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Wilhelm Lexis (17 July 1837, Eschweiler, Germany – 24 August 1914, Göttingen, Germany), full name Wilhelm Hector Richard Albrecht Lexis, was a German statistician, economist, and social scientist. The Oxford Dictionary of Statistics cites him as a "pioneer of the analysis of demographic time series".

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Lexis is largely remembered for two items that bear his name—the Lexis ratio and the Lexis diagram.

Life
Lexis was born in Eschweiler, Germany in 1837, the son of a medical doctor. He entered the University of Bonn in 1855, first studying law but soon moving to science and mathematics. Lexis graduated in 1859 after writing a thesis in analytical mechanics, On the General Laws of Motion. He remained in Bonn for a brief time afterwards, working as a teacher at a gymnasium. Moving to Heidelberg, Lexis spent another brief amount of time working at a chemical laboratory run by Robert Bunsen. But by 1861, he had relocated to Paris.

Little is known about Lexis's life during the 1860s (one of his obituarists described it as a "ten-year wandering"). But it was during this period that he began to study the social sciences. In doing so, Lexis became acquainted with the work of Adolphe Quetelet, whose quantitative approach to the social sciences was to guide much of Lexis's work. At first, his interest was in the field of economics and Lexis spent some time as an economics correspondent with a major German newspaper. And Lexis's first book-length work was in that field. Published in 1870, it addressed French exports in the years following the Bourbon Restoration.

When the Franco-Prussian War broke out in 1870, Lexis served as a non-combatant for the Prussian military and received the non-combatant version of the Iron Cross for that service. The war resulted in the Prussian annexation of Alsace-Lorraine and Lexis took up residence there. He first lived in Hagenau, where he worked as editor of the Official News for Alsace-Lorraine. The newspaper moved to Strasbourg and Lexis moved with it, continuing to edit the paper under its new name, Strasbourg News. But he stopped doing this in late 1872 when he became a teacher of economics at the newly-founded Kaiser Wilhem University. He remained in that position until 1874 and, while there, wrote his first major work on demography, Introduction to the Theory of Population Statistics (but it was not published until later). Upon leaving, the university awarded him an honorary doctorate.

From Strasbourg, Lexis moved to the Estonian city of Tartu (then called Dorpat), where he taught [WHAT] and [WHERE]. But his stay there was not very long and, in 1876, Lexis became the chair of the Economics Department at the University of Freiburg. The various papers written by him during his eight-year tenure at Freiburg were, in the eyes of statistics historian Stephen Stigler, "his most important statistical work". [GENERAL DISCUSSION OF WHAT HE WAS WORKING ON] Foremost among them was the 1879 paper "On the Theory of the Stability of Statistical Series", which introduced the quantity now often called the Lexis ratio.

Lexis moved on from Freiburg to the University of Breslau but stayed there only a few years (from 1884 to 1887). He then settled in Göttingen, taking a position at that city's University. In 1895, he established a course in actuarial science at the university, the first ever in Germany. In 1901, Lexis became a member of the Insurance Advisory Council for Germany's Federal Insurance Supervisory Office. He remained a member of the Council until his death in 1914. During this final period of his life, Lexis published two more books: Treatises on Population and Social Statistics (Jena: Gustav Fischer, 1903) and General Economics (Leipzig: Teubner, 1910). He was also the editor of a book on the German education system.

Work
Throughout his professional career, Lexis published books and articles on a wide variety of topics, including demography, economics and mathematical statistics. However, little of that work proved to have lasting significance. Today, Lexis is largely remembered for two items that bear his name—the Lexis ratio and the Lexis diagram. His theory of mortality has also enjoyed a recent revival of interest.

Lexis ratio
To Lexis, a time series was "stable" if the underlying probability giving rise to the observed rates remained constant from year to year (or, more generally, from one measurement period to the next). Using modern terminology, such a time series would be called a zero-order moving-average series (also known as a white noise process). Lexis was aware that many series were not stable. For non-stable series, he imagined that the underlying probabilities varied over time, being affected by what he called "physical" forces (as opposed to the random "non-essential" forces that would cause an observed rate to be different than the underlying probability). In his 1879 paper "On the Theory of the Stability of Statistical Series", Lexis set himself the task of devising a method for distinguishing between stable and non-stable time series.

To this end, Lexis created a test statistic equal to the ratio between (i) the probable error of the observed rates and (ii) the probable error that would be expected if the underlying probabilities for each of the observed rates were all equal to the average rate observed across all of the observations. He called this ratio Q. Lexis then reasoned that if Q was sufficiently close to 1, then the time series was exhibiting what he called "normal dispersion" and one could assume that it was stable. If Q was substantially greater than 1, then the series was exhibiting "supernomal dispersion" and one must conclude that physical forces were having a discernible effect on the variability of the observations. Lexis used a Q value of 1.41 (i.e., the square root of 2) as the dividing line between "normal" and "supernormal" dispersion.

"Stability of Statistical Series" is the only one of Lexis' works cited in his entry in the Oxford Dictionary of Statistics. It is also the only one that receives an extended discussion in Stigler's A History of Statistics. And yet, Stigler ends his discussion by labeling the work a failure. To Stigler, its chief value was the discussion that it generated from other researchers in the field. It was those other researchers, and not Lexis, who created the modern science of time-series analysis.

Lexis diagram
Although it can take various forms, the typical Lexis diagram is a graphical illustration of the lifetime of either an individual or a cohort of same-aged individuals. On the diagram, each such lifetime appears as a straight line in a two-dimensional plane, with one dimension representing time and the other representing age. The use of Lexis diagrams is very common amongst demographers, so much so that they often are used without being identified as Lexis diagrams.

Lexis introduced his diagram in his first book, Introduction to the Theory of Population Statistics (Strasbourg: Trubner, 1875). However, the notion of using a time vs. age diagram appears to have been developed more or less simultaneously by other authors.

Theory of mortality
In his 1877 book On the Theory of Mass Phenomena in Human Society (Freiburg: Wagnersche Buchhandlung), Lexis proposed that all human deaths could be classified into one of three types: (i) normal deaths, (ii) infant deaths and (iii) premature adult deaths. He also proposed that the normal deaths were subject to random forces such that, if all infant and other premature deaths were eliminated, the ages at which people died would exhibit a normal (i.e., Gaussian) distribution. Furthermore, the average of those ages would be equal to the age at which most adults are actually observed to die (i.e., the modal age at death), even though the actual observations are taking place in the presence of infant and other premature deaths.

In the adjacent diagram, the normal deaths are represented by the vertically-shaded bell-shaped area centered slightly above age 70; the infant deaths are represented by the unshaded area starting at age 0; the premature deaths are represented by the horizontally-shaded area bridging the infant and normal deaths.

Although Lexis's theory did generate some contemporaneous discussion, it never supplanted the traditional demographic measures of life expectancy and age-adjusted mortality rates. However, recent research suggests that the modal age at death might be a useful statistic for tracking changes in the lifespans of the elderly.

Legacy
Amongst Lexis's students at the insurance institute were Alfred Manes and Paul Moldenhauer. (Source: DNB)

Festgaben

Works

 * (Lexis's inaugural doctoral dissertation)
 * On the Theory of the Stability of Statistical Series (1879) (in German)
 * https://babel.hathitrust.org/cgi/pt?id=hvd.32044054997721;view=1up;seq=13
 * Treatises on Population and Social Statistics (1903) (in German)
 * 1904 text on the German education system (in English translation)
 * https://babel.hathitrust.org/cgi/pt?id=hvd.32044054997721;view=1up;seq=13
 * Treatises on Population and Social Statistics (1903) (in German)
 * 1904 text on the German education system (in English translation)
 * 1904 text on the German education system (in English translation)

Biographies of Lexis

 * Entry in the Neue Deutsche Biographie (in German)

Temp sources

 * https://babel.hathitrust.org/cgi/pt?id=uc1.a0003113883;view=1up;seq=340 (Klein obituary)
 * https://babel.hathitrust.org/cgi/pt?id=mdp.39015011063693;view=1up;seq=316 (Biographical yearbook, with list of obituaries)