Karl Pearsons Theoretical Errors and the Advances They Inspired

arXiv:0808.4032v1 [stat.ME] 29 Aug 2008 Statistical Science 2008, Vol. 23, No. 2, 261–271 DOI: 10.1214/08-STS256 c ⃝Institute of Mathematical Statistics, 2008 Karl Pearson’s Theoretical Errors and the Advances They Inspired Stephen M. Stigler Abstract. Karl Pearson played an enormous role in determining the content and organization of statistical research in his day, through his research, his teaching, his establishment of laboratories, and his ini- tiation of a vast publishing program. His technical contributions had initially and continue today to have a profound impact upon the work of both applied and theoretical statisticians, partly through their in- adequately acknowledged influence upon Ronald A. Fisher. Particular attention is drawn to two of Pearson’s major errors that nonetheless have left a positive and lasting impression upon the statistical world. Key words and phrases: Karl Pearson, R. A. Fisher, Chi-square test, degrees of freedom, parametric inference, history of statistics. 1. INTRODUCTION Karl Pearson surely ranks among the more pro- ductive and intellectually energetic scholars in his- tory. He cannot match the most prolific humanists, such as one of whom it has been said, “he had no unpublished thought,” but in the domain of quanti- tative science Pearson has no serious rival. Even the immensely prolific Leonhard Euler, whose collected works are still being published more than two cen- turies after his death, falls short of Pearson in sheer volume. A list of Pearson’s works fills a hardbound book; that book lists 648 works and is still incom- plete (Morant, 1939). My own moderate collection of his works—itself very far from complete (it omits his contributions to Biometrika)—occupies 5 feet of Stephen M. Stigler is the Ernest DeWitt Burton Distinguished Service Professor in the Department of Statistics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637, USA e-mail: stigler@uchicago.edu. This paper is based upon a talk presented at the Royal Statistical Society in March 2007, at a symposium celebrating the 150th anniversary of Karl Pearson’s birth. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in Statistical Science, 2008, Vol. 23, No. 2, 261–271. This reprint differs from the original in pagination and typographic detail. shelf space. And his were not casually constructed works: when a student or a new co-worker would do the laborious calculations for some statistical anal- ysis, Pearson would redo the work to greater accu- racy, as a check. An American visiting Pearson in the early 1930s once asked him how he found the time to write so much and compute so much. Pear- son replied, “You Americans would not understand, but I never answer a telephone or attend a commit- tee meeting” (Stouffer, 1958). Pearson’s accomplishments were not merely volu- minous; they could be luminously enlightening as well. Today the most famous of these are Pearson’s Product Moment Correlation Coefficient and the Chi- square test, dating respectively from 1896 and 1900 (Pearson, 1896, 1900a, 1900b). He was a driving force behind the founding of Biometrika, which he edited for 36 years and made into the first important journal in mathematical statistics. He also estab- lished another journal (the Annals of Eugenics) and several additional serial publications, two research laboratories, and a school of statistical thought. Pear- son pioneered in the use of machine calculation, and he supervised the calculation of a series of mathe- matical tables that influenced statistical practice for decades. He made other discoveries, less commonly associated with his name. He was in 1897 the first to name the phenomenon of “spurious correlation,” thus publicly identifying a powerful idea that made 1 2 S. M. STIGLER him and countless descendents more aware of the pitfalls expected in any serious statistical investi- gation of society (Pearson, 1897). And in a series of investigations of craniometry he introduced the idea of landmarks to the statistical study of shapes. Pearson was at one time well known for the Pear- son Family of Frequency Curves. That family is sel- dom referred to today, but there is a small fact (re- ally a striking discovery) he found in its early de- velopment that I would call attention to. When we think of the normal approximation to the binomial, we usually think in terms of large samples. Pear- son discovered that there is a sense in which the two distributions agree exactly for even the smallest number of trials. It is well known that the normal density is characterized by the differential equation d dx log(f(x)) = f ′(x) f(x) = −(x −µ) σ2 . Pearson discovered that p(k), the probability func- tion for the symmetric binomial distribution (n in- dependent trials, p = 0.5 each trial), satisfies the analogous difference equation exactly: p(k + 1) −p(k) (p(k + 1) + p(k))/2 = −(k + 1/2) −n/2 (n + 1) · 1/2 · 1/2 or rate of change p(k)

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