The Black Swan: The Impact of the Highly Improbable

To see how meaningless correlation can be outside of Mediocristan, take a historical series involving two variables that are patently from Extremistan, such as the bond and the stock markets, or two securities prices, or two variables like, say, changes in book sales of children’s books in the United States, and fertilizer production in China; or real-estate prices in New York City and returns of the Mongolian stock market. Measure correlation between the pairs of variables in different subperiods, say, for 1994, 1995, 1996, etc. The correlation measure will be likely to exhibit severe instability; it will depend on the period for which it was computed. Yet people talk about correlation as if it were something real, making it tangible, investing it with a physical property, reifying it.

The same illusion of concreteness affects what we call “standard” deviations. Take any series of historical prices or values. Break it up into subsegments and measure its “standard” deviation. Surprised? Every sample will yield a different “standard” deviation. Then why do people talk about standard deviations? Go figure.

Note here that, as with the narrative fallacy, when you look at past data and compute one single correlation or standard deviation, you do not notice such instability.

How to Cause Catastrophes

If you use the term statistically significant, beware of the illusions of certainties. Odds are that someone has looked at his observation errors and assumed that they were Gaussian, which necessitates a Gaussian context, namely, Mediocristan, for it to be acceptable.

To show how endemic the problem of misusing the Gaussian is, and how dangerous it can be, consider a (dull) book called Catastrophe by Judge Richard Posner, a prolific writer. Posner bemoans civil servants’ misunderstandings of randomness and recommends, among other things, that government policy makers learn statistics … from economists. Judge Posner appears to be trying to foment catastrophes. Yet, in spite of being one of those people who should spend more time reading and less time writing, he can be an insightful, deep, and original thinker; like many people, he just isn’t aware of the distinction between Mediocristan and Extremistan, and he believes that statistics is a “science,” never a fraud. If you run into him, please make him aware of these things.

QUÉTELET’S AVERAGE MONSTER

This monstrosity called the Gaussian bell curve is not Gauss’s doing. Although he worked on it, he was a mathematician dealing with a theoretical point, not making claims about the structure of reality like statistical-minded scientists. G. H. Hardy wrote in “A Mathematician’s Apology”:

The “real” mathematics of the “real” mathematicians, the mathematics of Fermát and Euler and Gauss and Abel and Riemann, is almost wholly “useless” (and this is as true of “applied” as of “pure” mathematics).

As I mentioned earlier, the bell curve was mainly the concoction of a gambler, Abraham de Moivre (1667-1754), a French Calvinist refugee who spent much of his life in London, though speaking heavily accented English. But it is Quételet, not Gauss, who counts as one of the most destructive fellows in the history of thought, as we will see next.

Adolphe Quételet (1796-1874) came up with the notion of a physically average human, l’homme moyen. There was nothing moyen about Quételet, “a man of great creative passions, a creative man full of energy.” He wrote poetry and even coauthored an opera. The basic problem with Quételet was that he was a mathematician, not an empirical scientist, but he did not know it. He found harmony in the bell curve.

The problem exists at two levels. Primo, Quételet had a normative idea, to make the world fit his average, in the sense that the average, to him, was the “normal.” It would be wonderful to be able to ignore the contribution of the unusual, the “nonnormal,” the Black Swan, to the total. But let us leave that dream for Utopia.

Secondo, there was a serious associated empirical problem. Quételet saw bell curves everywhere. He was blinded by bell curves and, I have learned, again, once you get a bell curve in your head it is hard to get it out. Later, Frank Ysidro Edgeworth would refer to Quételesmus as the grave mistake of seeing bell curves everywhere.

Golden Mediocrity