Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3-942944-20-5 © Oscar Sheynin 2012 NG Verlag Berlin 1 Contents 0. Introduction 0.1. General information 0.2. The pertinent disciplines and their stages 1. Prehistory 1.1. Randomness 1.2. Probability 1.3. Cause vs chance 1.4. Expectation 1.5. Astronomy 1.6. Astrology 1.7. Treatment of observations 2. Early history 2.1. Stochastic ideas in science and society 2.2. Mathematical investigations 3. Jakob Bernoulli and the law of large numbers 3.1. Bernoulli’s works 3.2. Art of Conjecturing : part four 3.3. Bernoulli’s contemporaries 4. De Moivre and the De Moivre – Laplace limit theorem 4.1. The Measurement of Chance (1712) 4.2. Life insurance 4.3. The Doctrine of Chances (1718, 1738, 1756) 4.4. The De Moivre – Laplace limit theorem 5. Bayes 5.1. The Bayes formula and induction 5.2. The limit theorem 5.3. Additional remark 6. Other investigations before Laplace 6.1. Stochastic investigations 6.2. Statistical investigations 6.3. Treatment of observations 7. Laplace 7.1. Theory of probability 7.2. Theory of errors 7.3. Critical considerations 8. Poisson 8.1. Probability and statistics 8.2. Theory of errors 8.3. Criminal statistics 8.4. Statistical physics 8.5. Medical statistics 9. Gauss, Helmert, Bessel 9.1. Gauss 9.2. Helmert 9.3. Bessel 10. The second half of the nineteenth century 10.1. Cauchy 10.2. Bienaymé 10.3. Cournot 10.4. Buniakovsky 10.5. Quetelet 10.6. Galton 10.7. Statistics 10.8. Statistics and natural sciences 10.9. Natural scientists 11. Bertrand and Poincaré 11.1. Bertrand: general information 11.2. Bertrand: the random chord 11.3. Poincaré 2 12. Chebyshev 12.1. His contributions 12.2. His lectures 12.3. Some general considerations 13. Markov, Liapunov, Nekrasov 13.1 Markov: personal traits 13.2. Markov: scientific issues 13.3. Markov: main investigations 13.4. Liapunov 13.5. Nekrasov 14. The birth of mathematical statistics 14.1. Stability of statistical series 14.2. The Biometric school 14.3. The merging of the two streams? Supplement: axiomatization Bibliography Index of names 3 0. Introduction I do feel how wrongful it was to work for so many years at statistics and neglect its history K. Pearson (1978, p. 1) 0.1. General Information. This book is intended for those interested in the history of mathematics or statistics and more or less acquainted with the latter. It will also be useful for statisticians. My exposition is based, in the first place, on my own investigations published over some 35 years and monograph (2009) and I stop at the axiomatization of probability and at the birth of the real mathematical statistics, i.e., at Fisher. In § 9.1.3 I succeeded in greatly simplifying Gauss’ mature justification of least squares. Among the preceding literature I single out the great work of Hald (1900; 1998). However, his second book does not touch on the contribution of the Continental direction of statistics (see my § 14.1) and only describes everything from a modern point of view. It is therefore only intended for highly qualified readers (my account is much easier to understand). Second, Hald does not describe the contents of any given contribution and the reader will not know what exactly was contained in, say, any of Laplace’s memoirs. At least in my field the situation is greatly worsened by bad reviewing. Reputable publishers sometimes reprint literature without consulting their authors so that the unsuspecting reader gets dated information. Gnedenko & Sheynin (1978/1992) reprinted in 2001 in the volume put out by Birkhäuser is a good example. Many bad books are also appearing because their manuscripts had not been properly reviewed. Subsequent comments are often no more than sweet nothings or downright misleading. An ignorant author who stated that Poisson had proved the strong law of large numbers was praised as a scholar of the first rank. In some cases the cause of such facts is apparently well described by the saying Scratch my back, and I’ll scratch yours . Incidentally, this is a consequence of publishers supplying free copies of their new books to editors of periodicals. Then, abstracting journals are as a rule publishing whatever they get from their reviewers. But, first and foremost, the scientific community wrongly does not set high store on that most important work. Even worse: Truth is dismissed as an old-fashioned superstition. This conclusion (Truesdell 1984, p. 292) which concerned scientific work in general fell on deaf ears. About 1985 the then Editor of Historia Mathematica visited Moscow and made a report at the (now, Vavilov) Institute for History of Natural Sciences and Technology. Answering a question, he said that only a few readers of his periodical read Russian. I do not think that that situation had changed, much to the detriment of science. For that matter, students of the humanities certainly become versed in older masters, but not in modernity. How, indeed, can we otherwise explain why did a young offspring of the British Royal family go to a fancy-dress party clad as a German officer of SS? With sincere gratitude I recall the late Professors Youshkevitch, without whose assistance I, living in Moscow, would have been unable to publish abroad, and Truesdell, the Editor of the Archive for History of Exact Sciences , who had to busy himself with my English and compelled me to 4 pay due attention to style. In 1991, after moving to Germany, I became able to continue my work largely because of Professor Pfanzagl’s warm support. He secured a grant for me (which regrettably dried up long ago) from Axel- Springer Verlag. Professor Strecker essentially helped me to prepare and publish both English editions of my Russian book (1990). In my papers, I had acknowledged the help of many colleagues including the late Doctors Chirikov (an able mathematician whose bad health thwarted his scientific career) and Eisenhart. Professor Herbert A. David (Iowa State University) and especially Professor Ulrich Krengel provided useful comments on this text. A final remark. According to some clever regulation, Bernstein, who published many contributions abroad and spelled his name in that way, should now be called Bernshtein. This is ugly and it corrupts his pen-name (if not real name). And why then are we not ordered to spell Markof, Chuprof? When I began publishing abroad, I had not chosen the best spelling of my name, but it became my pen-name, and I refuse to change it. Some explanation Abbreviation CLT – central limit theorem LLN – law of large numbers MLSq – method of least squares W-i – Gauss, Werke , Bd. i W/Erg-i – Gauss, Werke , Ergänzungsreihe, Bd. i Notation [ ab ] = a1b1 + ... + anbn (introduced by Gauss). ln x = log nat x, lg x = log 10 x References in text A double page number, e.g. 59/216, means that either the pertinent source has double paging, or a reference to a later edition, or that it was translated from Russian into English with p. 59 of the original contribution corresponding to p. 216 of the translation. 0.2. The Pertinent Scientific Disciplines and Their Stages My subject covers a great chronological period and is very wide since it includes the theory of probability and statistics, which are difficult to separate, while statistics itself is a vast subject which ought to be subdivided. I also have to explain the relation of the theory of errors to statistics. In addition, I subdivide the history of the development of these disciplines into stages to help the readers grasp at once their general outline. Theory of Probability 1. Its prehistory (from Aristotle to the mid-17th century). 2. Its early history (from Pascal and Fermat to Jakob Bernoulli). 3. The creation of its initial version (completed by Jakob Bernoulli, De Moivre and Bayes). 4. Its development as an applied mathematical discipline (from Bayes to Laplace and Poisson to Chebyshev). 5. A rigorous proof of its limit theorems (Chebyshev, Markov, Liapunov) and its gradual transition to the realm of pure mathematics. 5 6. Axiomatization. In the second half of the 19 th century and the first decades of the 20 th mathematicians barely recognized probability theory and perhaps to our day all but ignore the Gaussian theory of errors. Mathematical Statistics. It originated in the early years of the 20 th century in the Biometric school and the Continental direction of statistics and Fisher moved it to the realm of pure mathematics. Its aim is the systematizing, processing and utilizing statistical data, – information on the number of the specified objects (Kolmogorov & Prokhorov 1988/1990, p. 138). Unlike theoretical statistics , it does not include collection of data or exploratory data analysis which means revealing general structures in the data (e. g., blunders, systematic influences, deception). The Statistical Method. Usually, statistics is meant to study population and the term statistical method is applied in all other instances. The statistical method underwent three stages. At first, conclusions were being based on qualitative regularities conforming to the essence of ancient science. Indeed, a Roman scholar Celsus (1935, p.19) stated: Careful men noted what generally answered the better, and then began to prescribe the same for their patients. Thus sprang up the Art of medicine. During the second stage (Tycho in astronomy, Graunt in demography and medical statistics) statistical data became available. Conclusions were made by means of simple stochastic ideas and methods or even directly, as before. At the present stage, which dates back to Poisson, inferences are being checked by quantitative stochastic rules.