for whether its sign is meaningful. The competitors that matter are the lognormal, the exponential, the stretched exponential and the power law with an exponential cutoff.

What happens when you do this. Clauset and colleagues applied the procedure to 24 datasets previously conjectured to be power-law distributed. Stated as they stated it:

By goodness-of-fit alone, 17 of the 24 were consistent with a power law; the other seven had p-values small enough to rule it out firmly — HTTP connection sizes, earthquake intensities, web links, forest fire sizes, wealth, web hits and metabolic network degree.
There was only one case where the power law appeared truly convincing, an excellent fit with no alternative carrying any weight: the frequencies of words in English text.
The exponential could be ruled out in all cases but three, and the Poisson for every discrete dataset. But the lognormal was not ruled out for any dataset except one (HTTP connections): bird abundances, book sales, city populations, religious adherents, war intensities, citations, papers authored, protein degree and terrorism severities were all plausible power laws and plausible lognormals and stretched exponentials.
For roughly a dozen sets — forest fires, solar flares, earthquakes, web hits, web links, telephone calls, Internet degree, email address books, mammal species — the power law with an exponential cutoff was clearly favoured over the pure form.

Neither inflate that nor soften it. It does not say power laws are a myth. It says most published power-law claims, tested properly, cannot be distinguished from a lognormal; that a substantial minority are better described by a power law with a cutoff, which has a characteristic scale and so is not scale-free; and that a handful are simply wrong.

The network case stayed contentious. Anna Broido and Aaron Clauset, “Scale-free networks are rare” (Nature Communications, 2019), applied this machinery to nearly a thousand networks and concluded that strongly scale-free structure is empirically rare, with lognormals fitting most of them as well or better. Rebuttals followed immediately: Barabási circulated a response disputing the conclusions, and Voitalov, van der Hoorn, van der Hofstad and Krioukov argued in “Scale-free networks well done” that a regular-variation framework licenses a less restrictive definition under which far more networks qualify. Commentators broadly accepted the statistics while disputing what should count as scale-free — a disagreement about definitions as much as about data. It remains a live dispute, and either side reported as the final word is an overreach.

Where the evidence is reasonably good

Scepticism about the general claim is compatible with specific cases being well supported. Four, in descending order of confidence.

Word frequencies. Zipf’s law is the best-attested case in the Clauset survey and the only one they called convincing. Word-type frequencies in Moby Dick gave $\hat{\alpha} = 1.95 \pm 0.02$ with $\hat{x}_{\min} = 7 \pm 2$ , about 2,958 types in the fitted tail out of 18,855, and $p = 0.49$ . A density exponent near 2 is exactly the Zipf rank exponent $s\approx 1$ translated by $\alpha = 1+1/s$ , and Simon’s 1955 account supplies a mechanism. Note that $\hat\alpha < 2$ implies infinite mean — the fit describes a tail, not the whole vocabulary.

Earthquakes. The Gutenberg–Richter law, $\log_{10}N(\ge M) = a - bM$ with $b \approx 1$ (empirically 0.75 to 1.2), is among the most robust regularities in geophysics. Because moment magnitude is itself a logarithm of seismic moment, $M_w = \tfrac{2}{3}\log_{10}M_0 - 10.7$ , an exponential law in magnitude is a power law in energy: the exceedance exponent in moment is $\gamma = 2b/3 \approx 0.67$ , the density exponent $\alpha = 1+2b/3 \approx 1.67$ . With $\gamma<1$ , total released moment has no finite mean unless truncated at a maximum magnitude, which physically it must be. Two caveats: the exponent depends entirely on whether you count magnitude, amplitude or moment; and the Clauset survey’s California dataset, measured as ground-motion amplitude, failed its goodness-of-fit test outright ( $p = 0.00$ , $\hat\alpha = 1.64$ ).

City sizes. The survey fitted $\hat{\alpha} = 2.37 \pm 0.08$ above a cutoff near 52,000 inhabitants, with about 580 cities in the tail and $p = 0.76$ — a good fit. But the lognormal was not excluded, and Gibrat versus Champernowne predicts exactly those two competing shapes. Zipf’s law for cities would need $\gamma = 1$ ; the fitted $\hat\gamma = 1.37$ is steeper, while Gabaix’s reflecting-barrier model predicts slightly above 1. The tail is clearly heavy, clearly consistent with a power law, and not demonstrated to be one.

Upper-tail wealth. Pareto’s original case, still plausible with a qualification. The survey fitted the net worth of the 400 richest Americans in October 2003 at $\hat{\alpha} = 2.3 \pm 0.1$ above a cutoff near nine hundred million dollars — but with $p = 0.00$ , so the pure power law was ruled out there even though it beat the exponential. Fitted top-tail exponents from tax and estate data cluster between roughly 1.5 and 3, spanning both the finite- and infinite-variance regimes. The tail is heavy; the functional form is not established.

What it costs to get this wrong

Variance-based risk measures become meaningless. If losses have $\alpha < 3$ there is no population variance for a sample variance to estimate, so a volatility figure, a standard error, or a capital requirement set as a multiple of a standard deviation is a statement about your sample size. A category error, not a small correction.

Convergence of the mean slows or fails. For $\alpha>2$ the law of large numbers still holds, but for $\alpha \le 3$ the central limit theorem does not apply in its usual form: sample means converge to a stable law of index $\gamma$ at rate $n^{1-1/\gamma}$ rather than $n^{1/2}$ . At $\gamma = 1.5$ that is $n^{1/3}$ — halving your error takes eight times the data, not four. For $\alpha \le 2$ there is no population mean at all.

One observation can dominate the total. Take $n$ draws from a Pareto tail. The $k$ -th largest sits near the value exceeded by $k$ observations, $x_{(k)} \approx x_{\min}(n/k)^{1/\gamma}$ , so summing typical values,

$S_n \approx x_{\min}n^{1/\gamma}\sum_{k=1}^{n}k^{-1/\gamma}, \qquad M_n \approx x_{\min}n^{1/\gamma},$

and the largest observation’s share of the total is approximately

$\frac{M_n}{S_n} \approx \frac{1}{\sum_{k\ge 1}k^{-1/\gamma}} = \frac{1}{\zeta(1/\gamma)},$

independent of $n$ . The sum converges only when $1/\gamma>1$ , that is $\gamma<1$ — exactly the infinite-mean condition.

$\gamma$	$1/\gamma$	$\zeta(1/\gamma)$	Largest share of total
0.5	2.000	1.645	60.8%
0.67	1.500	2.612	38.3%
0.8	1.250	4.595	21.8%
0.9	1.111	9.585	10.4%

At the earthquake exponent $\gamma\approx 0.67$ , the single largest event accounts for something like a third to a half of all energy released, however long the record. Simulations with a hundred thousand draws give about 0.48 there, above the asymptotic estimate because convergence is slow, but the point is robust: lengthening the record does not dilute the largest event, it replaces it with a bigger one. Aggregate statistics on such data are statements about one observation in disguise.

Selection makes it worse. When your dataset consists of the entities that grew large enough to be recorded — funded startups, surviving funds, cities with names — you are looking at a sample conditioned on being in the tail. That conditioning can manufacture apparent power-law behaviour out of distributions that have none, by the same mechanism that drives survivorship bias in performance records. A tail exponent fitted to a selected sample estimates the selection rule at least as much as the process.

Tails are where you have the least data

The closing point is epistemic rather than mathematical. A claim about the shape of a tail is a claim about rare events, and rare events are by construction the observations you have fewest of. In the Clauset survey the fitted tails typically held a few hundred points out of datasets of tens or hundreds of thousands: Internet degree contributed about 770 tail observations from 22,688 nodes, city populations about 580 from 19,447. The exponent, which carries every consequence above, comes from that handful, and a standard error of $(\hat\alpha-1)/\sqrt{n}$ with $n$ in the hundreds runs to 0.05 or 0.1 at best — enough to leave the boundary between finite and infinite variance genuinely uncertain in many published fits.

There is nothing wrong with power laws. Scale invariance is a rigid and beautiful property, the functional equation characterising it admits no other solutions, the moment arithmetic is exact, and the generating mechanisms are real. What is wrong is the confidence. A straight-ish line over two decades on log-log axes — which is what most power-law claims amount to — is compatible with at least four distributions that differ enormously exactly where it matters, beyond the data the model was fitted to describe. Extrapolating a tail is the most consequential and least constrained thing a statistician does, and the appropriate posture is the one the subject has been slow to adopt: fit the model, test it against its rivals, report that the rivals could not be excluded, and say what follows under each.

Frequently asked

What is a power law distribution?

A power law distribution is one whose probability density falls off as a fixed negative power of the variable, so that p(x) is proportional to x to the minus alpha above some lower cutoff. Its defining property is scale invariance: multiplying x by a constant multiplies the density by a constant but leaves its shape unchanged, so the distribution has no characteristic size. This is the opposite of the normal distribution, where the standard deviation fixes a scale and observations far beyond it are effectively impossible.

What is the 80-20 rule?

The 80-20 rule, or Pareto principle, is the observation that a small share of cases accounts for most of the total. For a Pareto distribution the top q fraction of observations holds a share equal to q raised to the power one minus one over gamma, where gamma is the tail exponent. Setting that share to 0.8 at q equal to 0.2 gives gamma about 1.161. The 80-20 split is therefore one particular exponent rather than a law of nature, and Pareto himself typically fitted exponents near 1.5, which gives roughly a 58-20 split instead.

What is the difference between a power law and a lognormal distribution?

A power law has a density that is exactly straight on log-log axes, while a lognormal has a density whose log-log plot is a downward parabola with curvature equal to one over the variance of the logarithm. When that variance is large the curvature is tiny, so a lognormal looks straight over two or three decades. The generating mechanisms differ too: multiplicative random growth alone produces a lognormal, whereas multiplicative growth with a lower barrier produces a power law.

Why do power laws have infinite variance?

The k-th moment of a power law is an integral of x to the power k minus alpha, which converges at infinity only when k is less than alpha minus one. With the density exponent alpha between 2 and 3 the mean exists but the second moment does not, so the variance is infinite. The practical consequence is that a sample standard deviation computed on such data is not estimating anything: it grows without bound as the sample size increases rather than settling down.

How do you test whether data follows a power law?

Fit the exponent by maximum likelihood using the Hill estimator rather than by regression on a log-log histogram, choose the lower cutoff by minimising the Kolmogorov-Smirnov distance between the fitted and empirical distributions, compute a goodness-of-fit p-value by comparing the observed KS distance against synthetic data drawn from the fitted model, and then run likelihood-ratio tests against the lognormal, the exponential and the power law with an exponential cutoff. This is the procedure of Clauset, Shalizi and Newman, published in SIAM Review in 2009.