it could easily be called the 50/01 rule, that is, 50 percent of the work
comes from 1 percent of the workers. This formulation makes the world
look even more unfair, yet the two formulae are exactly the same. How?
Well, if there is inequality, then those who constitute the 20 percent in the
80/20 rule also contribute unequally—only a few of them deliver the lion's
share of the results. This trickles down to about one in a hundred contributing
a little more than half the total.
The 80/20 rule is only metaphorical; it is not a rule, even less a rigid
law. In the U.S. book business, the proportions are more like 97/20 (i.e.,
97 percent of book sales are made by 20 percent of the authors); it's even
worse if you focus on literary nonfiction (twenty books of close to eight
thousand represent half the sales).
Note here that it is not all uncertainty. In some situations you may
have a concentration, of the 80/20 type, with very predictable and tractable
2 3 6 THOSE GRAY SWANS OF EXTREMISTAN
properties, which enables clear decision making, because you can identify
beforehand where the meaningful 20 percent are. These situations are very
easy to control. For instance, Malcolm Gladwell wrote in an article in The
New Yorker that most abuse of prisoners is attributable to a very small
number of vicious guards. Filter those guards out and your rate of prisoner
abuse drops dramatically. (In publishing, on the other hand, you do
not know beforehand which book will bring home the bacon. The same
with wars, as you do not know beforehand which conflict will kill a portion
of the planet's residents.)
Grass and Trees
I'll summarize here and repeat the arguments previously made throughout
the book. Measures of uncertainty that are based on the bell curve simply
disregard the possibility, and the impact, of sharp jumps or discontinuities
and are, therefore, inapplicable in Extremistan. Using them is like focusing
on the grass and missing out on the (gigantic) trees. Although unpredictable
large deviations are rare, they cannot be dismissed as outliers
because, cumulatively, their impact is so dramatic.
The traditional Gaussian way of looking at the world begins by focusing
on the ordinary, and then deals with exceptions or so-called outliers as
ancillaries. But there is a second way, which takes the exceptional as a
starting point and treats the ordinary as subordinate.
I have emphasized that there are two varieties of randomness, qualitatively
different, like air and water. One does not care about extremes; the
other is severely impacted by them. One does not generate Black Swans;
the other does. We cannot use the same techniques to discuss a gas as we
would use with a liquid. And if we could, we wouldn't call the approach
"an approximation." A gas does not "approximate" a liquid.
We can make good use of the Gaussian approach in variables for
which there is a rational reason for the largest not to be too far away from
the average. If there is gravity pulling numbers down, or if there are physical
limitations preventing very large observations, we end up in Mediocristan.
If there are strong forces of equilibrium bringing things back rather
rapidly after conditions diverge from equilibrium, then again you can use
the Gaussian approach. Otherwise, fuhgedaboudit. This is why much of
economics is based on the notion of equilibrium: among other benefits, it
allows you to treat economic phenomena as Gaussian.
Note that I am not telling you that the Mediocristan type of randomTHE
BELL CURVE, THAT GREAT I N T E L L E C T U A L FRAUD 2 3 7
ness does not allow for some extremes. But it tells you that they are so rare
that they do not play a significant role in the total. The effect of such extremes
is pitifully small and decreases as your population gets larger.
To be a little bit more technical here, if you have an assortment of
giants and dwarfs, that is, observations several orders of magnitude apart,
you could still be in Mediocristan. How? Assume you have a sample of
one thousand people, with a large spectrum running from the dwarf to the
giant. You are likely to see many giants in your sample, not a rare occasional
one. Your average will not be impacted by the occasional additional
giant because some of these giants are expected to be part of your sample,
and your average is likely to be high. In other words, the largest observation
cannot be too far away from the average. The average will always
contain both kinds, giants and dwarves, so that neither should be too
rare—unless you get a megagiant or a microdwarf on very rare occasion.
This would be Mediocristan with a large unit of deviation.
Note once again the following principle: the rarer the event, the higher
the error in our estimation of its probability—even when using the Gaussian.
Let me show you how the Gaussian bell curve sucks randomness out
of life—which is why it is popular. We like it because it allows certainties!
How? Through averaging, as I will discuss next.
How Coffee Drinking Can Be Safe
Recall from the Mediocristan discussion in Chapter 3 that no single observation
will impact your total. This property will be more and more significant
as your population increases in size. The averages will become more
and more stable, to the point where all samples will look alike.
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
2 3 8 THOSE GRAY SWANS OF EXTREMISTAN
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."
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.
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.
FIGURE 7: How the Law of Large Numbers Works
THE BELL CURVE, THAT GREAT I N T E L L E C T U A L FRAUD 2 3 9
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.
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
2 4 0 THOSE GRAY SWANS OF EXTREMISTAN
As I mentioned earlier, the bell curve was mainly the concoction of a
gambler, Abraham de Moivre (1667-1754), a French Calvinist refugee
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.
QUéTELET'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 statisticalminded
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).
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 241
who spent much of his life in London, though speaking heavily accented
English. But it is Quételet, not Gauss, who counts as one of the most destructive
fellows in the history of thought, as we will see next.
Adolphe Quételet (1796-1874) came up with the notion of a physically
average human, l'homme moyen. There was nothing moyen about
Quételet, "a man of great creative passions, a creative man full of energy."
He wrote poetry and even coauthored an opera. The basic problem with
Quételet 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, Quételet 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. Quételet
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 Quételesmus as the
grave mistake of seeing bell curves everywhere.
