"Some heavy-tailed data look approximately power-law over a range. Recognizing the candidate is useful; declaring it from a log-log line is not."
Where Power Laws Are Considered
City sizes, earthquake measures, wealth tails, book sales, online degree counts, and network flow sizes can show strong skew and heavy tails. Power-law behavior is one candidate for some ranges of some datasets. Lognormal, stretched-exponential, truncated power-law, and mixture models can look similar in finite samples.
Define the Exponent Before Using It
For a Pareto tail written as P(X > x) ∝ x⁻ᵅ, α is the tail index. Under this convention the mean is finite only when α>1 and the variance only when α>2. If a source defines the density exponent instead, the thresholds shift by one. Always check the parameterization.
Power-law tails decay more slowly than exponential or normal tails. Extreme values remain rare, but they are less rare than a thin-tailed model predicts.
The 80/20 Rule Is Not a Law of Nature
The Pareto principle is a mnemonic for concentration, not a universal empirical ratio. An exact Pareto distribution can be scale-free over its modeled range, but the share held by the top fraction depends on α and on cutoffs. Real systems often have lower and upper scales, regime changes, and finite-size limits.
Means, Variances, and Honest Summaries
When the theoretical mean or variance is infinite, familiar convergence and standard-error formulas can fail. In finite real datasets, the sample mean is still a computable number, but it may be highly unstable and dominated by a few observations. Even when moments are finite, convergence can be slow.
That does not make reporting a mean malpractice. Means answer questions about totals and expectations. Pair them with medians, quantiles, tail shares, uncertainty, and a fitted-tail analysis suited to the decision.