Human height and personal wealth are both "distributions of a quantity across people", and they could hardly behave more differently. The tallest adult on record was a little under five times as tall as the shortest. The richest person alive has millions of times the wealth of a typical one.
The first is a normal distribution: outcomes cluster around an average and extremes are vanishingly rare. The second is a power law: most outcomes are small, a few are enormous, and those few dominate the total. Knowing which one you are dealing with changes almost every decision you make with the data.
Where bell curves come from
The central limit theorem explains the bell curve: add up many small, independent effects and the total tends towards a normal distribution, whatever the shape of each individual effect. Height is the sum of thousands of genetic and environmental nudges. Measurement error is the sum of many small disturbances. Both come out bell-shaped.
In a normal world, the rules of thumb are reassuring:
- About 68% of outcomes fall within one standard deviation of the mean, 95% within two, and 99.7% within three.
- A five-standard-deviation event in one direction has odds of roughly one in 3.5 million.
- The average is a stable, meaningful summary, and it settles down quickly as you collect data.
Where long tails come from
Power laws arise when effects feed on themselves instead of adding up independently:
- Rich get richer. In preferential attachment, new links go to nodes in proportion to the links they already have. Popular websites, well-cited papers and big cities grow faster because they are already big.
- Self-organised criticality. Systems such as forests, sandpiles and fault lines build up stress until a small trigger releases it. Most releases are tiny and a few are catastrophic, which is why earthquake magnitudes follow a power law.
- Multiplication. When independent factors multiply instead of add, the logarithm of the outcome is a sum, so the outcome itself is lognormal: another heavy-tailed shape that is easy to mistake for a power law.
Why the difference is expensive
For a power law with density proportional to x^-α, the mean is infinite when α ≤ 2 and the variance is infinite when α ≤ 3. In practice, "infinite" means the sample average never settles: each new extreme observation drags it somewhere new. Book sales, venture returns, insurance losses, wildfire sizes and software outages all behave this way.
Treat such data as normal and you will plan for the average and be wiped out by the tail. A risk model that calls a loss a "25-sigma event" is usually telling you that it assumed the wrong distribution, not that the world was unlucky.
A five-step test for your data
- Look at the histogram on ordinary axes. A symmetric hump suggests normal. A tall spike near zero with a long right tail suggests something heavy-tailed.
- Plot the tail on log-log axes. Use the complementary cumulative distribution: the share of observations larger than x. A roughly straight line over two or more orders of magnitude hints at a power law. A line that bends downwards suggests a lognormal or an exponential cut-off.
- Watch the running mean. Compute the average of the first 100, 1,000 and 10,000 observations. If it keeps jumping when a single large value arrives, the tail is heavy enough to matter.
- Fit properly. Don't fit a straight line to a log-log histogram by least squares; the estimate is biased. Use maximum likelihood for the exponent and compare the fit against a lognormal with a likelihood-ratio test, as Clauset, Shalizi and Newman describe in "Power-law distributions in empirical data" (2009).
- Ask about the mechanism. Are the effects additive and independent, or multiplicative and self-reinforcing? The story behind the data is often more reliable than a curve fit on a few hundred points.
Mixed and messy cases
Real datasets are often normal in the body and heavy in the tail, or they are a mixture of groups that each look normal on their own. A two-humped histogram usually means you are measuring two populations at once, for example new and returning customers. Split the data and test each group separately before reaching for an exotic distribution.
Deciding what to do
| If your data is… | Summarise with… | Plan for… |
|---|---|---|
| Normal | Mean and standard deviation | The average case, with a margin of a few standard deviations |
| Power law | Median, quantiles and the largest observations | The tail: caps, diversification, stress tests and insurance |
The distribution-checker skill walks an agent through the same decision. Before you summarise any dataset with an average, spend five minutes finding out which world it comes from.