for disease. Marmot's impressive project shows how social rank alone can
affect longevity. It was calculated that actors who win an Oscar tend to
live on average about five years longer than their peers who don't. People
live longer in societies that have flatter social gradients. Winners kill their
peers as those in a steep social gradient live shorter lives, regardless of
their economic condition.
I do not know how to remedy this (except through religious beliefs). Is
insurance against your peers' demoralizing success possible? Should the
Nobel Prize be banned? Granted the Nobel medal in economics has not
been good for society or knowledge, but even those rewarded for real
contributions in medicine and physics too rapidly displace others from
our consciousness, and steal longevity away from them. Extremistan is
here to stay, so we have to live with it, and find the tricks that make it
more palatable.
Chapter Fifteen
THE BELL CURVE, THAT GREAT
INTELLECTUAL FRAUD*
Not worth a pastis—Quételet's error—The average man is a monster—Let's
deify it—Yes or no—Not so literary an experiment
Forget everything you heard in college statistics or probability theory. If
you never took such a class, even better. Let us start from the very beginning.
THE GAUSSIAN AND THE MANDELBROTIAN
I was transiting through the Frankfurt airport in December 2001, on my
way from Oslo to Zurich.
I had time to kill at the airport and it was a great opportunity for me
to buy dark European chocolate, especially since I have managed to successfully
convince myself that airport calories don't count. The cashier
handed me, among other things, a ten deutschmark bill, an (illegal) scan
of which can be seen on the next page. The deutschmark banknotes were
going to be put out of circulation in a matter of days, since Europe was
* The nontechnical (or intuitive) reader can skip this chapter, as it goes into some details
about the bell curve. Also, you can skip it if you belong to the category of fortunate
people who do not know about the bell curve.
230 T H O S E G R A Y S W A N S O F E X T R E M I S T AN
The last ten deutschmark bill, representing Gauss and, to his right, the bell curve of
Mediocristan.
switching to the euro. I kept it as a valedictory. Before the arrival of the
euro, Europe had plenty of national currencies, which was good for printers,
money changers, and of course currency traders like this (more or less)
humble author. As I was eating my dark European chocolate and wistfully
looking at the bill, I almost choked. I suddenly noticed, for the first time,
that there was something curious about it. The bill bore the portrait of
Carl Friedrich Gauss and a picture of his Gaussian bell curve.
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 BELL CURVE, THAT GREAT I N T E L L E C T U A L FRAUD 2 3 1
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.*
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): l i n 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,
* I have fudged the numbers a bit for simplicity's sake.
2 3 2 THOSE GRAY SWANS OF EXTREMISTAN
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!*
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.)!
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
* One of the most misunderstood aspects of a Gaussian is its fragility and vulnerability
in the estimation of tail events. The odds of a 4 sigma move are twice that of
a 4.15 sigma. The odds of a 20 sigma are a trillion times higher than those of a 21 sigma!
It means that a small measurement error of the sigma will lead to a massive underestimation
of the probability. We can be a trillion times wrong about some events.
f My main point, which I repeat in some form or another throughout Part Three, is
as follows. Everything is made easy, conceptually, when you consider that there are
two, and only two, possible paradigms: nonscalable (like the Gaussian) and other
(such as Mandebrotian randomness). The rejection of the application of the nonscalable
is sufficient, as we will see later, to eliminate a certain vision of the world.
This is like negative empiricism: I know a lot by determining what is wrong.
T H E B E L L C U R V E , T H A T G R E A T I N T E L L E C T U A L F R A U D 233
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.
Recall the comparison between the scalable and the nonscalable in
Chapter 3. Scalability means that there is no headwind to slow you down.
Of course, Mandelbrotian Extremistan can take many shapes. Consider
wealth in an extremely concentrated version of Extremistan; there, if
you double the wealth, you halve the incidence. The result is quantitatively
different from the above example, but it obeys the same logic.
Fractal Wealth Distribution with Large Inequalities
People with a net worth higher than €1 million: 1 in 63
Higher than €2 million: 1 in 125
Higher than €4 million: 1 in 250
Higher than €8 million: 1 in 500
Higher than €16 million: 1 in 1,000
Higher than €32 million: 1 in 2,000
Higher than €320 million: 1 in 20,000
Higher than €640 million: 1 in 40,000
If wealth were Gaussian, we would observe the following divergence
away from €1 million.
Wealth Distribution Assuming a Gaussian Law
People with a net worth higher than €1 million: 1 in 63
Higher than €2 million: 1 in 127,000
Higher than €3 million: 1 in 14,000,000,000
Higher than €4 million: 1 in 886,000,000,000,000,000
Higher than €8 million:
1 in 16,000,000,000,000,000,000,000,000,000,000,000
Higher than €16 million: 1 in . . . none of my computers is capable of
handling the computation.
2 3 4 THOSE GRAY SWANS OF EXTREMISTAN
What I want to show with these lists is the qualitative difference in the
paradigms. As I have said, the second paradigm is scalable; it has no headwind.
Note that another term for the scalable is power laws.
Just knowing that we are in a power-law environment does not tell us
much. Why? Because we have to measure the coefficients in real life,
which is much harder than with a Gaussian framework. Only the Gaussian
yields its properties rather rapidly. The method I propose is a general
way of viewing the world rather than a precise solution.
What to Remember
Remember this: the Gaussian-bell curve variations face a headwind that
makes probabilities drop at a faster and faster rate as you move away
from the mean, while "scalables," or Mandelbrotian variations, do not
have such a restriction. That's pretty much most of what you need to
know. *
Inequality
Let us look more closely at the nature of inequality. In the Gaussian framework,
inequality decreases as the deviations get larger—caused by the increase
in the rate of decrease. Not so with the scalable: inequality stays the
same throughout. The inequality among the superrich is the same as the
inequality among the simply rich—it does not slow down.f
* Note that variables may not be infinitely scalable; there could be a very, very remote
upper limit—but we do not know where it is so we treat a given situation as
if it were infinitely scalable. Technically, you cannot sell more of one book than
there are denizens of the planet—but that upper limit is large enough to be treated
as if it didn't exist. Furthermore, who knows, by repackaging the book, you might
be able to sell it to a person twice, or get that person to watch the same movie several
times.
f As I was revising this draft, in August 2006,1 stayed at a hotel in Dedham, Massachusetts,
near one of my children's summer camps. There, I was a little intrigued
by the abundance of weight-challenged people walking around the lobby and causing
problems with elevator backups. It turned out that the annual convention of
NAFA, the National Association for Fat Acceptance, was being held there. As most
of the members were extremely overweight, I was not able to figure out which delegate
was the heaviest: some form of equality prevailed among the very heavy
(someone much heavier than the persons I saw would have been dead). I am sure
that at the NARA convention, the National Association for Rich Acceptance, one
person would dwarf the others, and, even among the superrich, a very small percentage
would represent a large section of the total wealth.
T H E B E L L C U R V E , T H A T G R E A T I N T E L L E C T U A L F R A U D 2 35
Consider this effect. Take a random sample of any two people from
the U.S. population who jointly earn $1 million per annum. What is
the most likely breakdown of their respective incomes? In Mediocristan, the
most likely combination is half a million each. In Extremistan, it would be
$50,000 and $950,000.
The situation is even more lopsided with book sales. If I told you that
two authors sold a total of a million copies of their books, the most likely
combination is 993,000 copies sold for one and 7,000 for the other. This
is far more likely than that the books each sold 500,000 copies. For any
large total, the breakdown will be more and more asymmetric.
Why is this so? The height problem provides a comparison. If I told
you that the total height of two people is fourteen feet, you would identify
the most likely breakdown as seven feet each, not two feet and twelve feet;
not even eight feet and six feet! Persons taller than eight feet are so rare
that such a combination would be impossible.
Extremistan and the 80/20 Rule
Have you ever heard of the 80/20 rule? It is the common signature of a
power law—actually it is how it all started, when Vilfredo Pareto made
the observation that 80 percent of the land in Italy was owned by 20 percent
of the people. Some use the rule to imply that 80 percent of the work
is done by 20 percent of the people. Or that 80 percent worth of effort
contributes to only 20 percent of results, and vice versa.
As far as axioms go, this one wasn't phrased to impress you the most:
