The Black Swan: The Impact of the Highly Improbable

I’ve had plenty of cups of coffee in my life (it’s my principal addiction). I have never seen a cup jump two feet from my desk, nor has coffee spilled spontaneously on this manuscript without intervention (even in Russia). Indeed, it will take more than a mild coffee addiction to witness such an event; it would require more lifetimes than is perhaps conceivable—the odds are so small, one in so many zeroes, that it would be impossible for me to write them down in my free time.

Yet physical reality makes it possible for my coffee cup to jump—very unlikely, but possible. Particles jump around all the time. How come the coffee cup, itself composed of jumping particles, does not? The reason is, simply, that for the cup to jump would require that all of the particles jump in the same direction, and do so in lockstep several times in a row (with a compensating move of the table in the opposite direction). All several trillion particles in my coffee cup are not going to jump in the same direction; this is not going to happen in the lifetime of this universe. So I can safely put the coffee cup on the edge of my writing table and worry about more serious sources of uncertainty.

FIGURE 7: How the Law of Large Numbers Works

In Mediocristan, as your sample size increases, the observed average will present itself with less and less dispersion—as you can see, the distribution will be narrower and narrower. This, in a nutshell, is how everything in statistical theory works (or is supposed to work). Uncertainty in Mediocristan vanishes under averaging. This illustrates the hackneyed “law of large numbers.”

The safety of my coffee cup illustrates how the randomness of the Gaussian is tamable by averaging. If my cup were one large particle, or acted as one, then its jumping would be a problem. But my cup is the sum of trillions of very small particles.

Casino operators understand this well, which is why they never (if they do things right) lose money. They simply do not let one gambler make a massive bet, instead preferring to have plenty of gamblers make series of bets of limited size. Gamblers may bet a total of $20 million, but you needn’t worry about the casino’s health: the bets run, say, $20 on average; the casino caps the bets at a maximum that will allow the casino owners to sleep at night. So the variations in the casino’s returns are going to be ridiculously small, no matter the total gambling activity. You will not see anyone leaving the casino with $1 billion—in the lifetime of this universe.

The above is an application of the supreme law of Mediocristan: when you have plenty of gamblers, no 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.