The Median Isn't the Message


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 
reading. I soon realized why she had offered that 
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 

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 
contemporary traditions, abetted by attitudes 
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 
story about the utility of dry, academic knowledge about 
science. Heart and head are focal points of one body, one 

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 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 

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 

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, 
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, and since attitude matters so 

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 

   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. 
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 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.