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Stephen
J. Gould
32
The Median Isn't the Message
MY
LIFE HAS RECENTLY intersected, in a most personal way, two of Mark
Twain’s famous quips. One I shall defer to the end of this essay. The other
(sometimes attributed to Disraeli) identifies three species of mendacity,
each worse than the one before—lies, damned lies, and statistics.
Consider the standard example of stretching truth with numbers—a case quite
relevant to my story. Statistics recognizes different measures of an “average,”
or central tendency. The mean represents our usual concept of an
overall average—add up the items and divide them by the number of sharers
(100 candy bars collected for five kids next Halloween will yield 20 for
each in a fair world). The median, a different measure of central
tendency, is the halfway point. If I line up five kids by height, the median
child is shorter than two and taller than the other two (who might have
trouble getting their mean share of the candy). A politician in power might
say with pride, “The mean income of our citizens is $15,000 per year.”
The leader of the opposition might retort, “But half our citizens make
less than $10,000 per year.” Both are right, but neither cites a statistic
with impassive objectivity. The first invokes a mean, the second a median.
(Means are higher than medians in such cases because one millionaire
may outweigh hundreds of poor people in setting a mean, but can balance
only one mendicant in calculating a median.)
The larger issue that creates a common distrust or contempt for
statistics is more troubling. Many people make an unfortunate and invalid
separation between heart and mind, or feeling and intellect. In some contemporary
traditions, abetted by attitudes
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A right-skewed distribution showing that means must be higher than medians,
and that the right side of the distribution extends out into a long tail.
BEN GAMIT.
sterotypically centered upon Southern California, feelings are exalted
as more “real” and the only proper basis for action, while intellect
gets short shrift as a hang-up of outmoded elitism. Statistics, in this
absurd dichotomy, often becomes the symbol of the enemy. As Hilaire BeIIoc
wrote, “Statistics are the triumph of the quantitative method, and the
quantitative method is the victory of sterility and death.’
This is a personal story of statistics, properly
interpreted, as profoundly nurturant and life-giving. It declares holy
war on the downgrading of intellect by telling a small story to illustrate
the utility of dry, academic knowledge about science. Heart and head are
focal points of one body, one personality.
In July 1982, I learned that I was suffering
from abdominal mesothelioma, a rare and serious cancer usually associated
with exposure to asbestos. When I revived after surgery, I asked my first
question of my doctor and chemotherapist: ‘What is the best technical literature
about mesothelioma?" She replied, with a touch of diplomacy (the
only departure she has ever made from direct frankness), that the medical
literature contained nothing really worth reading.
Of course, trying to keep an intellectual
away from literature
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THE MEDIAN ISN'T THE MESSAGE | 475
works about as well as recommending chastity to Homo sapiens, the
sexiest primate of all. As soon as I could walk, I made a beeline for Harvard's
Countway medical library and punched mesothelioma into the computer's bibliographic
search program. An hour later, surrounded by the latest literature on abdominal
mesothelioma, I realized with a gulp why my doctor had offered that humane
advice. The literature couldn't have been more brutally clear: Mesothelioma
is incurable, with a median mortality of only eight months after discovery.
I sat stunned for about fifteen minutes, then smiled and said to myself:
So that's why they didn't give me anything to read. Then my mind started
to work again, thank goodness.
If a little learning could ever be a dangerous
thing, I had encountered a classic example. Attitude clearly matters in
fighting cancer. We don't know why (from my old-style materialistic perspective,
I suspect that mental states feed back upon the immune system). But match
people with the same cancer for age, class, health, and socioeconomic status,
and, in general, those with positive attitudes, with a strong will and
purpose for living, with commitment to struggle, and with an active response
to aiding their own treatment and not just a passive acceptance of anything
doctors say tend to live longer. A few months later I asked Sir Peter Medawar,
my personal scientific guru and a Nobelist in immunology, what the best
prescription for success against cancer might be. “A sanguine personality,”
he replied. Fortunately (since one can't reconstruct oneself at short notice
and for a definite purpose), I am, if anything, even-tempered and confident
in just this manner.
Hence the dilemma for humane doctors: Since
attitude matters so critically, should such a somber conclusion be advertised,
especially since few people have sufficient understanding of statistics
to evaluate what the statements really mean? From years of experience with
the small-scale evolution of Bahamian land snails treated quantitatively,
I have developed this technical knowledge-and I am convinced that it played
a major role in saving my life. Knowledge is indeed power as Francis Bacon
proclaimed.
The problem may be briefly stated: What does
‘‘median mortality of eight months" signify in our vernacular? I suspect
that most people, without training in statistics, would read such a
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statement as "I will probably be dead in eight months"—the very conclusion
that must be avoided, both because this formulation is false, and because
attitude matters so much.
I was not, of course, overjoyed, but I didn't
read the statement in this vernacular way either. My technical training
enjoined a different perspective on “eight months median mortality.” The
point may seem subtle, but the consequences can be profound. Moreover,
this perspective embodies the distinctive way of thinking in my own field
of evolutionary biology and natural history.
We still carry the historical baggage of a
Platonic heritage that seeks sharp essences and definite boundaries. (Thus
we hope to find an unambiguous “beginning of life” or “definition of death,”
although nature often comes to us as irreducible continua.) This Platonic
heritage, with its emphasis on clear distinctions and separated immutable
entities, leads us to view statistical measures of central tendency wrongly,
indeed opposite to the appropriate interpretation in our actual world of
variation, shadings, and continua. In short, we view means and medians
as hard “realities,” and the variation that permits their calculation as
a set of transient and imperfect measurements of this hidden essence. If
the median is the reality and variation around the median just a device
for calculation, then “I will probably be dead in eight months” may pass
as a reasonable interpretation.
But all evolutionary biologists know that
variation itself is nature's only irreducible essence. Variation is the
hard reality, not a set of imperfect measures for a central tendency. Means
and medians are the abstractions. Therefore, I looked at the mesothelioma
statistics quite differently—and not only because I am an optimist who
tends to see the doughnut instead of the hole, but primarily because I
know that variation itself is the reality. I had to place myself amidst
the variation.
When I learned about the eight-month median,
my first intellectual reaction was: Fine, half the people will live longer;
now what are my chances of being in that half. I read for a furious and
nervous hour and concluded, with relief: damned good. I possessed every
one of the characteristics conferring a probability of longer life: I was
young; my disease had been recognized in a relatively early stage; I would
receive the nation's best medical treatment; I had the world to live for;
I knew how to read the data properly and not despair.
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Another technical point then added even more
solace. I immediately recognized that the distribution of variation about
the eight-month median would almost surely be what statisticians call “right
skewed.” (In a symmetrical distribution, the profile of variation to the
left of the central tendency is a mirror image of variation to the right.
Skewed distributions are asymmetrical, with variation stretching out more
in one direction than the other— left skewed if extended to the left, right
skewed if stretched out to the right.) The distribution of variation had
to be right skewed, I reasoned. After all, the left of the distribution
contains an irrevocable lower boundary of zero (since mesothelioma can
only be identified at death or before). Thus, little space exists for the
distribution's lower (or left) half—it must he scrunched up between zero
and eight months. But the upper (or right) half can extend out for years
and years, even if nobody ultimately survives. The distribution must be
right skewed, and I needed to know how long the extended tail ran—for I
had already concluded that my favorable profile made me a good candidate
for the right half of the curve.
The distribution was, indeed, strongly right
skewed, with a long tail (however small) that extended for several years
above the eight-month median. I saw no reason why I shouldn't be in that
small tail, and I breathed a very long sigh of relief. My technical knowledge
had helped. I had read the graph correctly. I had asked the right question
and found the answers. I had obtained, in all probability, that most precious
of all possible gifts in the circumstances—substantial time. I didn't have
to stop and immediately follow Isaiah’s injunction to Hezekiah—set thine
house in order: for thou shalt die, and not live. I would have time to
think, to plan, and to fight.
One final point about statistical distributions.
They apply only to a prescribed set of circumstances—in this case to survival
with most mesothelioma under conventional modes of treatment. If circumstances
change, the distribution may alter. I was placed on an experimental protocol
of treatment and, if fortune holds, will be in the first cohort of a new
distribution with high median and a right tail extending to death by natural
causes at advanced old age.*
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*So far so good.
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478 | BULLY FOR BRONTOSAURUS
It has become, in my view, a bit too trendy
to regard the acceptance of death as something tantamount to intrinsic
dignity. Of course I agree with the preacher of Ecclesiastes that there
is a time to love and a time to die—and when my skein runs out I hope to
face the end calmly and in my own way. For most situations, however, I
prefer the more martial view that death is the ultimate enemy—and I find
nothing reproachable in those who rage mightily against the dying of the
light.
The swords of battle are numerous, and none
more effective than humor. My death was announced at a meeting of my colleagues
in Scotland, and I almost experienced the delicious pleasure of reading
my obituary penned by one of my best friends (the so-and-so got suspicious
and checked; he too is a statistician, and didn't expect to find me so
far out on the left tail). Still, the incident provided my first good laugh
after the diagnosis. Just think, I almost got to repeat Mark Twain’s most
famous line of all: The reports of my death are greatly exaggerated.*
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*Since writing this, my death has actually been reported in two European
magazines, five years apart. Fama volai (and lasts a long
time). I squawked very loudly both times and demanded a retraction; guess
I just don't have Mr. Clemens's savoir faire.
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