The Black Swan: The Impact of the Highly Improbable

The last ten deutschmark bill, representing Gauss and, to his right, the bell curve of Mediocristan.

The striking irony here is that the last possible object that can be linked to the German currency is precisely such a curve: the reichsmark (as the currency was previously called) went from four per dollar to four trillion per dollar in the space of a few years during the 1920s, an outcome that tells you that the bell curve is meaningless as a description of the randomness in currency fluctuations. All you need to reject the bell curve is for such a movement to occur once, and only once—just consider the consequences. Yet there was the bell curve, and next to it Herr Professor Doktor Gauss, unprepossessing, a little stern, certainly not someone I’d want to spend time with lounging on a terrace, drinking pastis, and holding a conversation without a subject.

Shockingly, the bell curve is used as a risk-measurement tool by those regulators and central bankers who wear dark suits and talk in a boring way about currencies.

The Increase in the Decrease

The main point of the Gaussian, as I’ve said, is that most observations hover around the mediocre, the average; the odds of a deviation decline faster and faster (exponentially) as you move away from the average. If you must have only one single piece of information, this is the one: the dramatic increase in the speed of decline in the odds as you move away from the center, or the average. Look at the list below for an illustration of this. I am taking an example of a Gaussian quantity, such as height, and simplifying it a bit to make it more illustrative. Assume that the average height (men and women) is 1.67 meters, or 5 feet 7 inches. Consider what I call a unit of deviation here as 10 centimeters. Let us look at increments above 1.67 meters and consider the odds of someone being that tall.[50]

10 centimeters taller than the average (i.e., taller than 1.77 m, or 5 feet 10): 1 in 6.3

20 centimeters taller than the average (i.e., taller than 1.87 m, or 6 feet 2): 1 in 44

30 centimeters taller than the average (i.e., taller than 1.97 m, or 6 feet 6): 1 in 740

40 centimeters taller than the average (i.e., taller than 2.07 m, or 6 feet 9): 1 in 32,000

50 centimeters taller than the average (i.e., taller than 2.17 m, or 7 feet 1): lin 3,500,000

60 centimeters taller than the average (i.e., taller than 2.27 m, or 7 feet 5): 1 in 1,000,000,000

70 centimeters taller than the average (i.e., taller than 2.37 m, or 7 feet 9): 1 in 780,000,000,000

80 centimeters taller than the average (i.e., taller than 2.47 m, or 8 feet 1): 1 in 1,600,000,000,000,000

90 centimeters taller than the average (i.e., taller than 2.57 m, or 8 feet 5): 1 in 8,900,000,000,000,000,000

100 centimeters taller than the average (i.e., taller than 2.67 m, or 8 feet 9): 1 in 130,000,000,000,000,000,000,000

… and,

110 centimeters taller than the average (i.e., taller than 2.77 m, or 9 feet 1): 1 in 36,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

Note that soon after, I believe, 22 deviations, or 220 centimeters taller than the average, the odds reach a googol, which is 1 with 100 zeroes behind it.

The point of this list is to illustrate the acceleration. Look at the difference in odds between 60 and 70 centimeters taller than average: for a mere increase of four inches, we go from one in 1 billion people to one in 780 billion! As for the jump between 70 and 80 centimeters: an additional 4 inches above the average, we go from one in 780 billion to one in 1.6 million billion![51]

This precipitous decline in the odds of encountering something is what allows you to ignore outliers. Only one curve can deliver this decline, and it is the bell curve (and its nonscalable siblings).

The Mandelbrotian

By comparison, look at the odds of being rich in Europe. Assume that wealth there is scalable, i.e., Mandelbrotian. (This is not an accurate description of wealth in Europe; it is simplified to emphasize the logic of scalable distribution.)[52]

Scalable Wealth Distribution

People with a net worth higher than €1 million: 1 in 62.5

Higher than €2 million: 1 in 250

Higher than €4 million: 1 in 1,000

Higher than €8 million: 1 in 4,000

Higher than €16 million: 1 in 16,000

Higher than €32 million: 1 in 64,000

Higher than €320 million: 1 in 6,400,000

The speed of the decrease here remains constant (or does not decline)! When you double the amount of money you cut the incidence by a factor of four, no matter the level, whether you are at €8 million or €16 million. This, in a nutshell, illustrates the difference between Mediocristan and Extremistan.