What do polluting cars, Moon craters and fortunes have in common?

Imagine lining up every car in a city by how much exhaust it produces. In most everyday situations you'd expect the dirtiest cars to be only a bit worse than average. But car pollution doesn't work that way: it follows a power law, where a tiny share of cars accounts for most of the contamination, which means removing just those few cars from the road would cut total exhaust substantially, far more than removing a random handful. Moon craters, solar flares, word frequencies and Americans' fortunes all follow the same lopsided pattern.

That's the signature of a power law: a relationship where changing one quantity changes another by a proportional amount raised to a fixed exponent, no matter how big or small the starting quantities are. The same shape, not the same numbers, but the same lopsided shape, turns up again and again: in the sizes of craters on the Moon, in how often different words appear in a language, in the net worth of Americans, and in how big cities' populations compare to each other. This last case even has its own name, Zipf's law: the second-biggest city in a country tends to have about half the population of the biggest, the third-biggest about a third, and so on down the list.

Because a small number of cases can dominate the total, statistics built on variance and standard deviation, like regression analysis, become the wrong tool for power-law data; a typical case tells you much less than it would for, say, human heights. But the same skew that breaks ordinary statistics is also what makes targeting power-law problems so effective, whether that means a handful of polluting cars, a handful of cities, or a handful of anything else sitting at the extreme end of the curve.