Why Do Rare Events Keep Happening? The Ends of a Distribution

The tails of a distribution are its extreme ends, describing how likely the rarest values are, and they matter far more than the middle for any question about risk. In the familiar bell-shaped distribution the tails fall away extremely rapidly, so values several standard deviations from the average are not merely rare but effectively impossible, with a value ten standard deviations out having a probability so small that it should not occur in the lifetime of the universe. That property is why the distribution is so convenient and why assuming it is so dangerous when it does not hold, since a quantity whose tails fall away more slowly will produce extreme values at rates the bell curve says are inconceivable, and the discrepancy is not small but enormous.

The distinction tends to track how the quantity is generated:

• •Human height, measurement errors and many biological quantities follow it closely
• •Sums of many small independent contributions tend towards it, which is why it is so common
• •Wealth, city sizes, word frequencies and file sizes do not, following power laws instead
• •Earthquake magnitudes, flood levels and insurance losses have much heavier tails
• •Financial returns have heavier tails than the standard models assume
• •Quantities produced by multiplication rather than addition, or by feedback, tend away from the bell shape

The mechanisms producing heavy-tailed distributions are identifiable and several are common. Preferential attachment, where something already large grows faster because it is large, generates power laws directly and describes how cities, firms, citations and network connections grow. Multiplicative processes, where changes are proportional rather than absolute, produce distributions with long right tails. Systems near a critical point, where a small disturbance can trigger a cascade of any size, produce events with no characteristic scale, which describes earthquakes, avalanches and forest fires. Interdependence matters too, since the bell curve emerges from adding independent contributions, and correlated contributions do not average out, which is why a portfolio of assets that move together behaves nothing like the sum of independent bets.

Heavy tails break the intuitions people carry about averages, in ways worth stating explicitly. In a bell-shaped distribution the average describes a typical case and a sample converges on it quickly, so a few hundred observations give a reliable figure. In a heavy-tailed one the average is dominated by rare large values, so it describes nothing typical and a sample of any realistic size can be far from it, with the sample average jumping whenever a large value arrives rather than settling. Wealth is the standard illustration, since average wealth in a population is pulled upwards by a few holders and describes almost nobody, which is why the median is reported instead. For the heaviest tails the theoretical average may not exist at all, which sounds like a technicality and means that computing one from data produces a number with no stable meaning.

The consequences of the wrong assumption are concrete. Risk models in finance built on normal assumptions repeatedly described actual market moves as events of impossible rarity, with the same institutions experiencing several such impossibilities within a few years, which is a clear sign the model rather than the world was wrong. Flood defences designed around a historical record too short to contain the extremes have failed at rates exceeding their design specification. Insurance pricing that treats correlated claims as independent underestimates the cost of a single event affecting everyone at once. The practical response is to model the tail separately from the body using methods built for extremes, to stress test against scenarios rather than relying on a probability, and to treat any model's statement about a one-in-ten-thousand-year event with appropriate scepticism.

In the bell-shaped distribution extreme values are not merely rare but effectively impossible, which makes assuming it dangerous when a quantity does not follow it. Sums of many small independent contributions tend towards that shape, while preferential attachment, multiplication and cascading systems produce much heavier tails. Institutions repeatedly experiencing supposedly impossible events have the wrong model.