Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

  At the age of four, Galton claimed, he could read any book written in English. He could recite "all the Latin Substantives and adjectives and active verbs besides 52 lines of Latin poetry" and could multiply by 2, 3, 4, 5, 6, 7, 10.6

  He began to study medicine in Birmingham when he was 16 years old, but described his visits to the wards and the postmortems as "Horror-horror-horror!"7 After Charles Darwin advised him to "read Mathematics like a house on fire," Galton hdaded to Cambridge to study math and the classics.8

  Galton was 22 when his father died, leaving a substantial estate to his seven surviving children. Deciding that he could now do anything he liked, he soon chose to give up formal studies. Inspired by Darwin's voyage to the Galapagos, he made the first of two trips to Africa, sailing up the Nile and then traveling by camel to Khartoum-a total distance of a thousand miles. After his return to England, he idled away four years and then made a second trip to Africa. He wrote a book about Africa in 1853 that gained him membership in the Royal Geographic Society, which awarded him a gold medal, and won him acceptance by the scientific community. In 1856, he was made a fellow of the Royal Society.

  His second trip to Africa when he was 27 left Galton "rather used up in health," the result of a combination of physical exhaustion and bouts of depression that were to recur often though briefly throughout his life. He referred to himself on those occasions as someone with a "sprained brain."9

  Galton was an amateur scientist with a keen interest in heredity but with no interest in business or economics. Yet his studies of "the ideal mean filial type," "the parental type," and "the average ancestral type" led him to a statistical discovery that is essential to forecasting and to risk management.

  The study of heredity has to do with the transmission of key characteristics such as intelligence, eye color, size, and behavior from generation to generation. It takes note of the outliers-individuals whose characteristics do not conform to the norm-but it pays more attention to the tendency of all members of a species to look pretty much the same. Hidden within that tendency toward homogeniety-the tendency of the average to dominate-is a powerful statistical tool that relates to many aspects of risk management.

  Galton's primary goal was to understand how talent persists through generation after generation in certain families, including the Darwin family-and, not incidentally, the Bernoulli family. Galton had hoped to see that persistence of talent in his own progeny, but he and his wife were childless, as were both of his brothers and one of his sisters. Most of all, he sought to identify "natures preeminently noble" among members of the families he classified as the most highly talented.

  In 1883, he labeled this field of study "eugenics," a word whose Greek root means good or well. The adoption of the term a half-century later by the Nazis was associated with the extermination of millions of human beings whom they identified as utterly without talent, or any kind of worth.

  Whether Galton should be charged with responsibility for that evil outcome has been the subject of spirited debate. There is nothing about the man to suggest that he would have condoned such barbaric behavior. For him, the good society was a society that had an obligation to help and educate "highly gifted" individuals, regardless of their wealth, social class, or racial background. He proposed inviting and welcoming "emigrants and refugees from other lands" to Britain and encouraging their descendants to become citizens. Yet at the same time he seems to have been looking for ways to limit the reproduction of people who were less talented or ill; he suggests that the good society would also be a society "where the weak could find a welcome and a refuge in celibate monasteries or sisterhoods."10

  Regardless of the uses to which others put Galton's work in eugenics, its significance extends far beyond the parochial questions he addressed directly. In brief, it gave further credibility to the truism that variety is the spice of life. When Enobarbus paid homage to Cleopatra, he remarked, "Age cannot wither her, nor custom stale her infinite variety." Though always the same woman, she was alternately lover, friend, cool, hot, temptress, enemy, submissive, and demanding. One person can be many.

  We can recognize as an individual every one of 5.5 billion people alive today. Countless maples grow in the forests of Vermont, each of which is different from all the other maples, but.none of which could be mistaken for a birch or a hemlock. General Electric and Biogen are both stocks listed on the New York Stock Exchange, but each is influenced by entirely different kinds of risk.

  Which of the many guises of Cleopatra, of the billions of human beings alive today, of the maples, birches, and hemlocks in Vermont, or of the stocks listed on the New York Stock Exchange is the prototypical exemplar of its class? How much do the members of each class differ from one another? How much does a child in Uganda differ from an old woman in Stockholm? Are the variations systematic or merely the results of random influences? Again, what do we mean by normal anyway?

  In searching for the answers to such questions, Galton makes little mention of early mathematicians and ignores social statisticians like Graunt. He does, however, cite at great length a set of empirical studies carried out in the 1820s and 1830s by a Belgian scientist named Lambert Adolphe Jacques Quetelet. Quetelet was twenty years older than Galton, a dogged investigator into social conditions, and as obsessed with measurement as Galton himself."

  Quetelet was only 23 years old when he received the first doctorate of science to be awarded by the new University of Ghent. By that time, he had already studied art, written poetry, and co-authored an opera.

  He was also what the historian of statistics Stephen Stigler calls "an entrepreneur of science as well as a scientist."12 He helped found several statistical associations, including the Royal Statistical Society of London and the International Statistical Congress, and for many years he was regional correspondent for the Belgian government's statistical bureau. Around 1820, he became leader of a movement to found a new observatory in Belgium, even though his knowledge of astronomy at the time was scant. Once the observatory was established, he persuaded the government to fund a three-month stay in Paris so that he could study astronomy and meteorology and learn how to run an observatory.

  During his time in Paris, he met many of the leading French astronomers and mathematicians, from whom he learned a good bit about probability. He may even have met Laplace, who was then 74 years old and about to produce the final volume of his masterpiece, Mecanique celeste. Quetelet was fascinated by the subject of probability. He subsequently wrote three books on the subject, the last in 1853. He also put what he learned about it to good-and practical-use.

  Although Quetelet continued to work at the Royal Observatory in Brussels after he returned from Paris in 1820, he also carried on research relating to French population statistics and started to plan for the approaching census of 1829. In 1827, he published a monograph titled "Researches on population, births, deaths, prisons, and poor houses, etc. in the Kingdom of the Low Countries," in which he criticized the procedures used in gathering and analyzing social statistics. Quetelet was eager to apply a method that Laplace had developed back in the 1780s to estimate France's population. Laplace's method called for taking a random sample from a diversified group of thirty departements and using the sample as the basis for estimating the total population.

  A colleague soon persuaded Quetelet to abandon that approach. The problem was that the officials in charge of the French census would have no way of knowing how representative their sample might be. Each locality had certain customs and conventions that influenced the birth rate. Furthermore, as Halley and Price had discovered, the representative quality of a survey even in a small area could be affected by movements of the population. Unlike Enobarbus, Quetelet found too much variety in the French sociological structure for anyone to generalize on the basis of a limited sample. A complete census of France was decided upon.

  This experience led Quetelet to begin using social measurement in a search to explain why such differences exist among p
eople and placeswhence the variety that adds the spice? If the differences were random, the data would look about the same each time a sample was taken; if the differences were systematic, each sample would look different from the others.

  This idea set Quetelet off on a measurement spree, which Stigler describes as follows:

  He examined birth and death rates by month and city, by temperature, and by time of day.... He investigated mortality by age, by profession, by locality, by season, in prisons, and in hospitals. He considered ... height, weight, growth rate, and strength ... [and developed] statistics on drunkenness, insanity, suicides, and crime.13

  The result was A Treatise on Man and the Development of His Faculties, which was first published in French in 1835 and subsequently translated into English. The French expression Quetelet chose for "faculties" was "physique social." This work established Quetelet's reputation. The author of a three-part review of it in a leading scholarly journal remarked, "We consider the appearance of these volumes as forming an epoch in the literary history of civilization."14

  The book consisted of more than just dry statistics and plodding text. Quetelet gave it a hero who lives to this very day: l'homme moyen, or the average man. This invention captured the public imagination and added to Quetelet's growing fame.

  Quetelet aimed to define the characteristics of the average man (or woman in some instances), who then became the model of the particular group from which he was drawn, be it criminals, drunks, soldiers, or dead people. Quetelet even speculated that "If an individual at any epoch of society possessed all the qualities of the average man, he would represent all that is great, good, or beautiful."15

  Not everyone agreed. One of the harshest critics of Quetelet's book was Antoine-Augustin Cournot, a famous mathematician and economist, and an authority on probability. Unless we observe the rules of probability, Cournot maintained, "we cannot get a clear idea of the precision of measurements made in the sciences of observation ... or of the conditions leading to the success of commercial enterprises."16 Cournot ridiculed the concept of the average man. An average of all the sides of a bunch of right triangles, he argued, would not be a right triangle, and a totally average man would not be a man but some kind of monstrosity.

  Quetelet was undeterred. He was convinced that he could identify the average man for any age, occupation, location, or ethnic origin. Moreover, he claimed that he could find a method to predict why a given individual belonged in one group rather than in another. This was a novel step, for no one up to that point had dared to use mathematics and statistics to separate cause and effect. "[E]ffects are proportional to causes," he wrote, and then went on to italicize these words: "The greater the number of individuals observed, the more do peculiarities, whether physical or moral, become effaced, and allow the general facts to predominate, by which society exists and is preserved."" By 1836, Quetelet had expanded these notions into a book on the application of probability to the "moral and political sciences."

  Quetelet's study of causes and effects makes for fascinating reading. For example, he carried out an extended analysis of the factors that influence rates of conviction among people accused of crimes. An average of 61.4% of all people accused were convicted, but the probability was less than 50% that they would be convicted for crimes against persons while it was over 60% that they would be convicted for crimes against property. The probability of conviction was less than 61.4% if the accused was a woman older than thirty who voluntarily appeared to stand trial instead of running away and who was literate and well educated. Quetelet also sought to determine whether deviations from the 61.4% average were significant or random: he sought moral certainty in the trials of the immoral.

  Quetelet saw bell curves everywhere he looked. In almost every instance, the "errors," or deviations from the average, obediently distributed themselves according to the predictions of Laplace and Gaussin normal fashion, falling symmetrically along both sides of the average. That beautifully balanced array, with the peak at the average, was what convinced Quetelet of the validity of his beloved average man. It lay behind all the inferences he developed from his statistical investigations.

  In one experiment, for example, Quetelet took chest measurements on 5,738 Scottish soldiers. He concocted a normal distribution for the group and then compared the actual result with the theoretical result. The fit was almost perfect.18

  It had already been demonstrated that Gaussian normal distributions are typical throughout nature; now they appeared to be rooted in the social structures and the physical attributes of human beings. Thus, Quetelet concluded that the close fit to a normal distribution for the Scottish soldiers signified that the deviations around the average were random rather than the result of any systematic differences within the group. The group, in other words, was essentially homogeneous, and the average Scottish soldier was fully representative of all Scottish soldiers. Cleopatra was a woman before all else.

  One of Quetelet's studies, however, revealed a less than perfect fit with the normal distribution. His analysis of the heights of 100,000 French conscripts revealed that too many of them fell in the shortest class for the distribution to be normal. Since being too short was an excuse for exemption from service, Quetelet asserted that the measurements must have been distorted by fraud in order to accommodate draft-dodgers.

  Cournot's remark that the average man would be some sort of monstrosity reflected his misgivings about applying probability theory to social as opposed to natural data. Human beings, he argued, lend themselves to a bewildering variety of classifications. Quetelet believed that a normally distributed set of human measurements implied only random differences among the sample of people he was examining. But Cournot suspected that the differences might not be random. Consider, for example, how one might classify the number of male births in any one year: by age of parents, by geographical location, by days of the week, by ethnic origin, by weight, by time in gestation, by color of eyes, or by length of middle fingers, just to name a few possibilities. How, then, could you state with any confidence which baby was the average baby? Cournot claimed that it would be impossible to determine which data were significant and which were nothing more than the result of chance: "[T]he same size deviation [from the average] may lead to many different judgments."19 What Cournot did not mention, but what modern statisticians know well, is that most human measurements reflect differences in nutrition, which means that they tend to reflect differences in social status as well.

  Today, statisticians refer to the practice that stirred Cournot's misgivings as "data mining." They say that if you torture the data long enough, the numbers will prove anything you want. Cournot felt that Quetelet was on dangerous ground in drawing such broad generalizations from a limited number of observations. A second set of observations drawn from a group of the same size could just as likely turn up a different pattern from the first.

  There is no doubt that Quetelet's infatuation with the normal distribution led him to claim more than he should have. Nevertheless, his analysis was hugely influential at the time. A famous mathematician and economist of a later age, Francis Ysidro Edgeworth, coined the term "Quetelismus" to describe the growing popularity of discovering normal distributions in places where they did not exist or that failed to meet the conditions that identify genuine normal distributions. 20

  When Galton first came upon Quetelet's work in 1863, he was deeply impressed. "An Average is but a solitary fact," he wrote, "whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence. Some people hate the very name of statistics, but I find them full of beauty and interest."21

  Galton was enthralled by Quetelet's finding that "the very curious theoretical law of the deviation from the average"-the normal distribution-was ubiquitous, especially in such measurements as body height and chest measurements.22 Galton himself had found bell curves in the record of 7,634 grades in mathematics for Ca
mbridge students taking their final exam for honors in mathematics, ranging from highest to "one can hardly say what depth."23 He found similar statistical patterns in exam grades among the applicants for admission to the Royal Military College at Sandhurst.

  The aspect of the bell curve that impressed Galton most was its indication that certain data belonged together and could be analyzed as a relatively homogeneous entity. The opposite would then also be true: absence of the normal distribution would suggest "dissimilar systems." Galton was emphatic: "This presumption is never found to be belled.""

  But it was differences, not homogeneity, that Galton was pursuingCleopatra, not the woman. In developing his new field of study, eugenics, he searched for differences even within groups whose mea surable features seemed to fall into a normal distribution. His objective was to classify people by "natural ability," by which he meant

  ... those qualities of intellect and disposition, which urge and qualify a man to perform acts that lead to reputation.... I mean a nature which, when left to itself, will, urged by an inherent stimulus, climb the path that leads to eminence, and has strength to reach the summit. ... [M]en who achieve eminence, and those who are naturally capable, are, to a large extent, identical. "25

  Galton began with the facts. During the years 1866 to 1869, he collected masses of evidence to prove that talent and eminence are hereditary attributes. He then summarized his findings in his most important work, Hereditary Genius (which includes an appendix on Quetelet's work, as well as Galton's own caustic appraisal of the typical prickly Bernoulli personality). The book begins with an estimate of the proportion of the general population that Galton believed he could classify as "eminent." On the basis of obituaries in the London Times and in a biographical handbook, he calculated that eminence occurred among English people past middle age in a ratio of one to every 4,000, or about 5,000 people in Britain at that time.