Read Online Full House by Stephen Jay Gould - Free Book Online
Authors: Stephen Jay Gould
Tags: nonfiction
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philosophical error and dilemma that motivated this book. Most people view averages as basic reality and variation as a device for calculating a meaningful measure of central tendency. In this Platonic world, "eight months’ median mortality" can only signify: "I will most probably be dead in eight months"—about the most chilling diagnosis anyone could ever read.
But we make a serious mistake if we view a measure of central tendency as the most likely outcome for any single individual—though most of us commit this error all the time. Central tendency is an abstraction, variation the reality. We must first ask what "median" mortality signifies. A median is the third major measure of central tendency. (I discussed the other two in the last chapter—the mean, or average obtained by adding all the values and dividing by the number of cases; and the mode, or most common value.) The median, as etymology proclaims, is the halfway point in a graded array of values. In any population, half the individuals will be below the median, and half above. If, say, in a group of five children, one has a penny, one a dime, one a quarter, one a dollar, and one ten dollars, then the kid with the quarter is the median, since two have more money and two less. (Note that means and medians are not equal in this case. The mean wealth of $2.27—the total cash of $11.36 divided by five—lies between the fourth and fifth kids, for the tycoon with ten bucks overbalances all the paupers.) We favor medians in such cases, when extension at one end of the variation drags the mean so far in that direction. For mortality in mesothelioma and other diseases, we generally favor the median as a measure of central tendency because we want to know the halfway point in a series of similar outcomes graded in time. A higher mean might seem misleading in the case of mesothelioma because one or two people living a long time (the analog of the kid with ten bucks) might drag the mean to the right and convey a false impression that most people with the disease will live for more than eight months—whereas the median correctly informs us that half the afflicted population dies within eight months of diagnosis.
We now come to the crux of practice: I am not a measure of central tendency, either mean or median. I am one single human being with mesothelioma, and I want a best assessment of my own chances—for I have personal decisions to make, and my business cannot be dictated by abstract averages. I need to place myself in the most probable region of the variation based upon particulars of my own case; I must not simply assume that my personal fate will correspond to some measure of central tendency.
I then had the key insight that proved so life-affirming at such a crucial moment. I started to think about the variation and reasoned that the distribution of deaths must be strongly "right skewed" in statistical parlance—that is, asymmetrically extended around a chosen measure of central tendency, with a much wider spread to the right than to the left. After all, there just isn’t much room between the absolute minimum value of zero (dropping dead at the moment of diagnosis) and the median value of eight months. Half the variation must be scrunched up into this left half of the curve (see Figure 4) between the minimum and the median. But the right half may, in principle, extend out forever, or at least into extreme old age. (Statisticians refer to the ends of such distribution as "tails"—so I am saying that the left tail abuts a wall at zero survivorship, while the right tail has no necessary limit but the maximal human life span.)
I needed, above all, to know the form and expanse of variation, and my most probable position within the spread. I realized that all factors favored a potential location on the right tail — I was young, rarin’ to fight the bastard, located in a city offering the best possible medical treatment, blessed with a supportive family, and
But we make a serious mistake if we view a measure of central tendency as the most likely outcome for any single individual—though most of us commit this error all the time. Central tendency is an abstraction, variation the reality. We must first ask what "median" mortality signifies. A median is the third major measure of central tendency. (I discussed the other two in the last chapter—the mean, or average obtained by adding all the values and dividing by the number of cases; and the mode, or most common value.) The median, as etymology proclaims, is the halfway point in a graded array of values. In any population, half the individuals will be below the median, and half above. If, say, in a group of five children, one has a penny, one a dime, one a quarter, one a dollar, and one ten dollars, then the kid with the quarter is the median, since two have more money and two less. (Note that means and medians are not equal in this case. The mean wealth of $2.27—the total cash of $11.36 divided by five—lies between the fourth and fifth kids, for the tycoon with ten bucks overbalances all the paupers.) We favor medians in such cases, when extension at one end of the variation drags the mean so far in that direction. For mortality in mesothelioma and other diseases, we generally favor the median as a measure of central tendency because we want to know the halfway point in a series of similar outcomes graded in time. A higher mean might seem misleading in the case of mesothelioma because one or two people living a long time (the analog of the kid with ten bucks) might drag the mean to the right and convey a false impression that most people with the disease will live for more than eight months—whereas the median correctly informs us that half the afflicted population dies within eight months of diagnosis.
We now come to the crux of practice: I am not a measure of central tendency, either mean or median. I am one single human being with mesothelioma, and I want a best assessment of my own chances—for I have personal decisions to make, and my business cannot be dictated by abstract averages. I need to place myself in the most probable region of the variation based upon particulars of my own case; I must not simply assume that my personal fate will correspond to some measure of central tendency.
I then had the key insight that proved so life-affirming at such a crucial moment. I started to think about the variation and reasoned that the distribution of deaths must be strongly "right skewed" in statistical parlance—that is, asymmetrically extended around a chosen measure of central tendency, with a much wider spread to the right than to the left. After all, there just isn’t much room between the absolute minimum value of zero (dropping dead at the moment of diagnosis) and the median value of eight months. Half the variation must be scrunched up into this left half of the curve (see Figure 4) between the minimum and the median. But the right half may, in principle, extend out forever, or at least into extreme old age. (Statisticians refer to the ends of such distribution as "tails"—so I am saying that the left tail abuts a wall at zero survivorship, while the right tail has no necessary limit but the maximal human life span.)
I needed, above all, to know the form and expanse of variation, and my most probable position within the spread. I realized that all factors favored a potential location on the right tail — I was young, rarin’ to fight the bastard, located in a city offering the best possible medical treatment, blessed with a supportive family, and
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