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\noindent{\bf Wilhelm LEXIS}\\
b. 17 July 1837 - d. 24 August 1914
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{\bf 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.
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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 ({\it
Die Ausfuhrpr\"{a}mien 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 {\it Amtliche Nachrichten f\"ur 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 f\"ur 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, {\it Einleitung in die Theorie
der Bev\"olkerungsstatistik}, 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 {\it Jahrb\"ucher f\"ur National\"okonomie 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
{\it Handw\"orterbuch der Staatswissenschaften}. He was particularly
expert in the field of finance, publishing his {\it Er\"orterungen
\"uber die W\"ahrungsfragen}, among other works in 1881.
In 1884, he resigned his Chair in Freiburg for the Chair of Statistics
({\it Staatswissenschaften}) at the University of Breslau
(now Wroclaw in Poland). Finally, in 1887,
he moved to G\"ottingen 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 G\"ottingen in 1892. In
1895, Lexis founded the first actuarial institute in Germany
({\it K\"onigliches Seminar f\"ur 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 {\it Abhandlungen zur Theorie der Bev\"olkerungs-
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.
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\begin{thebibliography}{3}
\bibitem{1} Bauer, R. (1955). Die Lexissche Dispersionstheorie in ihren
Beziehungen zur modernen statistischen Methodenlehre, insbesondere zur
Streuungsanalyse (Analysis of Variance), {\it Mitteilungsblatt f\"ur
mathematische Statistik und ihre Anwendungsgebiete}, {\bf 7}, 25-45.
\bibitem{2} Bortkiewicz, L. von (1915). Wilhelm Lexis, {\it Bulletin de
l'Institut International de Statistique}, Tome 20, 1\`ere livraison, pp.
328-332.
\bibitem{3} Bortkiewicz, L. von (1904). Die Theorie der Bev\"olkerungs - und
Moralstatistik nach Lexis, {\it Jahrb\"ucher f\"ur National\"okonomie
und Statistik}, III. Folge, Bd. 27, pp. 230-254.
\bibitem{4} Heiss, K.-P. (1968). Lexis, Wilhelm, {\it International
Encyclopedia of the Social Sciences}, {\bf 9}, 271-276. Macmillan and the
Free Press, New York.
\bibitem{5} Heyde, C.C. \& Seneta, E. (1977). {\it I.J. Bienaym\'e:
Statistical Theory Anticipated}. Springer, New York, pp. 49-58.
\bibitem{6} Stigler, S. M. (1986). {\it The History of Statistics. The
Measurement of Uncertainty Before 1900}. Belknap Press, Harvard.
pp. 221-238.
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\hfill{S\'ebastien Hertz}
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