The Joy of x: A Guided Tour of Math, from One to Infinity by Strogatz Steven

Author:Strogatz, Steven [Strogatz, Steven]
Language: eng
Format: azw3, epub
Publisher: Houghton Mifflin Harcourt
Published: 2012-10-01T16:00:00+00:00

where x is a city’s size, y is how many cities have that size, C is a constant, and the exponent a (the power in the power law) is the negative of the straight line’s slope.

Power-law distributions have counterintuitive properties from the standpoint of conventional statistics. For example, unlike normal distributions’, their modes, medians, and means do not agree because of the skewed, asymmetrical shapes of their L-curves. President Bush made use of this property when he stated that his 2003 tax cuts had saved families an average of $1,586 each. Though that is technically correct, he was conveniently referring to the mean rebate, a figure that averaged in the whopping rebates of hundreds of thousands of dollars received by the richest 0.1 percent of the population. The tail on the far right of the income distribution is known to follow a power law, and in situations like this, the mean is a misleading statistic to use because it’s far from typical. Most families, in fact, got less than $650 back. The median was a lot less than the mean.

This example highlights the most crucial feature of power-law distributions. Their tails are heavy (also known as fat or long), at least compared to the puny little wisp of a tail on the normal distribution. So extremely large outliers, though still rare, are much more common for these distributions than they would be for normal bell curves.

On October 19, 1987, now known as Black Monday, the Dow Jones industrial average dropped by 22 percent in a single day. Compared to the usual level of volatility in the stock market, this was a drop of more than twenty standard deviations. Such an event is all but impossible according to traditional bell-curve statistics; its probability is less than one in 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (that’s 10 raised to the 50th power). Yet it happened . . . because fluctuations in stock prices don’t follow normal distributions. They’re better described by heavy-tailed distributions.

So are earthquakes, wildfires, and floods, which complicates the task of risk management for insurance industries. The same mathematical pattern holds for the numbers of deaths caused by wars and terrorist attacks, and even for more benign things like word frequencies in novels and the number of sexual partners people have.

Though the adjectives used to describe their prominent tails weren’t originally meant to be flattering, such distributions have come to wear them with pride. Fat, heavy, and long? Yeah, that’s right. Now who’s normal?

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