The Pareto distribution explains the 80:20 rule and why averages can explode
Vilfredo Pareto noticed that a small slice of society held most of its wealth. The mathematical curve named after him now describes everything from insurance claims to natural phenomena, and it has a strange property: set its shape parameter low enough and the average value becomes infinite.
The Pareto distribution is a power law. It starts at some minimum value and then tails off, with the chance of seeing a value larger than x shrinking in proportion to x raised to a negative power. Two numbers define it: the minimum, which sets the scale, and a shape parameter called alpha, also known as the tail index or, when describing wealth, the Pareto index. A smaller alpha means a fatter tail, so extreme values turn up more often.
Drawn on ordinary axes it forms a J-shaped curve that hugs both axes, and every stretch of the curve looks like a rescaled copy of any other. Switch to logarithmic scales on both axes and the swoop becomes a straight line sloping downward, which is the usual way to spot one in real data.
The famous 80:20 rule, the idea that 80 per cent of outcomes come from 20 per cent of causes, was named in Pareto's honour. It corresponds to a particular shape value, alpha of roughly 1.16. Push alpha to 1 or below and the expected value is infinite; between 1 and 2 the average exists but the variance does not. In practice that means sample averages from such data can jump wildly whenever a single enormous value arrives.
Another quirk is that the distribution looks the same from any vantage point. If you only consider values above some threshold, what remains is again a Pareto distribution with the same index, and its expected value grows in proportion to the threshold. Applied to lifetimes, this says the longer something has already lasted, the longer it can be expected to keep going, an idea known as the Lindy effect.
Source: Pareto distribution
