Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

  There was another factor at work. Two years before the publication of Graunt's Observations, Charles II had been recalled from exile in Holland. With the Restoration in full sway, the English were finally rid of the intellectual repression that the Puritans had imposed on the nation. The death of absolutism and Republicanism led to a new sense of freedom and progress throughout the country. Great wealth was beginning to arrive from the colonies across the Atlantic and from Africa and Asia as well. Isaac Newton, now 28 years old, was leading people to think in new ways about the planet on which they lived. Charles II himself was a free soul, a Merry Monarch who offered no apologies for enjoying the good things of life.

  It was time to stand up and look around. John Graunt did, and began counting.

  Although Graunt's book offers interesting bits for students of sociology, medicine, political science, and history, its greatest novelty is in its use of sampling. Graunt realized that the statistics available to him represented only a fraction of all the births and deaths that had ever occurred in London, but that failed to deter him from drawing broad conclusions from what he had. His line of analysis is known today as "statistical inference"-inferring a global estimate from a sample of data; subsequent statisticans would figure out how to calculate the probable error between the estimate and the true values. With his ground-breaking effort, Graunt transformed the simple process of gathering information into a powerful, complex instrument for interpreting the world-and the skies-around us.

  The raw material that Graunt gathered was contained in "Bills of Mortality" that the City of London had started collecting in 1603. That was only incidentally the year in which Queen Elizabeth died; it was also the year in which London suffered one of the worst infestations of the plague. Accurate knowledge of what was going on in the field of public health was becoming increasingly important.10

  The bills of mortality revealed the causes of death as well as the number of deaths and also listed the number of children christened each week. The accompanying illustration shows the documents for two weeks in the year 1665.* There were 7,165 deaths from plague in just the one week of September 12-19, and only four of 130 parishes were free of the disease.' 1

  Graunt was particularly interested in the causes of death, especially "that extraordinary and grand Casualty" the plague, and in the way people lived under the constant threat of devastating epidemic. For the year 1632, for example, he listed nearly sixty different causes of death, with 628 deaths coming under the heading of "aged." The others range from "affrighted" and "bit with mad dog" (one each) to "worms," "quinsie," and "starved at nurse." There were only seven "murthers" in 1632 and just 15 suicides.

  In observing that "but few are Murthered ... whereas in Paris few nights came without their Tragedie," Graunt credits the government and the citizen guard of the City of London. He also credits "the natural, and customary, abhorrence of that inhumane Crime, and all Bloodshed by most Englishmen," remarking that even "Usurpers" during English revolutions executed only a few of their countrymen.

  Graunt gives the number of deaths from plague for certain years; one of the worst was in 1603, when 82% of the burials were of plague victims. From 1604 to 1624, he calculated that 229,250 people had died of all diseases and "casualties," about a third of which were from children's diseases. Figuring that children accounted for half the deaths from other diseases, he concluded that "about thirty six per centum of all quick conceptions died before six years old." Fewer than 4,000 died of "outward Griefs, as of Cancers, Fistulaes, Sores, Ulcers, broken and bruised Limbs, Impostumes, King's evil, Loprosie, Scald-head, Swinepox, Wens, &c."

  Graunt suggests that the prevalence of acute and epidemical diseases might give "a measure of the state, and disposition of this Climate, and Air ... as well as its food." He goes on to observe that few are starved, and that the beggars, "swarming up and down upon this City ... seem to be most of them healthy and strong." He recommends that the state "keep" them and that they be taught to work "each according to his condition and capacity."

  After commenting on the incidence of accidents-most of which he asserts are occupation-related-Graunt refers to "one Casualty in our Bills, of which though there be daily talk, [but] little effect." This casualty is the French-Pox-a kind of syphilis-"gotten for the most part, not so much by the intemperate use of Venery (which rather causes the Gowt) as of many common Women."` Graunt wonders why the records show that so few died of it, as "a great part of men have, at one time or another, had some species of this disease." He concludes that most of the deaths from ulcers and sores were in fact caused by venereal disease, the recorded diagnoses serving as euphemisms. According to Graunt, a person had to be pretty far gone before the authorities acknowledged the true cause of death: "onely hated persons, and such, whose very Noses were eaten of, were reported ... to have died of this too frequent Maladie."

  Although the bills of mortality provided a rich body of facts, Graunt was well aware of the shortcomings in the data he was working with. Medical diagnosis was uncertain: "For the wisest person in the parish would be able to find out very few distempers from a bare inspection of the dead body," Graunt warned. Moreover, only Church of England christenings were tabulated, which meant that Dissenters and Catholics were excluded.

  Graunt's accomplishment was truly impressive. As he put it himself, having found "some Truths, and not commonly believed Opinions, to arise from my Meditations upon these neglected Papers, I proceeded further, to consider what benefit the knowledge of the same would bring to the world." His analysis included a record of the varying incidence of different diseases from year to year, movements of population in and out of London "in times of fever," and the ratio of males to females.

  Among his more ambitious efforts, Graunt made the first reasoned estimate of the population of London and pointed out the importance of demographic data for determining whether London's population was rising or falling and whether it "be grown big enough, or too big." He also recognized that an estimate of the total population would help to reveal the likelihood that any individual might succumb to the plague. And he tried several estimating methods in order to check on the reliability of his results.

  One of his methods began with the assumption that the number of fertile women was double the number of births, as "such women ... have scarce more than one child in two years."12 On average, yearly burials were running about 13,000-about the same as the annual nonplague deaths each year. Noting that births were usually fewer in number than burials, he arbitrarily picked 12,000 as the average number of births, which in turn indicated that there were 24,000 "teeming women." He estimated "family" members, including servants and lodgers, at eight per household, and he estimated that the total number of households was about twice the number of households containing a woman of childbearing age. Thus, eight members of 48,000 families yielded an estimated 384,000 people for the total population of London. This figure may be too low, but it was probably closer to the mark than the common assumption at the time that two million people were living in London.

  Another of Graunt's methods began with an examination of a 1658 map of London and a guess that 54 families lived in each 100 square yards-about 200 persons per acre. That assumption produced an estimate of 11,880 families living within London's walls. The bills of mortality showed that 3,200 of the 13,000 deaths occurred within the walls, a ratio of 1:4. Four times 11,880 produces an estimate of 47,520 families. Might Graunt have been figuring backwards from the estimate produced by his first method? We will never know.

  Graunt does not use the word "probability" at any point, but he was apparently well aware of the concept. By coincidence, he echoed the comment in the Port-Royal Logic about abnormal fears of thunderstorms:

  Whereas many persons live in great fear and apprehension of some of the more formidable and notorious diseases, I shall set down how many died of each: that the respective numbers, being compared with the total 229,520 [the mortality over twenty years], those persons m
ay the better understand the hazard they are in.

  Elsewhere he comments, "Considering that it is esteemed an even lay, whether any man lives ten years longer, I supposed it was the same, that one of any ten might die within one year."t3 No one had ever proposed this problem in this fashion, as a case in probability. Having promised "succinct paragraphs, without any long series of multiloquious deductions," Graunt does not elaborate on his reasoning. But his purpose here was strikingly original. He was attempting to estimate average expected ages at death, data that the bills of mortality did not provide.

  Using his assessment that "about thirty six per centum of all quick conceptions died before six years old" and a guess that most people die before 75, Graunt created a table showing the number of survivors from ages 6 to 76 out of a group of 100; for purposes of comparison, the right-hand column of the accompanying table shows the data for the United States as of 1993 for the same age levels.

  No one is quite sure how Graunt concocted his table, but his estimates circulated widely and ultimately turned out to be good guesses. They provided an inspiration for Petty's insistence that the government set up a central statistical office.

  Petty himself took a shot at estimating average life expectancy at birth, though complaining that "I have had only a common knife and a clout, instead of the many more helps which such a work requires." 14 Using the word "likelihood" without any apparent need to explain what he was talking about, Petty based his estimate on the information for a single parish in Ireland. In 1674, he reported to the Royal Society that life expectancy at birth was 18; Graunt's estimate had been 16.15

  The facts Graunt assembled changed people's perceptions of what the country they lived in was really like. In the process, he set forth the agenda for research into the country's social problems and what could be done to make things better.

  Graunt's pioneering work suggested the key theoretical concepts that are needed for making decisions under conditions of uncertainty. Sampling, averages, and notions of what is normal make up the struc ture that would in time house the science of statistical analysis, putting information into the service of decision-making and influencing the degrees of belief we hold about the probabilities of future events.

  Some thirty years after the publication of Graunt's Natural and Political Observations, another work appeared that was similar to Graunt's but even more important to the history of risk management. The author of this work, Edmund Halley, was a scientist of high repute who was familiar with Graunt's work and was able to carry his analysis further. Without Graunt's first effort, however, the idea of such a study might never have occurred to Halley.

  Although Halley was English, the data he used came from the Silesian town of Breslau-Breslaw, as it was spelled in those dayslocated in the easternmost part of Germany; since the Second World War the town has been part of Poland and is now known as Wrozlaw. The town fathers of Breslaw had a long-standing practice of keeping a meticulous record of annual births and deaths.

  In 1690 a local scientist and clergyman named Caspar Naumann went through the Breslaw records with a view to "disproving certain current superstitions with regard to the effect of the phases of the moon and the so-called `climacteric' years on health." Naumann sent the results of his study to Leibniz, who in turn sent them on to the Royal Society in London. 16

  Naumann's data soon attracted the attention of Halley. Halley was then only 35 years old but already one of England's most distinguished astronomers. Indeed, he was responsible for persuading Isaac Newton in 1684 to publish his Principia, the work in which Newton first set forth the laws of gravity. Halley paid all the costs of publication out of his own modest resources, corrected the page proofs, and put his own work aside until the job was done. The historian James Newman conjectures that the Principia might never have appeared without Halley's efforts.

  Widely recognized as a child genius in astronomy, Halley carried his 24-inch telescope with him when he arrived as an undergraduate at Queen's College, Oxford. He left Oxford without receiving a degree, however, and set off to study the heavens in the southern hemisphere; the results of that study established his reputation before he was even 20 years old. By the age of 22, he was already a member of the Royal Society. Oxford turned him down for a professorship in 1691 because he held "materialistic views" that did not match the religious orthodoxy of Oxford. But the dons relented in 1703 and gave him the job. In 1721, he became Royal Astronomer at Greenwich. Meanwhile, he had received his degree by the King's command.

  Halley would live to the age of 86. He appears to have been a jolly man, with an "uncommon degree of sprightliness and vivacity," and had many warm friendships that included Peter the Great of Russia. In 1705, in his pathbreaking work on the orbits of comets, Halley identified a total of 24 comets that had appeared between the years 1337 and 1698. Three seemed to be so similar that he concluded that all three were a single comet that had appeared in 1531, 1607, and 1682. Observations of this comet had been reported as far back as 240 BC. Halley's prediction that the comet would reappear in 1758 electrified the world when the comet arrived right on schedule. Halley's name is celebrated every 76 years as his comet sweeps across the skies.

  The Breslaw records were not exactly in Halley's main line of work, but he had promised the Royal Society a series of papers for its newly established scholarly journal, Transactions, and he had been scouting around for something unusual to write about. He was aware of certain flaws in Graunt's work, flaws that Graunt himself had acknowledged, and he decided to take the occasion to prepare a paper for Transactions on the Breslaw data by trying his hand at the analysis of social rather than heavenly statistics for a change.

  Graunt, lacking any reliable figure for the total population of London, had had to estimate it on the basis of fragmentary information. He had numbers and causes of deaths but lacked complete records of the ages at which people had died. Given the constant movement of people into and out of London over the years, the reliability of Graunt's estimate was now open to question.

  The data delivered by Leibniz to the Royal Society contained monthly data for Breslaw for the years 1687 through 1691, "seeming to be done," according to Halley, "with all the Exactness and Sincerity possible"; the data included age and sex for all deaths and the number of births each year. Breslaw, he pointed out, was far from the sea, so that the "Confluence of Strangers is but small." Births exceeded the "Funerals" by only a small amount and the population was much more stable than London's. All that was lacking was a number for the total population. Halley was convinced that the figures for mortality and birth were sufficiently accurate for him to come up with a reliable estimate of the total.

  He found an average of 1,238 births and 1,174 deaths a year over the five-year period, for an annual excess of about 64, which number, he surmised, "may perhaps be balanced by the Levies for the Emperor's Service in his Wars." Directing his attention to the 1,238 annual births and examining the age distribution of those who died, Halley calculated that "but 692 of the Persons born do survive Six whole Years," a much smaller proportion than Graunt's estimate that 64% of all births survived beyond six years. About a dozen of the deaths in Breslaw, on the other hand, occurred between the ages of 81 and 100. Combining a variety of estimates of the percentage of each age group who die each year, Halley worked back from the age distribution of the people dying annually to a grand estimate of 34,000 for the town's total population.

  The next step was to devise a table breaking down the population into an age distribution, "from birth to extream Old Age." This table, Halley asserts, offered manifold uses and gave "a more just Idea of the State and condition of Mankind, than any thing yet extant that I know of." For example, the table provided useful information on how many men were of the right age for military service-9,000-and Halley suggested that this estimate of 9/34ths of the population could "pass for a Rule for other places."

  Halley's entire analysis embodies the concept of probability and ultimately m
oves into risk management. Halley demonstrates that his table "shews the odds" that a "Party" of any given age "does not die in a Year." As an illustration, he offers the 25-year age group, which numbered 567, while the 26-year age group numbered 560. The difference of only 7 between the two age groups meant that the probability that a 25-year-old would die in any one year was 7/567, or odds of 80-to-1 that a 25-year-old would make it to 26. Using the same procedure of subtraction between a later age and a given age, and taking the given age as the base, the table could also show the odds that a man of 40 would live to 47; the answer in this instance worked out to odds of 5 1/2-to-1.

  Halley carried the analysis further: "[I]f it be enquired at what number Years, it is an even Lay that a Person of any Age shall die, this Table readily performs it." For instance, there were 531 people aged 30, and half that number is 265. One could then look through the table for the age group numbering 265, which appeared to be between 57 and 58. Hence, it would be "an even Wager that ... a Man of 30 may reasonably expect to live between 27 and 28 years."

  The next level of Halley's analysis was the most important of all. The table could be used to reckon the price of insuring lives at different ages, "it being 100 to 1 that a man of 20 dies not in a year, and but 38 to 1 for a Man of 50 Years of Age." On the basis of the odds of dying in each year, the table furnished the necessary information for calculating the value of annuities. At this point Halley launches into a detailed mathematical analysis of the valuation of annuities, including annuities covering two and three lives as well as one. He offers at the same time to provide a table of logarithms to reduce the "Vulgar Arithmetick" imposed by the mass of necessary calculations.