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The History of Statistics: The Measurement of Uncertainty Before 1900

Stephen M. Stigler
Belknap Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Scott Guthery
, on

A measurement is one of those scientific constructs like time and mass: you think your intuition has it covered until you start to build with the construct and discover that your intuition is not as well informed as you thought it was. Stephen Stigler’s The History of Statistics: The Measurement of Uncertainty before 1900 is the story of the discovery and bridging of gaps between intuitions about and reasoning with quantitative measurements.

The book begins with a question that bedeviled experimental philosophy: how do you combine two measurements of the same thing that are different? Book III of Newton’s Principia Mathematica is for the most part devoted to this problem. Regarding Problem XXI, From three observations given to determine the orbit of a Comet moving in a parabola, for example, Newton writes (in translation),

This being a problem of great difficulty, I try’d many methods of resolving it; and several of those problems, the composition whereof I have giv’n in the first book, tended to this purpose. But afterwards I contrived the following solution, which is something more simple.

Newton was analyzing observations of the comet of 1680 made by people with different (and largely unknown to Newton) levels of metrology expertise, using different mathematical instruments, and in different locations around the world.

Using detailed analysis of the works of Tobias Mayer on the libration of the moon, Euler on the orbits of Saturn and Jupiter, and Roger Boscovich on the shape of the earth, among others, Stigler recounts how the method of least squares slowly emerged as method for combining observations in a way that enhanced rather than dissipated their utility

This methodology frames each chapter in Stigler’s book. First, Stigler describes the then-current state of thinking about some aspect of measurement in a particular scientific context. Then he recounts examples of a conceptual conundrum that people encountered when they tried to extend the science based on their understanding. Next, comes an analysis of the various attempts to resolve the conundrum. And finally, there is a convergence to a consensus understanding of the nature of the barrier as well as an agreed-upon way of resolving it and moving to a new level of understanding.

After the combination of observations and least squares, Stigler takes up probability and the measurement of uncertainty, inverse probability and Bayes theory, the central limit theorem, sampling and the law of large numbers, distributions other than the normal, and, finally, correlation and regression. In each case Stigler uses detailed analyses of examples drawn from the literature of the day to explore the feedback loop between an intuition and its mathematization.

The book’s bibliography spans twenty-five pages and includes twenty-three articles by Stigler himself. (This is one of those rare situations wherein self-citation is to be lauded.) Together with four-pages of suggestions for further reading we have at hand many starting points for further exploration of the history of statistics. There are two appendices, each being a syllabus for a series of lectures by F. Y. Edgeworth at King’s College.

The strength of Stigler’s history is that we witness an aspect of doing mathematics that is, in my view, too infrequently written about; the difficulties in getting from scientific phenomena and their measurement to mathematical notations and their calculus. Chances of events, observation error, and correlated events are all aspects of measurement about which everyone has an intuitive understanding. There are huge difficulties attendant to rendering this understanding mathematically so that as you manipulate the mathematics the intuition comes along. Stigler’s book is a scholarly telling of the early history of statistics, but more than that: it is collection of case studies of how mathematics is really done that is as relevant today as it was when Newton was trying to figure out how to combine comet observations from Boston, Massachusetts, and Cambridge, England.

Scott Guthery is the author of Practical Purposes: Readers in Experimental Philosophy at the Boston Athenaeum (1827–1850), in which he uses the book borrowing registers of the Athenaeum to characterize the scientific and technical reading preferences of Boston’s antebellum mathematical practitioners. His previous book, A Motif of Mathematics, explored the history and application of the mediant and the Farey sequence. Guthery received a PhD in probability and statistics from Michigan State University and worked for Bell Laboratories, Schlumberger, and Microsoft before co-founding two of his own companies. He can be reached by e-mail at

  • Introduction
  • I. The Development of Mathematical Statistics in Astronomy and Geodesy before 1827
    • 1. Least Squares and the Combination of Observations
      • Legendre in 1805
      • Cotes’s Rule
      • Tobias Mayer and the Libration of the Moon
      • Saturn, Jupiter, and Euler
      • Laplace’s Rescue of the Solar System
      • Roger Boscovich and the Figure of the Earth
      • Laplace and the Method of Situation
      • Legendre and the Invention of Least Squares
    • 2. Probabilists and the Measurement of Uncertainty
      • Jacob Bernoulli
      • De Moivre and the Expanded Binomial
      • Bernoulli’s Failure
      • De Moivre’s Approximation
      • De Moivre’s Deficiency
      • Simpson and Bayes
      • Simpson’s Crucial Step toward Error
      • A Bayesian Critique
    • 3. Inverse Probability
      • Laplace and Inverse Probability
      • The Choice of Means
      • The Deduction of a Curve of Errors in 1772–1774
      • The Genesis of Inverse Probability
      • Laplace’s Memoirs of 1777–1781
      • The Error Curve of 1777
      • Bayes and the Binomial
      • Laplace the Analyst
      • Nonuniform Prior Distributions
      • The Central Limit Theorem
    • The Gauss–Laplace Synthesis
      • Gauss in 1809
      • Reenter Laplace
      • A Relative Maturity: Laplace and the Tides of the Atmosphere
      • The Situation in 1827
  • II. The Struggle to Extend a Calculus of Probabilities to the Social Sciences
    • 5. Quetelet’s Two Attempts
      • The de Keverberg Dilemma
      • The Average Man
      • The Analysis of Conviction Rates
      • Poisson and the Law of Large Numbers
      • Poisson and Juries
      • Comte and Poinsot
      • Cournot’s Critique
      • The Hypothesis of Elementary Errors
      • The Fitting of Distributions: Quetelismus
    • 6. Attempts to Revive the Binomial
      • Lexis and Binomial Dispersion
      • Arbuthnot and the Sex Ratio at Birth
      • Buckle and Campbell
      • The Dispersion of Series
      • Lexis’s Analysis and Interpretation
      • Why Lexis Failed
      • Lexian Dispersion after Lexis
    • 7. Psychophysics as a Counterpoint
      • The Personal Equation
      • Fechner and the Method of Right and Wrong Cases
      • Ebbinghaus and Memory
  • III. A Breakthrough in Studies of Heredity
    • 8. The English Breakthrough: Galton
      • Galton, Edgeworth, Pearson
      • Galton’s Hereditary Genius and the Statistical Scale
      • Conditions for Normality
      • The Quincunx and a Breakthrough
      • Reversion
      • Symmetric Studies of Stature
      • Data on Brothers
      • Estimating Variance Components
      • Galton’s Use of Regression
      • Correlation
    • 9. The Next Generation: Edgeworth
      • The Critics’ Reactions to Galton’s Work
      • Pearson’s Initial Response
      • Francis Ysidro Edgeworth
      • Edgeworth’s Early Work in Statistics
      • The Link with Galton
      • Edgeworth, Regression, and Correlation
      • Estimating Correlation Coefficients
      • Edgeworth’s Theorem
    • 10. Pearson and Yule
      • Pearson the Statistician
      • Skew Curves
      • The Pearson Family of Curves
      • Pearson versus Edgeworth
      • Pearson and Correlation
      • Yule, the Poor Law, and Least Squares: The Second Synthesis
      • The Situation in 1900
  • Appendix A. Syllabus for Edgeworth’s 1885 Lectures
  • Appendix B. Syllabus for Edgeworth’s 1892 Newmarch Lectures
  • Suggested Readings
  • Bibliography
  • Index