De Moivre Describes the Bell-Shaped Curve

Date 1733

Abraham de Moivre was the first person to describe the so-called normal curve, a symmetrical bell-shaped graph that symbolizes probability distribution. This graph of the average distribution of events resolved a serious issue that had been left hanging by the previous generation of mathematicians.

Locale London, England

Key Figures

• Abraham de Moivre (1667-1754), French-born English mathematician
• Jakob I Bernoulli (1655-1705), Swiss mathematician
• Nikolaus Bernoulli (1687-1759), Swiss mathematician, lawyer, and editor of Jakob’s posthumous text

Summary of Event

The earliest mathematical work on probability involved problems with dice. Throws of the dice could be described by a function called the binomial distribution, which provided the probability of any given result coming up a set number of times given a set number of tosses of the dice. For example, given thirty-six tosses, 7 is most likely to come up six times; the next most likely outcomes are five times or seven times, then four times or eight times, and so on. The binomial distribution provides the exact probability of each result.

Problems relating to the binomial distribution proved difficult to answer until the mathematician Abraham de Moivre found a graphical way to approximate the function. The approximation had the shape of a bell. To continue the example of the dice, rolling six 7’s would form the highest point on the bell, which would then slope downward to either side. This approximation enabled de Moivre to answer important questions about games of dice and other situations in which probability could be represented by the binomial distribution.

De Moivre was born in France to a Protestant family. He received his education in France but then left the country at the time of the revocation of the Edict of Nantes (1685) in view of the prospective danger to Protestants represented by Louis XIV’s abandonment of the policy of religious tolerance protected by that edict. De Moivre’s mathematical work was all done in England. Although he was a distinguished mathematician, however, he never fit into the English mathematical world. Most of his life he had to eke out a living by tutoring students and answering questions about the applications of probability.

Nevertheless, de Moivre was highly esteemed by his contemporaries, including Sir Isaac Newton, the most eminent mathematician of the period. Newton is said to have told students with questions to ask de Moivre on the grounds that de Moivre knew the material better than did Newton. There is no evidence that Newton was being ironical in making such a strong claim, and de Moivre’s name is still attached to an important theorem about powers of complex numbers.

De Moivre developed an interest in probability by reading some of the earliest treatments of the subject, which had only acquired mathematical respectability in the middle of the seventeenth century with the work of Blaise Pascal and Pierre de Fermat. He probably first read a treatise on probability by the Dutch mathematician Christiaan Huygens and shortly thereafter wrote one of the first English accounts of the subject, “De mensura sortis” (1711; on the measurement of chance), published in the Philosophical Transactions of the Royal Society. The progress of de Moivre’s subsequent work on probability can be measured by the later editions of this text, which appeared in English as the textbook The Doctrine of Chances: Or, A Method of Calculating the Probability of Events in Play (1718, 1738, 1756). De Moivre also had a strong philosophical interest in the application of probability, which led him to draw philosophical conclusions from his mathematical results.

After the appearance of de Moivre’s Latin text in the Royal Society’s Philosophical Transactions, the foundations of probability theory were transformed by the posthumous appearance of Jakob I Bernoulli’s Ars conjectandi (1713; the conjectural arts). The work was prepared for the press by Bernoulli’s nephew, Nikolaus Bernoulli. Nikolaus himself had used his uncle’s theories in a dissertation he submitted to the faculty of law at the University of Basel. The work was intended to illustrate the applications of probability to law, although it seems to have been of more interest to mathematicians than to lawyers.

One of the advances in Jakob I Bernoulli’s treatment of probability was his formulation of the binomial distribution. The distribution was produced by looking at a sequence of identical experiments, where each had one possible target outcome (called a success) and one or more other outcomes (which would be collected together and called a failure). For example, if one throws a die a number of times, one could call a 6 a success and any other number a failure.

Bernoulli showed that the relative frequency of an event with probability p in n independent trials converges to p as n gets bigger. In other words, the odds of rolling a 6 are one in six: One could easily roll a die six times and not roll a 6; however, if one were to roll the same die one thousand times, it would be surprising if 6 did not come up about one-sixth of the time (approximately 167 times), and if one were to roll the die 1 million times, it would be extremely surprising if roughly one-sixth of the rolls did not result in 6’s. The more times the die is rolled, the closer to the average or ideal theoretical results one’s actual results will be. This is known as the law of large numbers, and it furnished a basis for the application of probability theory to practical situations in the physical world.

Despite demonstrating the law of large numbers theoretically, Bernoulli was unable to find a manageable way to perform the necessary arithmetic calculations to determine the probability of specific ranges of outcomes when the number of trials became large. For example, it was easy to calculate the probability of two successes in six trials, but it was much harder to perform the arithmetic in the case of two hundred successes in six hundred trials. (In both cases, the probability is close to one in three, but it is actually slightly smaller. One needs to crunch the numbers to find the exact probability.) The difficulty of performing extended arithmetic calculations made it difficult in turn to extend the general results Bernoulli had obtained to any specific situation. The algebra of dealing with the sum of many terms of a polynomial did not have an obvious solution.

De Moivre was able to see both the importance of Bernoulli’s problem in this regard and the most fruitful direction to explore in order to find a solution. Earlier in his career, de Moivre had found a way to approximate factorials of large numbers. (Factorials are products of all the positive integers from 1 up to a certain number, so that 5 factorial, written 5!, is equal to 5 × 4 × 3 × 2 × 1.) De Moivre had given credit for his method to the Scottish mathematician James Stirling, even though de Moivre had figured it out before Stirling did. The use to which de Moivre put the so-called Stirling’s formula altered the course of probability theory thereafter. Indeed, de Moivre felt that the work was of such importance that he published it at his own expense.

De Moivre’s application of Stirling’s formula was published first in Latin as a pamphlet supplementing his Miscellanea analytica (1730), Approximation ad summam terminorum binomii (a + b) in seriem expansi: Supplement to “Miscellanea Analytica” (1733; approximation to the sum of the terms of a binomial [a + b] expanded in a series). It was later incorporated in English into subsequent editions of The Doctrine of Chances. What de Moivre accomplished in this pamphlet was the introduction of a curve known to mathematicians as the normal distribution and more popularly as the bell-shaped curve, or simply the bell curve. This curve would have been impossible to conceive without the calculus, but de Moivre was able to use the techniques of the calculus to make a number of statements about what the curve was like. He did not actually write down what mathematicians now regard as the strict mathematical definition of the curve, but his results indicated that he understood it well enough to use it.

The bell-shaped curve enabled de Moivre to develop a good approximation of the probability of ranges of outcomes in the binomial distribution, thereby solving the problem that had plagued Bernoulli. One of the major consequences of the curve was that it made it possible to determine the rough probability of a range of outcomes clustered around the center of the distribution for large numbers of trials. For example, if one flipped a coin 600 times, one could use de Moivre’s technique to determine the likelihood of getting a number of heads between 250 and 350. Even more important, de Moivre’s curve enabled the work of Bernoulli to be expressed in a more concrete, quantitative form. In evaluating the series that he obtained at values that were multiples of the square root of the number of trials, de Moivre concluded that the natural unit for measuring the deviation from the center would be that square root.

Significance

The appearance of the normal distribution in the work of de Moivre permanently altered the emerging science of probability theory and its applications. One of the difficulties that Jakob Bernoulli had encountered in his work was in applying his results to statistical inference. The binomial distribution was the easiest distribution to describe mathematically, making it the best suited for creating a mathematical theory of statistical inference. De Moivre’s curve was a necessary stepping stone to such a theory, which was created in the next generation by Pierre-Simon Laplace. Laplace also added a few details that de Moivre had omitted (such as a formal proof of his main result).

The bell-shaped curve has made its appearance in all sorts of investigations and has been liable to misuse as well as useful applications. The conditions underlying the proper application of the curve have been studied at length and just as studiously ignored by those who have seen it as the one necessary ingredient for a probabilistic analysis. The normal distribution has probably been cursed by students who are under the impression that it was responsible for “grading on a curve.” Nevertheless, the language for measuring errors and deviations from a set standard has depended for many years on the normal distribution.

Bibliography

Daston, Lorraine. Classical Probability in the Enlightenment. Princeton, N.J.: Princeton University Press, 1988. Investigates de Moivre’s use of probability (including the normal distribution) to tackle questions of religion and philosophy.

David, Florence Nightingale. Games, Gods, and Gambling. London: Charles Griffin, 1962. Appendix 5 offers the English version of de Moivre’s presentation of the normal distribution.

Hald, Anders. A History of Mathematical Statistics from 1750 to 1950. New York: John Wiley and Sons, 1998. The second chapter summarizes the state of probability in the middle of the eighteenth century after de Moivre.

‗‗‗‗‗‗‗. A History of Probability and Statistics and Their Applications Before 1750. New York: John Wiley, 1990. Extensive reconstruction of de Moivre’s text, including what de Moivre’s proof of the normal approximation might have been.

Schneider, Ivo. “Abraham de Moivre.” Statisticians of the Centuries. New York: Springer, 2001. Schneider is the author of the definitive study of de Moivre.

Stigler, Stephen M. The History of Statistics. Cambridge, Mass.: Harvard University Press, 1986. Most definitive assessment of de Moivre’s work for subsequent developments in probability.