# Lexis, Wilhelm

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Wilhelm LEXIS

b. 17 July 1837 - d. 24 August 1914

Summary. In a hostile germanic environment Lexis established statistics as a highly mathematical subject based on the probability calculus, by means of his dispersion theory.

Wilhelm Lexis, the son of a physician, was born in Eschweiler near Aachen, in Germany. He studied at the University of Bonn from 1855, first devoting himself to law, and later to mathematics and the natural sciences. He was awarded his doctorate in philosophy in 1859 for a thesis on analytical mechanics. For some time, he taught secondary school mathematics at the Bonn Gymnasium. He also held a job in the Bunsen chemical laboratory in Heidelberg.

Lexis' departure for Paris in 1861 marked a turning point in his career. It was there that he developed his interest in the social sciences and political economy, as well as familiarizing himself with the works of Quetelet (q.v.). His first major work, published in Bonn in 1870, was a detailed study of the evolution of France's foreign trade after the restoration of the monarchy (Die Ausfuhrprämien im Zusammenhang mit der Tarifgeschichte und Handelsentwicklung Frankreichs seit der Restauration). In it Lexis stressed the importance of basing economic theories on quantitative data, while not hesitating to make use of mathematics.

The Franco-Prussian war of 1870-71 forced him to return to Germany. While editing the Amtliche Nachrichten für Elass-Lothringen at Hagenau, then the seat of the general government of Alsace-Lorraine, he befriended Friedrich Althof, who was to become director of higher education in the Prussian Ministry of Education and Culture. This friendship was at the basis of Lexis' active participation in the exchange of ideas and reforms of German universities.

In the autumn of 1872, he was very appropriately appointed as professor extraordinarius (Associate Professor) in political economy at the newly created University of Strasbourg, then one semester old, where Althof was also teaching. It was in the same year that he took part in the formation of the Verein für Sozialpolitik, a movement of university members (the Kathedersozialisten), an offshoot of the historical school whose aim was the promotion of social politics. It was in the Alsatian capital that he wrote his impressive introduction to the theory of statistical demography, Einleitung in die Theorie der Bevölkerungsstatistik, published in 1875.

By then he had already left Strasbourg for Dorpat, but not without recognition by award of Doctor rerum politicorum honoris causa in 1874. In Dorpat (now Tartu in Estonia), a town in the Russian Empire where the language of university instruction was German till 1895, he held the Chair as full professor in Geography, Ethnography and Statistics. He spent only two years there, returning to the banks of the Rhine as Chair of Political Economy at the University of Freiburg im Breisgau from 1876 to 1884. This was undoubtedly his most productive period. His publications of the time, most of them appearing in Jahrbücher für Nationalökonomie und Statistik, of which he was chief editor beginning from 1891, propelled him to the front rank in the field of theoretical statistics, and revealed him as the leader of a group working on the application of the calculus of probabilities to statistical data.

Lexis simultaneously continued his research in political economy, editing the first German encyclopedia of economic and social sciences Handwörterbuch der Staatswissenschaften. He was particularly expert in the field of finance, publishing his Erörterungen über die Währungsfragen, among other works in 1881.

In 1884, he resigned his Chair in Freiburg for the Chair of Statistics (Staatswissenschaften) at the University of Breslau (now Wroclaw in Poland). Finally, in 1887, he moved to Göttingen where he held the Chair of Statistics until his death, a few days after the start of the First World War. Bortkiewicz (q.v.) was his student in Göttingen in 1892. In 1895, Lexis founded the first actuarial institute in Germany (Königliches Seminar für Versicherungswissenschaften), which trained its candidates in both political economy and mathematics. His scholarship in both fields allowed him to manage its direction, and to provide part of the teaching in economics and statistics, while G. Bohlmann took charge of the teaching of mathematics.

Lexis left his mark on the history of statistics through his pioneering work on dispersion, which led on to the analysis of variance. Lexis' plan was to measure and compare the fluctuations for different statistical time series. In a sense, he followed Quetelet in applying urn models to statistical series. But by stressing fluctuations, he corrected Quetelet's work, which aimed to set every series within a unique "normal" model by assuming quite erroneously their homogeneity and stability. Similarly, using a binomial urn model to represent the annual number of male births, he derived a dispersion coefficient $Q$ (in homage to Quetelet) which is the ratio of the empirical variance of the series considered to the assumed theoretical variance. An analogous coefficient of divergence had been independently constructed by the French actuary Emile Dormoy in 1874. In the ideal case, Lexis refers to a normal" dispersion when the fluctuations are purely due to chance, and the coefficient is equal to 1. But in most cases the coefficient is different from 1, and thus differs from the binomial model. The fluctuations then indicate a physical" rather than a chance component. Lexis classified these dispersions into two categories, hypernormal" and hyponormal" according as to whether $Q > 1$ or $Q < 1$. He also showed that series of social data usually have a hypernormal dispersion.

His studies on the ratio of sexes at birth, his stability theory of statistical series with his famous $Q$ coefficient of dispersion were re-examined in his large treatise entitled "Abhandlungen zur Theorie der Bevölkerungs- und Moralstatistik(1903). In a review of it, Bortkiewicz in 1904 concludes that (Lexis) has known how to clarify and synthesize the most general problems of moral and demographic statistics, insofar as their conditions, methods and tasks are concerned; he has also shown that if this science has had to renounce its status as social physics" to which Quetelet tried to raise it, it remains nevertheless far more than the simple social accounting which some modern, and excessively timid, practitioners of the discipline would have us believe."

Lexis' coefficient foreshadowed the statistics of K.Pearson (q.v.) and R.A. Fisher (q.v.), in particular ${\chi}^2$ for the analysis of variance. However, it suffered from certain weaknesses which his more mathematical and younger contemporaries did not fail to point out and attempt to correct, among them Chuprov (q.v.), Markov (q.v.) and Bortkiewicz. In publications up to the period between the two World Wars, the Continental School of mathematical statistics tended to follow the dispersion theory of Lexis, but both eventually gave way together to the Anglo-Saxon developments in this area. Lexis' statistical views, however, did not disappear from view as they had the dubious distinction of being singled out for attack on their reactionary and bourgeois nature within the Soviet Union, by guardians of ideology such as Yastremsky.

#### References

 [1] Bauer, R. (1955). Die Lexissche Dispersionstheorie in ihren Beziehungen zur modernen statistischen Methodenlehre, insbesondere zur Streuungsanalyse (Analysis of Variance), Mitteilungsblatt für mathematische Statistik und ihre Anwendungsgebiete, 7, 25-45. [2] Bortkiewicz, L. von (1915). Wilhelm Lexis, Bulletin de l'Institut International de Statistique, Tome 20, 1ère livraison, pp. 328-332. [3] Bortkiewicz, L. von (1904). Die Theorie der Bevölkerungs - und Moralstatistik nach Lexis, Jahrbücher für Nationalökonomie und Statistik, III. Folge, Bd. 27, pp. 230-254. [4] Heiss, K.-P. (1968). Lexis, Wilhelm, International Encyclopedia of the Social Sciences, 9, 271-276. Macmillan and the Free Press, New York. [5] Heyde, C.C. & Seneta, E. (1977). I.J. Bienaymé: Statistical Theory Anticipated. Springer, New York, pp. 49-58. [6] Stigler, S. M. (1986). The History of Statistics. The Measurement of Uncertainty Before 1900. Belknap Press, Harvard. pp. 221-238.

Reprinted with permission from Christopher Charles Heyde and Eugene William Seneta (Editors), Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.

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Lexis, Wilhelm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lexis,_Wilhelm&oldid=39224