Full House +xtras

by Stephen Jay Gould

  The common error lies in failing to recognize that apparent trends can be generated as by-products, or side consequences, of expansions and contractions in the amount of variation within a system, and not by anything directly moving anywhere. Average values may, in fact, stay constant within the system (as average batting percentages have done in major-league baseball, and as the bacterial mode has remained for life)—while our (mis)perception of a trend may represent only our myopic focus on rare objects at one extreme in a system's variation (as this periphery expands or contracts). And the reasons for expansion or contraction of a periphery may be very different from causes for a change in average values. Thus, if we mistake the growth or shrinkage of an edge for movement of an entire mass, we may devise a backwards explanation. I will show that the disappearance of 0.400 hitting marks the shrinkage of such an edge caused by increasing excellence in play, not the extinction of a cherished entity (which would surely signify degeneration of something, and a _loss of excellence).

  Let me illustrate this unfamiliar concept with a simple (and silly) example to show how, in two cases, an apparent trend may arise only by expansion or contraction of variation. In both cases we tend to misinterpret a phenomenon because we maintain such strong preferences for viewing trends as entities moving somewhere.

  The one hundred inhabitants of a mythical land subsist on an identical diet and all weigh one hundred pounds. In my first case, an argument about nutrition develops, with some folks pushing a new (and particularly calorific) brand of cake, and others advocating increased abstemiousness. Most members of the population don't give a damn and stay where they are, but ten folks eat copious amounts of cake and now average 150 pounds, while ten others run and starve to reach an average weight of fifty pounds. The mean of the population hasn't altered at all, remaining right at its old value of one hundred pounds—but variation in weight has expanded markedly (and symmetrically in both directions).

  Cake-makers, pushing the aesthetic beauty of the new and fuller look, might celebrate a trend to greater weight by focusing on the small subset of people under their influence, and ignoring the others—just as the running-and-dieting moralists might exalt twigginess and praise a supposed trend in this direction by isolating their own small subset. But no general trend has occurred at all, at least in the usual sense. The average of the population has not altered by a single pound, and most people (80 percent in this case) have not varied their weight by an ounce. The only change has been a symmetrical expansion of variation on both sides of a constant mean weight. (You may recognize this increased spread as significant, of course, but we usually don't describe such nondirectional changes as "trends.”)

  You may choose to regard this example as both silly and transparent. Few of us would have any trouble identifying the actual changes, and we would laugh the shills of both cake-makers and runner-dieters out of town, if they tried to pass off the changes in their small subset as a general trend. But bear with me, for I shall show that many phenomena often perceived as trends, and either celebrated or lamented with gusto and acres of printer's ink—the disappearance of 0.400 hitting among them—also represent symmetrical changes of variation around constant mean values, and therefore display the same fallacy, though better hidden.

  My second case features a totalitarian society ruled by the runner-dieters. They have been pushing their line for so long that everyone has succumbed to social pressure and weighs fifty pounds. A more liberal regime takes over and permits free discussion about ideal weights. Fine, but for one catch imposed by physiology rather than politics: fifty pounds is the lower limit for sustaining life, and no one can get any thinner. Therefore, although citizens are now free to alter their weight, only one direction of change is possible. The great majority of inhabitants remain content with the old ways and elect to maintain themselves at fifty pounds. Fifteen percent of the population revels in its newfound freedom and begins to gain weight with abandon. Six months later, these fifteen individuals average seventy-five pounds; after a year, one hundred pounds; and after two years, 150 pounds.

  The statistical spin doctors for the fat fifteen now step in. They argue that their clients' point of view is sweeping through the whole society, as clearly indicated by the steady increase of mean weight for the entire population. And who can deny their evidence? They even present a fancy graph (shown here as Figure 3). Before the liberation, average weight stood at fifty pounds; after six months the mean rises to 53.8 pounds (the average for eighty-five remaining at fifty pounds, and fifteen rising to seventy-five pounds); after a year to 57.5 pounds; and after two years to sixty-five pounds (an increase of 30 percent from the original fifty)—a steady, unreversed, and substantial rise.

  Again, you may view this example as silly (and purposely chosen to illustrate the obvious nature of the point, once you understand the whole system and its variation). Few people would be fooled, so long as they grasped the totality of the story, and knew that most members of the population had not changed their weight, and that the steady increase in mean values arises as an artifact produced by amalgamating two entirely different subpopulations—a majority of stalwarts with a minority of revolutionaries. But suppose you didn't appreciate the whole tale, and only listened to the statistical spin doctors for the fat fifteen. Suppose, in addition, that you tended to imbue mean values (as I fear most of us do) with a reality transcending actual individuals and the variation among them. You might then be persuaded from Figure 3 that a general trend has swept through the population, thrusting it _as a whole_ toward greater average weights.

  We are more likely to be fooled by the second case, where limits to variation on one side of the average permit change in only one direction. The rise of mean values isn't "false" in this second case, but the supposed trend is surely misleading in the sense of Mark Twain's or Disraeli's famous line (the quote has been attributed to both) about three kinds of falsification—"lies, damned lies, and statistics." I will present the technicalities later, but let me quickly state why such false impressions can emerge from correct data in this case—as so often exploited by economic pundits and political spin doctors. As in the cliché about skinning cats, there is more than one way to represent an "average." The most common method, technically called the _mean_, instructs us to add up all the values and divide by the number of cases. If ten kids have ten dollars among them, the mean wealth per kid is one dollar. But means can be grossly misleading—and never more so than in the type of example purposely chosen above: when variation can expand markedly in one direction and little or not at ail in the other. For means will then drift toward the open end and give an impression (often quite false) that the whole population has moved in that direction.

  After all, one kid may have a ten-dollar bill, and the other nine nothing. One dollar per kid would still be the mean value, but would such a figure accurately characterize the population? Similarly, to be serious about real cases, spin doctors for politicians in power often use mean incomes to paint dishonestly bright pictures. Suppose that, under a super-Reaganomic system with tax breaks only for the rich, a few millionaires add immense wealth while a vast mass of people at the poverty line either gain nothing or become poorer. The mean income may rise because one tycoon's increase from, say, $6 million to $600 million per year may balance several million paupers. If one man gains $594 million and one hundred million people lose five dollars each (for a total of $500 million), mean income for the whole population will still rise—but no one would dare say (honestly) that the average person was making more money.

  Statisticians have developed other measure of average, or "central tendency," to deal with such cases. One alternative, called the _mode_, is denned as the most common value in the population. No mathematical rule can tell us which measure of central tendency will be most appropriate for any particular problem. Proper decisions rest upon knowledge of all factors in a given case, and upon basic honesty.

  Would anyone dispute a claim that modes, rather
than means, provide a better understanding of all the examples presented above: The modal amount of money for the ten kids is zip. The modal income for our population remains constant (or falls slightly), while the mean rises because one tycoon makes an immense killing. The modal weight for the population of my second silly example remains at fifty pounds. The fifteen gainers increase steadily (and the mean of the whole population therefore rises), but who would deny that stability of the majority best characterizes the population as a whole? (At the very least, allow me that you cannot represent the population by the rising mean values of Figure 3 if, for whatever personal reason, you choose to focus on the gainers—and that you must identify the stability of the majority as a major phenomenon.) I belabor this point because my second focal example, progress in the history of life, emerges as a delusion on precisely the same grounds. A few creatures have evolved greater complexity in the only direction open to variation. The mode has remained rock-solid on bacteria throughout the history of life—and bacteria, by any reasonable criterion, were in the beginning, are now, and ever shall be the most successful organisms on earth.

  Variation as Universal Reality

  I have tried to show how an apparent trend in a whole system—traditionally read as a "thing" (the population's average, for example) moving somewhere—can represent a false reading based only on expansion or contraction of variation within the system. We make such errors either because we focus myopically upon the small set of changing extreme values and falsely read their alteration as a trend in the whole system (my first case, to be illustrated by 0.400 hitting in baseball)—or because variation can expand or contract in only one direction, and we falsely characterize the system by a changing mean value, while a stable mode suggests a radically different interpretation (my second case, to be illustrated by the chimera of progress as the primary thrust of life's history).

  I am not saying that all trends fall victim to this error (genuine "things" do move somewhere sometimes), or that this "fallacy of reified variation"[2] exceeds in importance the two more commonly recognized errors of confusing trends with random sequences, or conflating correlation with causality. But the variational fallacy has caused us to read some of our most important, and most intensely discussed, cultural trends in an ass-backwards manner. I am also intrigued by this fallacy because our general misunderstanding or undervaluation of variation raises a much deeper issue about the basic perception of physical reality.

  [2. _Reificarion_ is an unfamiliar word, but this term describes the fallacy so well that I hasten to use (and explain) it. As coined by philosophers and social scientists in the mid-nineteenth century, reification refers to "the mental conversion of a person or abstract concept into a thing" (_Oxford English Dictionary_). The word comes from the Latin _res_, meaning "thing" (a republic, or res publica, is the people's thing). When committing the error discussed in this book, we abstract the variation within a system into some measure of central tendency, like the mean value—and then make the mistake of reifying this abstraction and interpreting the mean as a concrete "thing"; we then compound our error by assuming that changes in the mean must, _ipso facto_, be read as an entity moving somewhere. Or, in another version of the same fallacy, we focus on extremes in variation and falsely reify these values as separate things, rather than treating them as an inextricable part of the entire system's variation.]

  We often portray taxonomy as the dullest of all fields, as expressed in a variety of deprecatory metaphors: hanging garments on nature's coat-rack; placing items into pigeonholes; or (in an image properly resented by philatelists) sticking stamps into the album of reality. All these images clip the wings of taxonomy and reduce the science of classification to the dullest task of keeping things neat and tidy. But these portrayals also reflect a cardinal fallacy: the assumption of a fully objective nature "out there" and visible in the same way to any unprejudiced observer (the same image that I criticized in the first section of this chapter as "Huxley's chessboard"). If such a vision could be sustained, I suppose that taxonomy would become the most boring of all sciences, for nature would then present a set of obvious pigeonholes, and taxonomists would search for occupants and shove them in—an enterprise requiring diligence, perhaps, but not much creativity or imagination.

  But classifications are not passive ordering devices in a world objectively divided into obvious categories. Taxonomies are human decisions imposed upon nature—theories about the causes of nature's order. The chronicle of historical changes in classification provides our finest insight into conceptual revolutions in human thought. Objective nature does exist, but we can converse with her only through the structure of our taxonomic systems.

  We may grant this general point, but still hold that certain fundamental categories present so little ambiguity that basic divisions must be invariant across time and culture. Not so—not for these, or for any subjects. Categories are human impositions upon nature (though nature's factuality offers hints and suggestions in return). Consider, as an example, the "obvious" division of humans into two sexes.

  We may view male versus female as a permanent dichotomy, as expressions of two alternative pathways in embryological development and later growth. How else could we possibly classify people? Yet this "two-sex model" has only recently prevailed in Western history (see Laqueur, 1990; Gould, 1991), and could not hold sway until the mechanical philosophy of Newton and Descartes vanquished the Neoplatonism of previous worldviews. From classical times to the Renaissance, a "one-sex model" was favored, with human bodies ranged on a continuum of excellence, from low earthiness to high idealization. To be sure, people might clump into two major groups, called male and female, along this line, but only one ideal or archetypal body existed, and all actual expressions (real persons) had to occupy a station along a single continuum of metaphysical advance. This older system is surely as sexist as the later "two-sex model" (which posits innate and predetermined differences of worth from the start), but for different reasons—and we need to understand this history of radically altered taxonomy if we wish to grasp the depth of oppression through the ages. (In the "one-sex model," conventional maleness, by virtue of more heat, stood near the apex of the single sequence, while the characteristic female form, through relative weakness of the same generating forces, ranked far down the single ladder.)

  This book treats the even more fundamental taxonomic issue of what we designate as a thing or an object in the first place. I will argue that we are still suffering from a legacy as old as Plato, a tendency to abstract a single ideal or average as the "essence" of a system, and to devalue or ignore variation among the individuals that constitute the full population. (Just consider our continuing hang-ups about "normality." When I was a new father, my wife and I bought a wonderful book by the famous pediatrician T. Berry Brazelton. He wrote to combat every parent's excessive fear that one standard of normality exists for a child's growth, and that anything your particular baby does must be judged against this unforgiving protocol. Brazelton used the simple device of designating three perfectly fine pathways, each exemplified by a particular child—one hellion, one in the middle, and one shy baby who, in gentle euphemism, was labeled "slow to warm up." Even three, instead of one, doesn't capture the richness of normal _variation_, but what a fine start in the right direction.)

  In his celebrated analogy of the cave, Plato (in the _Republic_) held that actual organisms are only shadows on the cave's wall (empirical nature)—and that an ideal realm of essences must exist to cast the shadows. Few of us would maintain such an unbridled Platonism today, but we have never put aside this distinctive view that populations of actual individuals form a set of accidents, a collection of flawed examples, each necessarily imperfect and capable only of approaching the ideal to a certain extent. One might survey this pool of accidents and form some idea of the essence by cobbling together the best parts—the most symmetrical nose from this person, the most oval eyes from a second, the roundest navel from a third, and
the best-proportioned toe from a fourth—but no actual individual can stand for the category's deeper reality.

  Only by acknowledging this lingering Platonism can I understand the fatal inversion that we so often apply to calculated averages. In Darwin's post-Platonic world, variation stands as the fundamental reality and calculated averages become abstractions. But we continue to favor the older and opposite view, and to regard variation as a pool of inconsequential happenstances, valuable largely because we can use the spread to calculate an average, which we may then regard as a best approach to an essence. Only as Plato's legacy can I grasp the common errors about trends that make this book necessary: our misreading of expanding or contracting variation within a system as an average (or extreme) value moving somewhere.

  I spoke in chapter 2 about completing Darwin's revolution. This intellectual upheaval included many components—in part (and already accomplished among educated people during Darwin's lifetime), the simple acceptance of evolution as an alternative to divine creation; in part (and still unfulfilled), Freud's pedestal-smashing recognition of _Homo sapiens_ as only a recent twiglet on an ancient and enormous genealogical bush. But, in an even more fundamental sense, Darwin's revolution should be epitomized as the substitution of variation for essence as the central category of natural reality (see Mayr, 1963, our greatest living evolutionist, for a stirring defense of the notion that "population thinking," as a replacement for Platonic essentialism, forms the centerpiece of Darwin's revolution). What can be more discombobulating than a full inversion, or "grand flip," in our concept of reality: in Plato's world, variation is accidental, while essences record a higher reality; in Darwin's reversal, we value variation as a defining (and concrete earthly) reality, while averages (our closest operational approach to "essences") become mental abstractions.