DISCOVER / JUNE 1985

ESSAY

The Median Isn't the Message

STEPHEN JAY GOULD

Stephen Jay Gould teaches biology, geology,
and the history of science at Harvard.

In 1982, I learned I was suffering from a rare
and serious cancer. After surgery, I asked my
doctor what the best technical literature on the
cancer was. She told me, with a touch of di-
plomacy, that there was nothing really worth
humane advice: my cancer is incurable, with a
median mortality of eight months after discovery.

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 statement
as "I will probably be dead in eight months" -
the very conclusion that must be avoided, since
it isn't so.

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 is our usual concept of an overall
averag - 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 just 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 he 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
stereotypically centered upon Southern California,
feelings are exalted as more "real" and the only proper
basis for action - if it feels good, do it - while intellect gets
short shrift as a hang-up of outmoded elitism. Statistics,
in this absurd dichotomy, often become the symbol of the
enemy. As Hilaire Belloc 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
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

Of course, trying to keep an intellectual away from
literature 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, socioeconomic
status, and, in general, those with positive attitudes, with
a strong will and purpose for living, with commitment to
struggle, 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 sombre 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, in
Bacon's proverb.

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,
in eight months" - the very conclusion that must be
avoided, since it isn't so, and since 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 is a subtle one, but
profound - for it 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 the 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 its calculation, the "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.

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. In skewed distributions, variation to one side of the
central tendency is more stretched out - left skewed if
extended to the 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 there isn't much room for the distribution's lower
(or left) half - it must be 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 that part 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.
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 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.

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.