The Black Swan: The Impact of the Highly Improbable

Poincaré himself was quite suspicious of the Gaussian. I suspect that he felt queasy when it and similar approaches to modeling uncertainty were presented to him. Just consider that the Gaussian was initially meant to measure astronomic errors, and that Poincaré’s ideas of modeling celestial mechanics were fraught with a sense of deeper uncertainty.

Poincaré wrote that one of his friends, an unnamed “eminent physicist,” complained to him that physicists tended to use the Gaussian curve because they thought mathematicians believed it a mathematical necessity; mathematicians used it because they believed that physicists found it to be an empirical fact.

Eliminating Unfair Influence

Let me state here that, except for the grocery-store mentality, I truly believe in the value of middleness and mediocrity—what humanist does not want to minimize the discrepancy between humans? Nothing is more repugnant than the inconsiderate ideal of the Ubermensch! My true problem is epistemological. Reality is not Mediocristan, so we should learn to live with it.

“The Greeks Would Have Deified It”

The list of people walking around with the bell curve stuck in their heads, thanks to its Platonic purity, is incredibly long.

Sir Francis Galton, Charles Darwin’s first cousin and Erasmus Darwin’s grandson, was perhaps, along with his cousin, one of the last independent gentlemen scientists—a category that also included Lord Cavendish, Lord Kelvin, Ludwig Wittgenstein (in his own way), and to some extent, our überphilosopher Bertrand Russell. Although John Maynard Keynes was not quite in that category, his thinking epitomizes it. Galton lived in the Victorian era when heirs and persons of leisure could, among other choices, such as horseback riding or hunting, become thinkers, scientists, or (for those less gifted) politicians. There is much to be wistful about in that era: the authenticity of someone doing science for science’s sake, without direct career motivations.

Unfortunately, doing science for the love of knowledge does not necessarily mean you will head in the right direction. Upon encountering and absorbing the “normal” distribution, Galton fell in love with it. He was said to have exclaimed that if the Greeks had known about it, they would have deified it. His enthusiasm may have contributed to the prevalence of the use of the Gaussian.

Galton was blessed with no mathematical baggage, but he had a rare obsession with measurement. He did not know about the law of large numbers, but rediscovered it from the data itself. He built the quincunx, a pinball machine that shows the development of the bell curve—on which, more in a few paragraphs. True, Galton applied the bell curve to areas like genetics and heredity, in which its use was justified. But his enthusiasm helped thrust nascent statistical methods into social issues.

“Yes/No” Only Please

Let me discuss here the extent of the damage. If you’re dealing with qualitative inference, such as in psychology or medicine, looking for yes/no answers to which magnitudes don’t apply, then you can assume you’re in Mediocristan without serious problems. The impact of the improbable cannot be too large. You have cancer or you don’t, you are pregnant or you are not, et cetera. Degrees of deadness or pregnancy are not relevant (unless you are dealing with epidemics). But if you are dealing with aggregates, where magnitudes do matter, such as income, your wealth, return on a portfolio, or book sales, then you will have a problem and get the wrong distribution if you use the Gaussian, as it does not belong there. One single number can disrupt all your averages; one single loss can eradicate a century of profits. You can no longer say “this is an exception.” The statement “Well, I can lose money” is not informational unless you can attach a quantity to that loss. You can lose all your net worth or you can lose a fraction of your daily income; there is a difference.

This explains why empirical psychology and its insights on human nature, which I presented in the earlier parts of this book, are robust to the mistake of using the bell curve; they are also lucky, since most of their variables allow for the application of conventional Gaussian statistics. When measuring how many people in a sample have a bias, or make a mistake, these studies generally elicit a yes/no type of result. No single observation, by itself, can disrupt their overall findings.

I will next proceed to a sui generis presentation of the bell-curve idea from the ground up.

A (LITERARY) THOUGHT EXPERIMENT ON WHERE THE BELL CURVE COMES FROM