Power Law Distribution: Definition, Formula, Worked Example and How to Fit It

A power law distribution is a probability distribution whose tail falls off as a power of the value, so that large observations are rare but far more common than a bell curve allows. This guide gives the formula with every symbol explained, a fully worked estimate of the exponent, the moment conditions that decide whether a mean or variance even exists, and the checks that stop a straight line on a log-log plot from being mistaken for proof.

Definition of a power law distribution

A quantity x follows a power law when its probability density has the form p(x) = C x^−α, with a constant C and a scaling exponent α (Clauset, Shalizi and Newman, 2009). Because this density diverges as x approaches zero, the form can only hold above a lower bound x_min; the distribution describes the tail of the data, not necessarily all of it. A power law is also called a scale-free distribution: Newman (2005) notes that it is the only distribution that looks the same whatever scale it is viewed on, which is why rescaling x changes the constant but not the shape.

Power law formula

For a continuous variable with α > 1, normalising the density over x ≥ x_min gives (Clauset et al., 2009, Eq. 2.2):

p(x) = ((α − 1) / x_min) · (x / x_min)^−α

• p(x): the probability density at value x.
• α: the scaling exponent (α > 1). Larger α means a thinner tail.
• x_min: the smallest value for which the power law is assumed to hold.

Integrating the density gives the complementary cumulative distribution function (CCDF), the probability of exceeding x: P(X ≥ x) = (x / x_min)^{−(α − 1)}. Plotting the CCDF on log-log axes is preferred to a linear-bin histogram, which Newman (2005) warns against because the sparse tail bins are noisy.

Power law vs Pareto vs Zipf

The Pareto distribution is the same family written for the CCDF: if P(X ≥ x) = (x / x_min)^−k, the density exponent is α = k + 1. Zipf’s law is the rank-frequency version for discrete data; see the CASRAI guide to Zipf’s distribution. Pareto-type tails that appear in extreme-value modelling are covered in the guide to the generalized Pareto distribution. A log-normal distribution can look almost straight over a few orders of magnitude on a log-log plot while being a different model.

Worked example: estimating the exponent

The continuous maximum-likelihood estimator is (Clauset et al., 2009, Eq. 3.1):

α̂ = 1 + n · [ Σ ln(x_i / x_min) ]⁻¹, with standard error σ = (α̂ − 1) / √n.

Here n is the number of observations with x_i ≥ x_min. The ten values below are an illustrative set chosen to show the arithmetic, not real data. Take x_min = 1.

xi	1.2	1.5	1.8	2.4	3.1	4.5	6.8	11	24	60
ln(xi)	0.1823	0.4055	0.5878	0.8755	1.1314	1.5041	1.9169	2.3979	3.1781	4.0943

Because x_min = 1, ln(x_i/x_min) = ln(x_i). The sum is 16.274. Then α̂ = 1 + 10 / 16.274 = 1 + 0.6145 = 1.614, and the standard error is 0.614 / √10 = 0.614 / 3.162 = 0.194.

In real work x_min is not given. Clauset et al. (2009) choose the x_min that minimises the Kolmogorov-Smirnov distance between the empirical tail and the fitted power law, refitting α̂ for each candidate. Discrete data, such as counts, use a normalising constant based on the Hurwitz zeta function and have no closed-form estimator, so a numerical routine is needed.

Interpretation: what α means for the mean and variance

Integrating x·p(x) and x²·p(x) from x_min to infinity shows when the moments exist. For the density above, the mean is x_min(α − 1)/(α − 2), which is finite only when α > 2, and the variance is x_min²(α − 1)/[(α − 2)²(α − 3)], finite only when α > 3 (derived from the formula above).

Exponent α	Mean	Variance	Reading
1 < α ≤ 2	Does not exist (diverges)	Does not exist	Sample mean keeps growing with sample size
2 < α ≤ 3	Finite: xmin(α − 1)/(α − 2)	Does not exist	Mean is usable; standard deviation and ordinary confidence intervals are not
α > 3	Finite	Finite	Classical moment-based summaries apply, though the tail is still heavy

The worked estimate α̂ = 1.61 falls in the first row, so a sample mean of those values would not settle on a stable number. With only ten points this is an illustration of the reading, not a finding about any real system.

Assumptions and common mistakes

• A straight log-log line is not evidence. Clauset et al. (2009, Appendix A) show that least-squares fits to a log-log histogram give biased exponents and that a high r² can occur even when the data are not a power law.
• Regression standard errors do not apply to those fits, because the noise in the logarithm of the bin counts is not Gaussian.
• Test, then compare. Their recipe fits α and x_min, computes a goodness-of-fit p-value (a p-value above 0.1 makes the power law plausible; otherwise it is rejected), then compares it with alternatives such as the exponential or log-normal by a likelihood-ratio test.
• Ignoring x_min treats the whole data set as a power law when only the tail may be.
• Long tail is not the same as power law. Many heavy-tailed distributions have a long tail; only some have a power-law one.

To simulate power-law data for checking a fitting routine, Newman (2005) gives x = x_min(1 − r)^{−1/(α − 1)} for r drawn uniformly on [0, 1). For related diagnostics, see the guides to Q-Q plots and the generalized extreme value distribution.

How to report a power law fit (APA example)

Report the model, the estimator, x_min and how it was chosen, n, the estimate with its standard error, the goodness-of-fit result and the alternatives tested. Example wording using the illustrative numbers above:

A continuous power-law model was fitted by maximum likelihood to the n = 10 observations at or above x_min = 1, giving α̂ = 1.61 (SE = 0.19). Because α̂ ≤ 2, the mean of the fitted distribution is not finite.

Use italics for statistical symbols such as n and SE, and state the software and version used. Add the goodness-of-fit p-value and the likelihood-ratio comparison whenever you claim a power law.

How to calculate it in Excel, R or Python

Excel (values in A2:A11, x_min in D1; exponent and standard error):

=1+COUNT(A2:A11)/SUMPRODUCT(LN(A2:A11/D1))
=(D2-1)/SQRT(COUNT(A2:A11))   (with the exponent in D2)

R:

x <- c(1.2,1.5,1.8,2.4,3.1,4.5,6.8,11,24,60); xmin <- 1
n <- length(x); alpha <- 1 + n / sum(log(x / xmin))
se <- (alpha - 1) / sqrt(n); c(alpha, se)   # 1.614 0.194

Python (NumPy):

import numpy as np
x = np.array([1.2,1.5,1.8,2.4,3.1,4.5,6.8,11,24,60]); xmin = 1.0
n = len(x); alpha = 1 + n / np.sum(np.log(x / xmin))
se = (alpha - 1) / np.sqrt(n)
print(round(alpha, 3), round(se, 3))   # 1.614 0.194

These cover the continuous case with a known x_min. Clauset’s accompanying code and the Python powerlaw package handle x_min selection, discrete data and likelihood-ratio tests.

FAQ

What is a power law distribution?

It is a distribution whose density above a lower bound x_min is proportional to x^−α. Frequency drops as a fixed power of size, so the tail is heavy and has no characteristic scale.

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

They describe the same family. The Pareto form is usually written for the CCDF with exponent k; the power-law density exponent is α = k + 1.

Does every power law have a finite mean?

No. The mean is finite only when α > 2 and the variance only when α > 3.

How do I know if my data follow a power law?

Fit α and x_min by maximum likelihood, test goodness of fit, and compare against alternatives such as log-normal, rather than relying on a straight line or r².

Sources

• Clauset, A., Shalizi, C. R., and Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661-703. arxiv.org/abs/0706.1062 (read 7 November 2026).
• Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf’s law. Contemporary Physics, 46(5), 323-351. arxiv.org/abs/cond-mat/0412004 (read 7 November 2026).