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# The Rise of Probabilistic and Statistical Thinking

## Overview

The rise of probabilistic and statistical thinking is considered one of the most important accomplishments of nineteenth-century science. Probability and statistics played a major role as science changed the way our world was understood. At the beginning of the nineteenth century, science involved the search for universal and absolute truths. By the end of the century, scientists were dealing with the notion that some things could only be known to varying degrees of certainty. This degree of certainty was measured by probability.

## Background

Although the roots of probability extend far back into history, the beginning of mathematical probability is usually traced to a series of letters exchanged between the French mathematicians Pierre de Fermat (1601-1665) and Blaise Pascal (1623-1662) in the seventeenth century. This correspondence was based on the problem of how to distribute an amount of money wagered on a game if that game is interrupted before its conclusion. The correspondence between Pascal and Fermat influenced Christiaan Huygens (1629-1695), who published the first text on probability in 1657. Much of the writing on mathematical probability in the next 100 years was presented in gambling terms and examples.

The changes brought about by the development of probability in the nineteenth century are best understood in the context of developments in the previous century. The eighteenth century was a time of great expectations for science. Using the incredible work of Isaac Newton (1642-1727) as a starting point, scientists were making important discoveries in all parts of the natural world. The success of Newtonian principles in the physical sciences led others to apply similar principles to their own fields of study. Specialists in the biological sciences, earth sciences, and even social sciences had hopes of quantifying their studies following Newtonian principles. During this flurry of quantification, the field of probability began to take shape.

Other mathematicians contributed to the theory of probability before the nineteenth century. James Bernoulli's The Art of Conjecturing, published posthumously in 1713, addressed problems in probability related to gambling. Abraham De Moivre's Doctrine of Chances became the first English-language work on probability in 1718. In this work Moivre (1667-1754) introduced the famous normal curve, usually referred to as the bell curve today. Despite these early discussions of probability, it was not until the latter part of the eighteenth and early part of the nineteenth centuries that probability and statistics began to seriously influence science and society as a whole.

In addition to questions addressing gambling, the emerging business of insurance required an ever-increasing understanding of probability and statistics. Insurance rates had been based on guesswork and myth. Eventually, actuaries (mathematicians who worked with insurance statistics) began to pay increased attention to such things as mortality tables. A mortality table provided data on life expectancies based on the current age of the insured party. Thanks to work done by mathematicians such as Charles Babbage (1792-1871) in England, statistics and probability gradually became an important part of the insurance business.

## Impact

One of the leaders in the development of the theory of probability in the nineteenth century was French mathematician Pierre-Simon Laplace (1749-1827). In 1812 Laplace published a very influential work called Analytical Theory of Probability. In this and other works on probability, Laplace described the method of inverse probability and gave a formal proof of the least squares rule. The least squares rule is a method for fitting data to a curve. This method first appeared in a work by Adrien-Marie Legendre (1752-1833) in 1805, although Carl Friedrich Gauss (1777-1855) later claimed that he had been using the method since 1795. Regardless of who invented the method of least squares, it became an important tool in probability. Although initially applied to errors occurring in astronomical data, the use of least squares quickly spread to many other fields. Mathematical advances such as the method of least squares made probability and statistics applicable to a diverse collection of problems.

Laplace, like other mathematicians working on probability, developed his techniques for specific applications. In Laplace's case his work in probability was used in astronomy. Laplace noticed that the errors involved in astronomical observations followed the normal distribution introduced by De Moivre almost a century earlier. This normal distribution was to become a foundation for many of the applications of statistics and probability in the nineteenth century.

Laplace also influenced the way probability was perceived in the nineteenth century. Historically, there had been considerable resistance to the incorporation of probability into scientific disciplines. The perception was that probabilistic knowledge was imperfect knowledge. Laplace, and others, applied probability when absolute knowledge was not possible. Laplace believed that probability addressed "the important questions of life" for which complete knowledge was "problematical." This philosophy, known as classical probability, maintained that probability was a measure of the degree of certainty or rational belief. In this philosophy, probability was a measure of man's ignorance. However, probability was also a guide that would help a rational man plan his actions. The acceptance of probabilistic knowledge in place of absolute knowledge in science represented a fundamental change in Western culture.

The classical conception of probability was applied to a variety of life's problems. Its uses ranged from fairly obvious applications such as life insurance to less apparent applications such as deciding the credibility of a trial witness or the rationality of administering the smallpox vaccine. Probability affected the lives of nineteenth-century humans in many ways. It was not until late in the nineteenth century that the classical interpretation of probability gave way to the frequentist view. This view maintained that probabilistic events were not simply a result of man's lack of certain knowledge. Instead, it was believed that probability was an inherent part of the world's structure. In other words, there were some things that could not be known absolutely but only probabilistically.

Ever since Newton quantified astronomy and mechanics, other sciences had attempted to follow a similar pattern of quantification. Even in areas traditionally qualitative, such as the study of man and society, the Newtonian system was looked upon as a model to emulate. In the nineteenth century, this model was applied to statistics describing man and society.

Previous attempts had been made to apply probability to the statistics of society. However, it was Belgian astronomer Adolphe Quetelet (1796-1874) who developed extensive applications for probability in analyzing social statistics. Quetelet noticed that statistics as varied as the height and weight of individuals and the number of crimes and suicides that occurred in a society fit a normal curve. This normal curve, which had already been applied to astronomy, was used by Quetelet to develop what he called a "social physics." Quetelet's social physics was an attempt to quantify the social sciences in the same way that Newton had quantified the physical sciences. He was interested in subjecting the seemingly chaotic data of society to statistical laws. Quetelet looked for these laws using the same techniques as astronomers used in analyzing errors in data. By accumulating statistics related to man's physical dimensions and his personal characteristics, Quetelet believed he could describe the "average man." The average man represented the true type of the human species. Any deviations from the average were considered "errors." These terms were chosen by Quetelet to emphasize the similarity to statistical work in astronomy.

Quetelet showed that births, deaths, marriages, crimes, and suicides were almost constant in any given country, independent of individual actions. This seemed to Quetelet to confirm that stability might be found in any data given a large enough set of statistical information. This stability represented an example of the law of large numbers applied to social conditions. The phrase "law of large numbers" was coined by Simeon-Denis Poisson in 1835. It stated that over a long period of time, the frequency of an event must become ever closer to the probability of its outcome. For example, the probability that a coin flip will result in heads is one half. If a coin is flipped repeatedly, the law of large numbers says that the frequency of heads will approach fifty percent. Quetelet held that important conclusions might be drawn from the statistics of society using the law of large numbers. He believed that the conclusions of social physics could be used as a moral standard and that society might be improved through an understanding of sickness, crime, and other societal ills.

Quetelet is also important because of his influence on other scientists who used probability to revolutionize their fields. Three such men were Francis Edgeworth, Francis Galton (1822-1911), and Karl Pearson (1857-1936). Edge-worth applied error analysis and the normal curve to topics in economics, and Galton applied the same concepts to the study of heredity. Galton also developed an important method in statistics called correlation. Correlation measures the tendency of one variable to change as a related variable changes. For instance, correlation would measure the tendency of lung cancer cases to increase as the number of smokers in a population increases. Pearson further developed the work of Galton and in the process created the field of biometrics, which is the application of statistics to the biological sciences.

Quetelet's work also exerted a strong influence over physicist James Clerk Maxwell (1831-1879). Maxwell, along with Ludwig Boltzmann (1844-1906), developed a statistical theory of gases that revolutionized physics and created the field of statistical mechanics. Maxwell's theory of gases relied upon the idea that something that cannot be understood individually (the motion of a single molecule) may be understood as an aggregate (the motion of an accumulation of molecules). This idea mirrored the way probability was used in Quetelet's social physics.

These examples are by no means the only areas influenced by probability and statistics. Psychology, agronomy, medicine, and numerous other fields of study adopted probability as a key tool for evaluating data. Most importantly, though, the rise of probabilistic and statistical thinking forever changed the way humankind viewed knowledge.

TODD TIMMONS