Against the Gods: The Remarkable Story of Risk Page 18

  Although Galton said that he did not care to occupy himself with people whose gifts fell below average, he did estimate the number of "idiots and imbeciles" among Britain's twenty million inhabitants as 50,000, or one in 400, making them about ten times as prevalent as his eminent citizens.26 But it was the eminent ones he cared about. "I am sure," he concluded, that no one "can doubt the existence of grand human animals, of natures preeminently noble, of individuals born to be kings of men."27 Galton did not ignore "very powerful women" but decided that, "happy perhaps for the repose of the other sex, such gifted women are rare."28

  Galton was convinced that if height and chest circumference matched Quetelet's hypotheses, the same should be true of head size, brain weight, and nerve fibers-and to mental capacity as well. He demonstrated how well Quetelet's findings agreed with his own estimates of the range of Britons from eminence at one end to idiocy at the other. He arrived at "the undeniable, but unexpected conclusion, that eminently gifted men are raised as much above mediocrity as idiots are depressed below it."29

  But beyond all that, Galton wanted to prove that heredity alone was the source of special talents, not "the nursery, the school, the university, [or] professional careers."30 And heredity did seem to matter, at least within the parameters that Galton laid out. He found, for example, that a ratio of one out of nine close relatives of 286 judges were father, son, or brother to another judge, a ratio far greater than in the general population. Even better, he found that many relatives of judges were also admirals, generals, novelists, poets, and physicians.* (Galton explicitly excluded clergymen from among the eminent). He was disappointed to note that his "finger marks" failed to distinguish between eminent men and "congenital idiots."31

  Yet Galton discovered that eminence does not last long; as physicists would put it, eminence has a short half life. He found that only 36% of the sons of eminent men were themselves eminent; even worse, only 9% of their grandsons made the grade. He attempted to explain why eminent families tend to die out by citing their apparent habit of marrying heiresses. Why blame them? Because heiresses must come from infertile families, he argued; if they had had a large number of siblings with whom to share the family wealth, they would not have inherited enough to be classified as heiresses. This was a surprising suggestion, in view of the comfort in which Galton lived after sharing his father's estate with six other siblings.

  After reading Hereditary Genius, Charles Darwin told Galton, "I do not think I ever in my life read anything more interesting and original ... a memorable work."32 Darwin suggested that he go on with his analysis of the statistics of heredity, but Galton needed little encouragement. He was now well on his way to developing the science of eugenics and was eager to discover and preserve what he considered to be the best of humanity. He wanted the best people to have more offspring and the lowly to exercise restraint.

  But the law of the deviation from the mean stood stubbornly in his way. Somehow he had to explain differences within the normal distribution. He realized that the only way he could do so was to figure out why the data arranged themselves into a bell curve in the first place. That search led him to an extraordinary discovery that influences most of the decisions we make today, both small and large.

  Galton reported the first step in an article published in 1875, in which he suggested that the omnipresent symmetrical distribution around the mean might be the result of influences that are themselves arrayed according to a normal distribution, ranging from conditions that are most infrequent to conditions that are most frequent and then down to a set of opposite kinds of influences that again are less frequent. Even within each kind of influence, Galton hypothesized, there would be a similar range from least powerful to most powerful and then down again to least powerful. The core of his argument was that "moderate" influences occur much more often than extreme influences, both good and bad.

  Galton demonstrated this idea with a gadget he called the Quincunx to the Royal Society around 1874.33 The Quincunx looked a lot like an up-ended pinball machine. It had a narrow neck like an hour-glass, with about twenty pins stuck into the neck area. At the bottom, where the Quincunx was at its widest, was a row of little compartments. When shot were dropped through the neck, they hit the pins at random and tended to distribute themselves among the compartments in classic Gaussian fashion-most of them piled up in the middle, with smaller numbers on either side, and so on in diminishing numbers.

  In 1877, in conjunction with his reading of a major paper titled "Typical Laws of Heredity," Galton introduced a new model of the Quincunx. (We do not know whether he actually built one). This model contained a set of compartments part way down, into which the shot fell and arrayed themselves as they had in the bottom compartments in the first model. When any one of these midway compartments was opened, the shot that had landed in it fell into the bottom compartments where they arrayed themselves-you guessed it-in the usual normal distribution.

  The discovery was momentous. Every group, no matter how small and no matter how distinct from some other group, tends to array itself in accordance with the normal distribution, with most of the observations landing in the center, or, to use the more familiar expression, on the average. When all the groups are merged into one, as Quincunx I demonstrated, the shot also array themselves into a normal distribution. The grand normal, therefore, is an average of the averages of the small subgroups.

  Quincunx II provided a mechanical version of an idea that Galton had discovered in the course of an experiment proposed by Darwin in 1875. That experiment did not involve dice, stars, or even human beings. It was sweet peas-or peas in the pod. Sweet peas are hardy and prolific, with little tendency to cross-fertilize. The peas in each pod are essentially uniform in size. After weighing and measuring thousands of sweet peas, Galton sent ten specimens of each of seven different weights to nine friends, including Darwin, throughout the British Isles, with instructions to plant them under carefully specified conditions.

  After analyzing the results, Galton reported that the offspring of the seven different groups had arrayed themselves, by weight, precisely as the Quincunx would have predicted. The offspring of each individual set of specimens were normally distributed, and the offspring of each of the seven major groups were normally distributed as well. This powerful result, he claimed, was not the consequence of "petty influences in various combinations" (Galton's italics). Rather, "[T]he processes of heredity ... [were] not petty influences, but very important ones."34 Since few individuals within a group of humans are eminent, few of their offspring will be eminent; and since most people are average, their offspring will be average. Mediocrity always outnumbers talent. The sequence of small-large-small distributions among the sweet peasaccording to the normal distribution-confirmed for Galton the dominance of parentage in determining the character of offspring.

  The experiment revealed something else, as the accompanying table of diameters of the parent peas and their offspring shows.

  Note that the spread of diameters among the parents was wider than the dispersion among the offspring. The average diameter of the parents was 0.18 inches within a range of 0.15 to 0.21 inches, or 0.03 on either side of the mean. The average diameter of the offspring was 0.163 inches within a range of 0.154 to 0.173 inches, or only about 0.01 inches on either side of the mean. The offspring had an overall distribution that was tighter than the distribution of the parents.

  This experiment led Galton to propound a general principle that has come to be known as regression, or reversion, to the mean: "Reversion," he wrote, "is the tendency of the ideal mean filial type to depart from the parental type, reverting to what may be roughly and perhaps fairly described as the average ancestral type."36 If this narrowing process were not at work-if large peas produced ever-larger offspring and if small peas produced ever-smaller offspring-the world would consist of nothing but midgets and giants. Nature would become freakier and freakier with every generation, going completely haywire or running out to e
xtremes we cannot even conceive of.

  Galton summarized the results in one of his most eloquent and dramatic paragraphs:

  The child inherits partly from his parents, partly from his ancestry. ... [T]he further his genealogy goes back, the more numerous and varied will his ancestry become, until they cease to differ from any equally numerous sample taken at haphazard from the race at large. ... This law tells heavily against the full hereditary transmission of any gift.... The law is even-handed; it levies the same successiontax on the transmission of badness as well as of goodness. If it discourages the extravagant expectations of gifted parents that their children will inherit all their powers, it no less discountenances extravagant fears that they will inherit all their weaknesses and diseases.37

  This was bad news for Galton, no matter how elegantly he articulated it, but it spurred him on in his efforts to promote eugenics. The obvious solution was to maximize the influence of "the average ancestral type" by restricting the production of offspring at the low end of the scale, thereby reducing the left-hand portion of the normal distribution.

  Galton found further confirmation of regression to the mean in an experiment that he reported in 1885, on the occasion of his election to the presidency of the British Association for the Advancement of Science. For this experiment, he had gathered an enormous amount of data on humans, data that he had received in response to a public appeal backed by an offer of cash. He ended up with observations for 928 adult children born of 205 pairs of parents.

  Galton's focus in this case was on height, or, in the language of his times, stature. His goal was similar to that in the sweet-pea experiment, which was to see how a particular attribute was passed along by heredity from parents to children. In order to analyze the observations, he had to adjust for the difference in height between men and women; he multiplied the female's height in each case by 1.08, summed the heights of the two parents, and divided by two. He referred to the resulting entities as "mid-parents." He also had to make sure that there was no systematic tendency for tall men to marry tall women and for short men to marry short women; his calculations were "close enough" for him to assume that there were no such tendencies.38

  The results were stunning, as the accompanying table reveals. The diagonal structure of the numbers from lower left to upper right tells us at once that taller parents had taller children and vice versa-heredity matters. The clusters of larger numbers toward the center reveal that each height group among the children was normally distributed and that each set of children from each parental height group also was normally distributed. Finally, compare the furthest right-hand column to the furthest left-hand column. ("Median" means that half the group were taller and half were shorter than the number shown.) The midparents with heights of 68.5 inches and up all had children whose median heights were below the height of the mid-parents; the mid-parents who were shorter than 68.5 inches all had children who tended to be taller than they were. Just like the sweet peas.

  The consistency of normal distributions and the appearance of regression to the mean enabled Galton to calculate the mathematics of the process, such as the rate at which the tallest parents tend to produce children that are tall relative to their peers but shorter relative to their parents. When a professional mathematician confirmed his results, Galton wrote, "I never felt such a glow of loyalty and respect towards the sovereignty and magnificent sway of mathematical analysis."39

  Galton's line of analysis led ultimately to the concept of correlation, which is a measurement of how closely any two series vary relative to one another, whether it be size of parent and child, rainfall and crops, inflation and interest rates, or the stock prices of General Motors and Biogen.

  Karl Pearson, Galton's principal biographer and an outstanding mathematician himself, observed that Galton had created "a revolution in our scientific ideas [that] has modified our philosophy of science and even of life itself. "40 Pearson did not exaggerate: regression to the mean is dynamite. Galton transformed the notion of probability from a static concept based on randomness and the Law of Large Numbers into a dynamic process in which the successors to the outliers are predestined to join the crowd at the center. Change and motion from the outer limits toward the center are constant, inevitable, foreseeable. Given the imperatives of this process, no outcome other than the normal distribution is conceivable. The driving force is always toward the average, toward the restoration of normality, toward Quetelet's homme moyen.

  Regression to the mean motivates almost every variety of risk-taking and forecasting. It is at the root of homilies like "What goes up must come down," "Pride goeth before a fall," and "From shirtsleeves to shirtsleeves in three generations." Joseph had this preordained sequence of events in mind when he predicted to Pharaoh that seven years of famine would follow seven years of plenty. It is whatJ.P. Morgan meant when he observed that "the market will fluctuate." It is the credo to which so-called contrarian investors pay obeisance: when they say that a certain stock is "overvalued" or "undervalued," they mean that fear or greed has encouraged the crowd to drive the stock's price away from an intrinsic value to which it is certain to return. It is what motivates the gambler's dream that a long string of losses is bound to give way to a long string of winnings. It is what my doctor means when he predicts that "tincture of time" will cure my complaints. And it is what Herbert Hoover thought was going to happen in 1931, when he promised that prosperity was just around the corner-unhappily for him and for everyone else, the mean was not where he expected it to be.

  Francis Galton was a proud man, but he never suffered a fall. His many achievements were widely recognized. He ended a long, full life as a widower traveling and writing in the company of a much younger female relative. He never allowed his fascination with numbers and facts to blind him to the wonders of nature, and he delighted in diversity:

  It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once.41

  egression to the mean provides many decision-making systems with their philosophical underpinnings. And for good reason. There are few occasions in life when the large are likely to become infinitely large or when the small are likely to become infinitely small. Trees never reach the sky. When we are tempted-as we so often are-to extrapolate past trends into the future, we should remember Galton's peapods.

  Yet if regression to the mean follows such a constant pattern, why is forecasting such a frustrating activity? Why can't we all be as prescient as Joseph in his dealings with Pharaoh? The simplest answer is that the forces at work in nature are not the same as the forces at work in the human psyche. The accuracy of most forecasts depends on decisions being made by people rather than by Mother Nature. Mother Nature, with all her vagaries, is a lot more dependable than a group of human beings trying to make up their minds about something.

  There are three reasons why regression to the mean can be such a frustrating guide to decision-making. First, it sometimes proceeds at so slow a pace that a shock will disrupt the process. Second, the regression may be so strong that matters do not come to rest once they reach the mean. Rather, they fluctuate around the mean, with repeated, irregular deviations on either side. Finally, the mean itself may be unstable, so that yesterday's normality may be supplanted today by a new normality that we know nothing about. It is perilous in the extreme to assume that prosperity is just around the corner simply because it always has been just around the corner.

  Regression to the mean is most slavishly followed on the stock market. Wall Street folklore is full of such catch phrases as "Buy low and sell high," "You never get poor taking a profit," and "The bulls get something and the bears get something but the hogs get nothing." All are variations on a simple theme: i
f you bet that today's normality will extend indefinitely into the future, you will get rich sooner and face a smaller risk of going broke than if you run with the crowd. Yet many investors violate this advice every day because they are emotionally incapable of buying low or selling high. Impelled by greed and fear, they run with the crowd instead of thinking for themselves.

  It is not all that easy to keep the peapods in mind. Since we never know exactly what is going to happen tomorrow, it is easier to assume that the future will resemble the present than to admit that it may bring some unknown change. A stock that has been going up for a while somehow seems a better buy than a stock that has been heading for the cellar. We assume that a rising price signifies that the company is flourishing and that a falling price signifies that the company is in trouble. Why stick your neck out?

  Professionals are just as likely as amateurs to try to play it safe. For example, in December 1994, analysts at the brokerage firm of Sanford C. Bernstein & Co. found that professionals who tend to forecast a higherthan-average growth rate for a company consistently overestimate the actual results, while pessimists consistently underestimate them.* "[O]n average," the analysts reported, "expectations are not met."'

  The consequences are clear: stocks with rosy forecasts climb to unreal heights while stocks with dismal forecasts drop to unreal lows. Then regression to the mean takes over. The more realistic and stouthearted investors buy as others rush to sell, and sell as others rush to buy. The payoff comes when the actual earnings surprise those who followed the trend.