The Median Isn’t the Message
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| Statistics recognizes different measures of an “average,” or central tendency. The mean is 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 just world). The median, a different measure of central tendency, is the half-way 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).
In 1982, I learned that I was suffering from mesothelioma, a rare and serious cancer. When I revived after surgery, as soon as I could, I made a beeline for Harvard’s Countway medical library and punched mesothelioma into the computer’s search program. 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 my mind started to work again. 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 (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. 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? The problem may be briefly stated: What does “median mortality of eight months” signify? 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 much. We view means and medians as the hard “realities,” and the variation that permits their calculation as a set of transient and imperfect measurements. But all evolutionary biologists know that variation itself is nature’s 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 - 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. 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. 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. |
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| For the full essay click here: "The Median Isn't The Message" - Steven J. Gould | |||||||||||
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