Golden Mediocrity
Quételet provided a much needed product for the ideological appetites of
his day. As he lived between 1796 and 1874, so consider the roster of his
contemporaries: Saint-Simon (1760-1825), Pierre-Joseph Proudhon
(1809-1865), and Karl Marx (1818-1883), each the source of a different
version of socialism. Everyone in this post-Enlightenment moment was
longing for the aurea mediocritas, the golden mean: in wealth, height,
weight, and so on. This longing contains some element of wishful thinking
mixed with a great deal of harmony and . . . Platonicity.
I always remember my father's injunction that in medio stat virtus,
"virtue lies in moderation." Well, for a long time that was the ideal; mediocrity,
in that sense, was even deemed golden. All-embracing mediocrity.
But Quételet took the idea to a different level. Collecting statistics,
he started creating standards of "means." Chest size, height, the weight
of babies at birth, very little escaped his standards. Deviations from the
norm, he found, became exponentially more rare as the magnitude of the
deviation increased. Then, having conceived of this idea of the physical
characteristics of l'homme moyen, Monsieur Quételet switched to
2 4 2 THOSE GRAY SWANS OF EXTREMISTAN
social matters. L'homme moyen had his habits, his consumption, his
methods.
Through his construct of l'homme moyen physique and l'homme
moyen moral, the physically and morally average man, Quételet created a
range of deviance from the average that positions all people either to the
left or right of center and, truly, punishes those who find themselves occupying
the extreme left or right of the statistical bell curve. They became
abnormal. How this inspired Marx, who cites Quételet regarding this concept
of an average or normal man, is obvious: "Societal deviations in
terms of the distribution of wealth for example, must be minimized," he
wrote in Das Kapital.
One has to give some credit to the scientific establishment of Quételet's
day. They did not buy his arguments at once. The philosopher/mathematician/
economist Augustin Cournot, for starters, did not believe that one
could establish a standard human on purely quantitative grounds. Such a
standard would be dependent on the attribute under consideration. A
measurement in one province may differ from that in another province.
Which one should be the standard? L'homme moyen would be a monster,
said Cournot. I will explain his point as follows.
Assuming there is something desirable in being an average man, he
must have an unspecified specialty in which he would be more gifted than
other people—he cannot be average in everything. A pianist would be better
on average at playing the piano, but worse than the norm at, say,
horseback riding. A draftsman would have better drafting skills, and so
on. The notion of a man deemed average is different from that of a man
who is average in everything he does. In fact, an exactly average human
would have to be half male and half female. Quételet completely missed
that point.
God's Error
A much more worrisome aspect of the discussion is that in Quételet's day,
the name of the Gaussian distribution was la loi des erreurs, the law of errors,
since one of its earliest applications was the distribution of errors in
astronomic measurements. Are you as worried as I am? Divergence from
the mean (here the median as well) was treated precisely as an error! No
wonder Marx fell for Quételet's ideas.
This concept took off very quickly. The ought was confused with the
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 243
is, and this with the imprimatur of science. The notion of the average man
is steeped in the culture attending the birth of the European middle class,
the nascent post-Napoleonic shopkeeper's culture, chary of excessive
wealth and intellectual brilliance. In fact, the dream of a society with compressed
outcomes is assumed to correspond to the aspirations of a rational
human being facing a genetic lottery. If you had to pick a society to be
born into for your next life, but could not know which outcome awaited
you, it is assumed you would probably take no gamble; you would like to
belong to a society without divergent outcomes.
One entertaining effect of the glorification of mediocrity was the creation
of a political party in France called Poujadism, composed initially of
a grocery-store movement. It was the warm huddling together of the semifavored
hoping to see the rest of the universe compress itself into their
rank—a case of non-proletarian revolution. It had a grocery-store-owner
mentality, down to the employment of the mathematical tools. Did Gauss
provide the mathematics for the shopkeepers?
Poincaré to the Rescue
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.
2 4 4 THOSE GRAY SWANS OF EXTREMISTAN
"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 ùberphilosopher 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
THE BELL CURVE, THAT GREAT I N T E L L E C T U A L F R A U D 2 4 5
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
Consider a pinball machine like the one shown in Figure 8. Launch 32
balls, assuming a well-balanced board so that the ball has equal odds of
falling right or left at any juncture when hitting a pin. Your expected outcome
is that many balls will land in the center columns and that the number
of balls will decrease as you move to the columns away from the
center.
Next, consider a gedanken, a thought experiment. A man flips a coin
and after each toss he takes a step to the left or a step to the right, depending
on whether the coin came up heads or tails. This is called the random
walk, but it does not necessarily concern itself with walking. You could
identically say that instead of taking a step to the left or to the right, you
would win or lose $1 at every turn, and you will keep track of the cumulative
amount that you have in your pocket.
Assume that I set you up in a (legal) wager where the odds are neither in
your favor nor against you. Flip a coin. Heads, you make $1, tails, you
lose $ 1 .
At the first flip, you will either win or lose.
At the second flip, the number of possible outcomes doubles. Case one:
2 4 6 THOSE GRAY SWANS OF EXTREMISTAN
FIGURE 8: THE QUINCUNX (SIMPLIFIED)—A PINBALL MACHINE
Drop balls that at every pin, randomly fall right or left. Above is the most probable
scenario, which greatly resembles the bell curve (a.k.a. Gaussian disribution). Courtesy
of Alexander Taleb.
win, win. Case two: win, lose. Case three: lose, win. Case four: lose, lose.
Each of these cases has equivalent odds, the combination of a single win
and a single loss has an incidence twice as high because cases two and
three, win-lose and lose-win, amount to the same outcome. And that is the
key for the Gaussian. So much in the middle washes out—and we will see
that there is a lot in the middle. So, if you are playing for $1 a round, after
two rounds you have a 25 percent chance of making or losing $2, but a
50 percent chance of breaking even.
Let us do another round. The third flip again doubles the number of
cases, so we face eight possible outcomes. Case 1 (it was win, win in the
second flip) branches out into win, win, win and win, win, lose. We add a
win or lose to the end of each of the previous results. Case 2 branches out
into win, lose, win and win, lose, lose. Case 3 branches out into lose, win,
win and lose, win, lose. Case 4 branches out into lose, lose, win and lose,
lose, lose.
We now have eight cases, all equally likely. Note that again you can
group the middling outcomes where a win cancels out a loss. (In Galton's
quincunx, situations where the ball falls left and then falls right, or vice
THE BELL CURVE, THAT GREAT I N T E L L E C T U A L FRAUD 2 4 7
versa, dominate so you end up with plenty in the middle.) The net, or
cumulative, is the following: 1) three wins; 2) two wins, one loss, net one
win; 3) two wins, one loss, net one win; 4) one win, two losses, net one loss;
5) two wins, one loss, net one win; 6) two losses, one win, net one loss; 7) two
losses, one win, net one loss; and, finally, 8) three losses.
Out of the eight cases, the case of three wins occurs once. The case of
three losses occurs once. The case of one net loss (one win, two losses) occurs
three times. The case of one net win (one loss, two wins) occurs three
times.
Play one more round, the fourth. There will be sixteen equally likely
outcomes. You will have one case of four wins, one case of four losses,
four cases of two wins, four cases of two losses, and six break-even cases.
The quincunx (its name is derived from the Latin for five) in the pinball
example shows the fifth round, with sixty-four possibilities, easy to
track. Such was the concept behind the quincunx used by Francis Galton.
Galton was both insufficiently lazy and a bit too innocent of mathematics;
instead of building the contraption, he could have worked with simpler
algebra, or perhaps undertaken a thought experiment like this one.
Let's keep playing. Continue until you have forty flips. You can perform
them in minutes, but we will need a calculator to work out the number
of outcomes, which are taxing to our simple thought method. You will
have about 1,099,511,627,776 possible combinations—more than one
thousand billion. Don't bother doing the calculation manually, it is two
multiplied by itself forty times, since each branch doubles at every juncture.
(Recall that we added a win and a lose at the end of the alternatives
