Fat Tails
A fat-tailed distribution has higher probability of extreme events than a normal distribution. The normal distribution’s tails drop exponentially — events more than four standard deviations from the mean are vanishingly rare (1 in 31,574). Fat-tailed distributions drop more slowly, following power laws. Extreme events remain uncommon but possible, even dominant.
The difference matters enormously. In a normal distribution, the extremes contribute little to the average. In fat-tailed distributions, the extremes contain most of the action. A single observation can exceed all previous observations combined. The largest earthquake, the worst pandemic, the biggest financial crash — these dominate their categories.
Benoit Mandelbrot documented fat tails in cotton prices in 1963. Price changes didn’t follow the bell curve — large moves happened far more often than the normal distribution predicted. The October 1987 crash was a 20+ standard deviation event, theoretically impossible. The 2008 crisis included several “impossible” daily moves. The models failed because reality was fatter-tailed than the models assumed.
Power laws generate fat tails. If the probability of exceeding value x falls as x^(-α), you get a fat-tailed distribution. With α < 2, variance is infinite — no sample gives a reliable estimate of spread. With α < 1, even the mean is infinite. Real-world phenomena with power-law behavior: city sizes, word frequencies, website traffic, earthquake magnitudes, forest fire sizes, war deaths.
Fat tails break standard statistical intuitions. Sample averages don’t converge reliably to population averages. More data doesn’t necessarily give better estimates — a single outlier can dominate. Historical data understates future extremes because the biggest event hasn’t happened yet.
The “Turkey problem” captures this: a turkey fed every day learns that farmers are benevolent. Each day of evidence strengthens the belief. Then comes Thanksgiving. The turkey had no way to infer the extreme event from the history. Fat-tailed domains are Turkey domains.
Nassim Taleb argues that most social and economic phenomena live in “Extremistan” — fat-tailed domains where standard risk models fail. Finance, entrepreneurship, media, technology, war. The rare extreme event matters more than typical events. Planning for the average guarantees surprise at the extreme.
The strategic implications: avoid fragility to negative extremes, seek exposure to positive extremes. Don’t optimize for normal conditions at the cost of blowing up in tail events. Build in slack, redundancy, optionality. The rare event, when it comes, will define the outcome.
Go Deeper
Books
- The Black Swan by Nassim Nicholas Taleb — The accessible treatment of extreme events and why we miss them.
- Statistical Consequences of Fat Tails by Nassim Nicholas Taleb — The technical version. Free PDF on arXiv. For those who want the math.
- The (Mis)Behavior of Markets by Benoit Mandelbrot — The discoverer of fat tails in finance explains his findings accessibly.
Essays
- “The Variation of Certain Speculative Prices” by Benoit Mandelbrot (1963) — The original paper documenting fat tails in cotton prices.
Related: [[risk]], [[ergodicity]], [[risk-vs-uncertainty]], [[antifragility]]