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Authors: Stephen Jay Gould
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lucky that my disease had been discovered relatively early in its course. I was therefore far more interested in the right tail (my probable residence) than in any measure of central ten - dency (an abstraction with no special relevance to my case). What, then, could possibly be more uplifting than an inference that the spread of variation would be strongly right skewed? I then checked the data and confirmed my supposition: the variation was markedly right skewed, with a few people living a long time. I saw no reason why I shouldn’t be able to reside among these inhabitants of the right tail.
This insight gave me no guarantee of normal longevity, but at least I had obtained that most precious of all gifts at a crucial moment: the prospect of substantial time—to think, to plan, and to fight. I would not immediately have to follow Isaiah’s injunction to King Hezekiah: "Set thine house in order: for thou shalt die, and not live." I had made a good statistical inference about the importance of variation and the limited utility of averages, and I had been able to confirm this suspicion with actual data. I had used knowledge and gained succor. (This story boasts an even more favorable outcome. I was destined for the right tail anyway, but an experimental treatment worked and has now probably removed the disease entirely. Old distributions offer no predictions for new situations. I trust that I am now headed for the right tail of a new distribution based on this successful treatment: death at a ripe old age in two high figures— maybe even three low ones.)
I present this tale not only for the pleasure of retelling a crucial yarn about my life, but because it encapsulates all the principles that form the core of this book. First of all, my story illustrates the importance of variation within whole systems as an ultimate reality—and the limited utility (and abstract nature) of averages. Moreover, in a didactic sense for this book, my story embodies the three terms and concepts that form the conceptual apparatus for all the examples to follow. Let me try, then, to present these principles in a formal way, and in a context that will not seem too dry or forbidding.
FIGURE 4 A right-skewed distribution for time of death for an illness with a median mortality of eight months. Each individual must be considered as a separate entity and the entire distribution cannot be characterized by its median value.
THE SKEW OF A DISTRIBUTION. If we decide to treat variation as a principal reality, then we must consider the standard terms and pictures for portraying populations and their spread. We all know the conventional icon, called a frequency distribution, with the horizontal axis scaled as a graded series for the measure under consideration (height, weight, age, survivorship in disease, batting average, anatomical complexity, etc.), and the vertical axis scaled for the number of individuals in each interval of horizontal values (those weighing between ten and twenty pounds, between twenty and thirty, etc.; those between ten and fifteen years of age, between fifteen and twenty, etc.). Frequency distributions may be symmetrical— that is, with an identical shape and number on either side of the central tendency. The ubiquitous and idealized "normal distribution" or "bell curve" of current notoriety (Figure 5) is defined as symmetrical in this manner. We have all seen normal curves so often that we have been subtly led to treat natural systems as though they longed to display this ideal form. But most actual populations are not so simple or tidy. (Systems with purely random variation around a mean value will be symmetrical—as variation falls with equal probability on either side of the mean, with any single case more likely to lie close to the mean than far away. Runs of heads or tails in coin tossing, for example, form normal distributions. We regard the normal distribution as canonical because we tend to view systems as having
This insight gave me no guarantee of normal longevity, but at least I had obtained that most precious of all gifts at a crucial moment: the prospect of substantial time—to think, to plan, and to fight. I would not immediately have to follow Isaiah’s injunction to King Hezekiah: "Set thine house in order: for thou shalt die, and not live." I had made a good statistical inference about the importance of variation and the limited utility of averages, and I had been able to confirm this suspicion with actual data. I had used knowledge and gained succor. (This story boasts an even more favorable outcome. I was destined for the right tail anyway, but an experimental treatment worked and has now probably removed the disease entirely. Old distributions offer no predictions for new situations. I trust that I am now headed for the right tail of a new distribution based on this successful treatment: death at a ripe old age in two high figures— maybe even three low ones.)
I present this tale not only for the pleasure of retelling a crucial yarn about my life, but because it encapsulates all the principles that form the core of this book. First of all, my story illustrates the importance of variation within whole systems as an ultimate reality—and the limited utility (and abstract nature) of averages. Moreover, in a didactic sense for this book, my story embodies the three terms and concepts that form the conceptual apparatus for all the examples to follow. Let me try, then, to present these principles in a formal way, and in a context that will not seem too dry or forbidding.
FIGURE 4 A right-skewed distribution for time of death for an illness with a median mortality of eight months. Each individual must be considered as a separate entity and the entire distribution cannot be characterized by its median value.
THE SKEW OF A DISTRIBUTION. If we decide to treat variation as a principal reality, then we must consider the standard terms and pictures for portraying populations and their spread. We all know the conventional icon, called a frequency distribution, with the horizontal axis scaled as a graded series for the measure under consideration (height, weight, age, survivorship in disease, batting average, anatomical complexity, etc.), and the vertical axis scaled for the number of individuals in each interval of horizontal values (those weighing between ten and twenty pounds, between twenty and thirty, etc.; those between ten and fifteen years of age, between fifteen and twenty, etc.). Frequency distributions may be symmetrical— that is, with an identical shape and number on either side of the central tendency. The ubiquitous and idealized "normal distribution" or "bell curve" of current notoriety (Figure 5) is defined as symmetrical in this manner. We have all seen normal curves so often that we have been subtly led to treat natural systems as though they longed to display this ideal form. But most actual populations are not so simple or tidy. (Systems with purely random variation around a mean value will be symmetrical—as variation falls with equal probability on either side of the mean, with any single case more likely to lie close to the mean than far away. Runs of heads or tails in coin tossing, for example, form normal distributions. We regard the normal distribution as canonical because we tend to view systems as having
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