The Black Swan: The Impact of the Highly Improbable

single gambler will impact the total more than minutely.

The consequence of this is that variations around the average of the Gaussian, also called “errors”, are not truly worrisome. They are small and they wash out. They are domesticated fluctuations around the mean.

Love of Certainties

If you ever took a (dull) statistics class in college, did not understand much of what the professor was excited about, and wondered what “standard deviation” meant, there is nothing to worry about. The notion of standard deviation is meaningless outside of Mediocristan. Clearly it would have been more beneficial, and certainly more entertaining, to have taken classes in the neurobiology of aesthetics or postcolonial African dance, and this is easy to see empirically.

Standard deviations do not exist outside the Gaussian, or if they do exist they do not matter and do not explain much. But it gets worse. The Gaussian family (which includes various friends and relatives, such as the Poisson law) are the only class of distributions that the standard deviation (and the average) is sufficient to describe. You need nothing else. The bell curve satisfies the reductionism of the deluded.

There are other notions that have little or no significance outside of the Gaussian: correlation and, worse, regression. Yet they are deeply ingrained in our methods; it is hard to have a business conversation without hearing the word correlation.

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.

QUETELET’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 Fermat 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 Quetelet, not Gauss, who counts as one of the most destructive fellows in the history of thought, as we will see next.

Adolphe Quetelet (1796-1874) came up with the notion of a physically average human, l’homme moyen. There was nothing moyen about Quetelet, “a man of great creative passions, a creative man full of energy”. He wrote poetry and even coauthored an opera. The basic problem with Quetelet 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, Quetelet 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. Quetelet 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 Quetelesmus as the grave mistake of seeing bell curves everywhere.

Golden Mediocrity

Quetelet provided a much needed product for the ideological appetites of his day. As he lived between 1796 and 1874, so consider the roster of his contemporaries: Saint-Simon (1760-1825), Pierre-Joseph Proudhon (1809- 1865), and Karl Marx (1818-1883), each the source of a different version of socialism. Everyone in this post- Enlightenment moment was longing for the aurea mediocritas, the golden mean: in wealth, height, weight, and so on. This longing contains some element of wishful thinking mixed with a great deal of harmony and … Platonicity.

I always remember my father’s injunction that in medio stat virtus, “virtue lies in moderation”. Well, for a long time that was the ideal; mediocrity, in that sense, was even deemed golden. All- embracing mediocrity.

But Quetelet took the idea to a different level. Collecting statistics, he started creating standards of “means”. Chest size, height, the weight of babies at birth, very little escaped his standards. Deviations from the norm, he found, became exponentially more rare as the magnitude of the deviation increased. Then, having conceived of this idea of the physical characteristics of l’homme moyen, Monsieur Quetelet switched to social matters. L’homme moyen had his habits, his consumption, his methods.

Through his construct of l’homme moyen physique and l’homme moyen moral, the physically and morally average man, Quetelet created a range of deviance from the average that positions all people either to the left or right of center and, truly, punishes those who find themselves occupying the extreme left or right of the statistical bell curve. They became abnormal. How this inspired Marx, who cites Quetelet regarding this concept of an average or normal man, is obvious: “Societal deviations in terms of the distribution of wealth for example, must be minimized”, he wrote in Das Kapital.

One has to give some credit to the scientific establishment of Quetelet’s day. They did not buy his arguments at once. The philosopher/mathematician/economist Augustin Cournot, for starters, did not believe that one could establish a standard human on purely quantitative grounds. Such a standard would be dependent on the attribute under consideration. A measurement in one province may differ from that in another province. Which one should be the standard? L’homme moyen would be a monster, said Cournot. I will explain his point as follows.