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*THE HISTORY OF STATISTICS The Measurement of Uncertainty Before 1900*

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SCIENCE &TECHNOLOGY; Figures for Reformers

By MORRIS KLINE; MORRIS KLINE, A PROFESSOR EMERITUS OF MATHEMATICS AT NEW YORK UNIVERSITY, IS THE AUTHOR OF ''MATHEMATICS: THE LOSS OF CERTAINTY'' AND ''MATHEMATICS AND THE SEARCH FOR KNOWLEDGE.''

THE RISE OF STATISTICAL THINKING #1820-1900. By Theodore M. Porter. 333 pp. New Jersey: Princeton University Press. $35. THE HISTORY OF STATISTICS The Measurement of Uncertainty Before 1900. By Stephen M. Stigler. Illustrated. 410 pp. Cambridge, Mass.: The Belknap Press/Harvard University Press. $25.

These two books dealing with the history of statistics up to 1900 differ considerably in their aims, but each succeeds admirably in what it sets out to do. Since Theodore M. Porter is an assistant professor of history at the University of Virginia, it is not surprising that he scants the mathematical aspects in ''The Rise of Statistical Thinking: 1820-1900,'' though he does not ignore them. Not only will the reader not need mathematics; until about 1830 neither did the statistician. The German word ''Statistik,'' from which the subject derives its name, was used by its coiner in 1749 to refer to a descriptive science of state. It was a geography that revealed the state of production and consumption of one or several nations for several successive points of time. Generally, during the early 19th century numbers were regarded as secondary. But by 1830 statistics was acquiring its association with the collection and analysis of numerical data.

Beginning in the second quarter of that century, the collection of data became a wide-ranging enterprise. The data included figures on population, climate, trade, poverty, education and crime. The motivation for statistical investigations was often reformist; it rested on the belief that statistics would make it possible to erect a scientific basis for a progressive social policy. It was in that vein that Jean Jacques Rousseau invoked the statisticians: ''Experts in calculation! I leave it to you to count, to measure, to compare.'' The members of the several statistical societies formed in England in the mid-1800's were more likely to have reformist interests than scientific ones.

Adolphe Quetelet, who was the leading precursor of the modern statistician, shared the concerns of the reformers but believed that more than just facts were needed. His aim was to erect a numerical social science that would bring order to social chaos. Coming to statistics from astronomy, he was familiar with mathematical probability theory and convinced of the importance Continued on next page of mathematics in attaining his goal. Actually, Quetelet almost never used mathematics. He compiled and arranged statistical data to discover what could be learned about phenomena like birth and death rates, marriages and divorces and crime. His great innovation was to apply to real variations in nature the probabilistic error law that had been developed by mathematicians to deal with errors arising in the taking of measurements. Such random errors occur for several reasons, including the observer's mistakes and the expansion or contraction of the substance being measured and the measuring instrument. The error law states that repeated measurement of an object will result in varying numbers that, when graphed, form a bell-shaped error curve, now more familiarly known as the normal curve. QUETELET showed that variations in human heights, for example, conformed to this law of errors, and it was his perception of the wider applicability of the error law that provided the inspiration for the important work in statistics done in the late 19th century. His lasting contribution to science was to establish the concept of a statistical law - the notion that true facts about a mass can be discovered even when information about the constituent individuals is unattainable.

Clerk Maxwell and Ludwig Boltzmann, who developed the kinetic theory of gases, were among the scientists who profited from Quetelet's approach. Mr. Porter writes: ''Doubtless it would be too brave to argue that statistical gas theory only became possible after social statistics had accustomed scientific thinkers to the possibility of stable laws of mass phenomena with no dependence on predictability of individual events. Still, the actual history of the kinetic gas theory is fully consistent with such a claim.''

The book presents the substantial contributions of Frances Galton and Karl Pearson at the end of the century, which led to present-day mathematical statistics. Here too the general reader can readily follow the exposition, for it almost totally avoids mathematical expressions. Modern statisticians date the beginning of their discipline from 1889, when Galton published his book on biological inheritance and evolution and when his method of correlation appeared. In 1895 Pearson introduced the correlation coefficient as well as the terms ''normal curve'' and ''standard deviation.'' Quantitative genetics, the area in which both Galton and Pearson worked and which generated their innovative methodologies, remains the best example of a science whose very theory is built out of the concepts of statistics.

An outstanding feature of Mr. Porter's book is its depiction of the interrelationships between statistics and certain intellectual and social movements. Determinism had been the reigning philosophical doctrine since the 18th century, enthroned by the mathematical spirit of that age. The regularity of figures on crime and suicide seemed to indicate that there were statistical laws governing human behavior, just as the the law of gravitation determined certain motions. The existence of such statistical laws was disturbing, for it gave the lie to free will and human responsibility. A defense could be made that a statistical law did not apply to individuals. But an even stronger argument for free will came from statistics itself. The new probabilistic developments in physics led to the recognition of chance as a fundamental aspect of the world. A consequence was that determinism gave way to indeterminism.

Mr. Porter's book is unfailingly interesting. It is equipped with an excellent index, and the scholar will find that it is thoroughly footnoted. In ''The History of Statistics: The Measurement of Uncertainty Before 1900,'' Stephen M. Stigler, who is a professor of statistics at the University of Chicago, has written a more technical work. The content of the book, with its extensive bibliography, will be of special interest to students of mathematics and statistics and to those who use statistics. However, the book is a model of exposition; the presentation is so clear and thorough that the general reader will find most of the book comprehensible and interesting even if he cannot follow the mathematics.

A good example of Mr. Stigler's style and content is his discussion of the contrasting approaches taken by a mathematician and a statistician. Leonhard Euler, one of the greatest mathematicians of all time, and Tobias Mayer, an astronomer, faced similar problems involving astronomical observations. Euler in 1748 was reduced to groping for an answer and did not succeed. Mayer in 1749 devised a sensible statistical solution. In explaining this disparity, Mr. Stigler not only reproduces the table of 27 equations Mayer derived from his observations and the calculations performed on them but he elucidates every step of the procedures. And to make sure that the reader understands the astronomer's goal, Mr. Stigler precedes this discussion with a lesson, complete with diagram, on spherical trigonometry. Would that my colleagues in mathematics took such pains when writing a textbook. Mr. Stigler similarly demonstrates Euler's attack and ends with an explanation of why the mathematical approach was doomed to failure. ''T HE HISTORY OF STATISTICS'' is divided into three parts. Part One deals with the development of the method of least squares and the probability curve. Attention is focused on the work of Adrien Marie Legendre, Jacob Bernoulli, Abraham De Moivre, Thomas Bayes, Pierre Simon Laplace and Carl Friedrich Gauss, though the accomplishments of others are also discussed. This mathematical account culminates in the Gauss-Laplace synthesis of 1810, which ''brought together two well-developed lines - one the combination of observations through the aggregation of linearized equations of condition, the other the use of mathematical probability to assess uncertainty and make inferences - into a coherent whole.''

Parts Two and Three are concerned with the application to the social sciences of this synthesis of the method of least squares and the theory of errors. It forms an enormously detailed history covering the period 1827 to 1900. Over and above a full account of developments in statistical methodology, the book provides biography, critiques by the author and by contemporaries of the inventors, and discussions of social factors, intellectual responses and scientific developments related to the interpretation of statistical data. The researchers featured in this period are Quetelet, Simeon Poisson, Wilhelm Lexis, Gustav Fechner, Hermann Ebbinghaus, Francis Galton, Karl Pearson, Francis Edgeworth and George Udny Yule.

Mr. Stigler concludes with the observation that the advances in scientific logic that took place in statistics before 1900 were to be every bit as influential as those associated with the names of Newton and Darwin. One is tempted to say that the history of statistics in the 19th century will be associated with the name Stigler.

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The Rise of Statistical Thinking, 1820-1900
Paper | 1988 | |