Lexis, Wilhelm

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Lexis, Wilhelm





Wilhelm Lexis (1837-1914), a German statistician and economist, made major contributions to the theory of statistics and its application, particularly in population research and economic time series. As a mathematician Lexis was deeply skeptical about the state of mathematical economics in his time. His criticism of certain contemporary work in mathematical economics led him to some fundamental observations on economic events and their interdependence.

Lexis was born in Eschweiler, near Aachen, Germany. His studies were widespread, and his interests ranged from law to the natural sciences and mathematics. He graduated from the University of Bonn in 1859, having written a thesis on analytical mechanics; he also obtained a degree in mathematics. For some time he did research in Bunsen’s chemical laboratories in Heidelberg. In 1861, Lexis went to Paris to study the social sciences, and his studies led to his first major publication (1870), a treatise on French export policies. This work displays the feature that characterizes his later economic writings: a skepticism toward “pure economics” and toward the application of supposedly descriptive mathematical models which have no reference to economic reality. Even in this early work he insisted that economic theory should be founded on quantitative economic data. In Lexis’ view an elaborate general economic equilibrium analysis, whose main problem was to match the unknowns with an equal number of equations, could make no contribution to the understanding or solution of economic problems and therefore should not be taken too seriously. He was one of a number of mathematically trained students of economics who, in the second half of the nineteenth century, became alienated from that discipline; another, more famous one was Max Planck, who, after attempting to read Marshall’s work, threw it away and changed his course of study for good (see Schumpeter 1954, pp. 957-958).

Lexis was appointed to the University of Strassburg in 1872. While there, he wrote his introduction to the theory of population (1875). From Strassburg he went to Dorpat in 1874, as professor of geography, ethnology, and statistics, and then to Freiburg in 1876, as professor of economics. His major contributions to statistics were made while he was at Freiburg (1876; 1877; 1879a), and he also published papers on economics at this time (1879b; 1881; 1882a; 1882 b>). After an interlude at the University of Breslau from 1884 to 1887, he was appointed professor of political science at the University of Göttingen.

Lexis’ activities in his later years were remarkably diverse. An editor of the Handworterbuch der Staatswissenschaften, the major German economic encyclopedia, he was also an active contributor, and he was director of the first institute of actuarial sciences in Germany. In the 1890s he published and edited several volumes pertaining to education, particularly to the university system (1893; 1901; 1902; 1904a; 1904b). Lexis’works on population research, economics, and statistics during the subsequent decade bring together and refine some of his earlier arguments (see 1903; 1906a; 1906b; 1908; 1914). He died in Gottingen during the very first days of World War I.


Lexis’ major contributions were in statistics. His statistical work originated in problems he encountered in population research (1875; 1879b; 1891; 1903), sociology (1877), and economics (1870; 1879b; 1908; 1914). In connection with his studies of social mass phenomena (1877) and the time series encountered in several of the social sciences (1879 b), Lexis came upon problems concerning statistical homogeneity which apparently had been neglected up to then, although, as Bortkiewicz pointed out (1918), Dormoy (1874; 1878) developed similar ideas at about the same time. (Other possible forerunners of Lexis were Bienayme [1855], Cournot [1843], and R. Campbell [1859].) Lexis credited Dormoy with having anticipated some of his ideas; nevertheless, it was Lexis who more or less independently gave a new direction to the analysis of statistical series and led statisticians in the shift of emphasis from the purely mathematical approach, with which Laplace is associated, to an empirical or inductive approach (see Keynes 1921, pp. 392 ff.). He also initiated the analysis of dispersion and variance in his attempts to develop statistics with which to evaluate qualitative changes in populations over time (see Keynes 1921; Polya 1919).

Lexis showed that in the universe of social mass phenomena the conditions of statistical homogeneity (random sampling from a stable distribution) are seldom, if ever, fulfilled (1877; 1879a; 1879b). The underlying probability structure may well differ from one part of the sample to another because of special circumstances related to dispersion in space, time, or other factors. A universe in which individual samples are drawn from potentially different populations is now known as a Lexis universe (see, for example, Herdan 1966). To some extent Lexis’ work was a reaction to the uncritical assumptions of homogeneity made in statistical work before his time—for instance, by Quetelet. As Keynes (1921) pointed out, Quetelet and others simply asserted, with little evidence, the probabilistic stability from year to year of various social statistics.

Lexis’ work centered on the dispersion of observations around their local means and on the behavior of the means and dispersions over time. He devised statistics to measure the degree of stability of such time series and arrived at the useful generalization that these statistics would either confirm statistical homogeneity, indicating a Bernoulli series, or diverge from it, indicating a Poison series or a Lexis series. (This terminology was developed by C. V. L. Charlier; see A. Fisher 1915, p. 117).

Lexis considered only dichotomous variates (male-female, living-dead, etc.), but the argument he advanced holds equally for numerical variates in the ordinary sense (see also Polya 1919). In the following, Lexis’ ideas are given in a generalized form.

Let Xij (i = 1,..., n, j — 1, ... , m) be a set of n samples with m observations each, and let the arithmetic mean of the xij in sample i be ̄Xi and the arithmetic mean of the xij over all n samples be x. Similarly, let aij = E(xij), the expectation of Xij, so that ̄ai = E(̄Xi) and ̄a = E(̄x). Lexis considered the following quadratic forms which measure dispersion in three different senses:

where has rank (degrees of freedom)rδ= n — 1, has rank rw = n(m- 1), and s2 has rank r = rδ +rw = nm — 1. (The subscripts “w” and “b” are used to indicate that comprises within sample dispersion and between sample dispersion.) Furthermore,

If the n samples of m objects each are drawn at random from the same population, then the expected value of each observation equals the expected value of the sample mean and the expected value of the mean of all observations, that is,

aij = ̄ai = ̄a,

and statistical homogeneity is present. Repeated independent measurements of a distance and n x m drawings of balls from an urn with each ball returned after it is drawn are examples of such series. In this case the three quadratic forms, multiplied by appropriate constants, have the same expectations. Specifically, when statistical homogeneity holds,

and the common value is 1/m times the variance of the underlying population. A set of samples drawn under such conditions is known as a Bernoulli series (see, e.g., A. Fisher 1915).

It may, however, be the case that statistical homogeneity holds within the samples (ai1 = ai2 = ... = aim) but not between samples — that is, the n samples may be random samples from different populations. In this case a Lexis series is generated (a supernormal series, in Lexis’ terminology). Such a series is expected when, for example, each set of balls (one sample) is drawn from a different urn. Other examples of such series explain further the importance of the Lexis series:m ×n observations made at time t0 , another m observations made at time ti , and so forth, up to tn , will give rise to a time series of m x n observations, where the zth sample (covering one period, ti) may well come from a single population but where between different periods such changes occurred that statistical homogeneity is no longer preserved— that is, the samples come from different populations.

Similarly, social or economic samples of m observations drawn from n different geographical regions (nations) are likely to come from different statistical populations, although in each region (nation) the m observations of the sample come from the same distribution (interregional series, international series) . In short, if the over-all dispersion is caused not only by chance variations about a constant but also by trends and other systematic factors varying between samples, then a Lexis series will be generated. (A comprehensive and elementary treatment of Lexis series is given in Poly a 1919.)

We expect in this case that the variance between the samples will contribute relatively more to the over-all variance of the m ×n observations than will the variance within the samples and that the expected value of an observation will equal the sample mean, whereas the sample mean is expected to differ from the mean of all observations. Further, although aij = ̄ai, ̄a,īa for at least some i, and

Statistical homogeneity is not preserved.

Much less realistic, but a formal complement to the Bernoulli and Lexis series, is the Poisson series (a subnormal series, in Lexis’ terminology). The Poisson model was developed as one that would generate higher within sample than between sample variability. In this case the jth observation of each sample is drawn from a fixed population, but the populations differ according to j. In short, aij = a2j = ... =anj, but for a fixed i the aij are not all equal. Hence ̄ai = ̄a, and there is no between sample variability coming from the a’s. It follows that

Other kinds of models leading to subnormal dispersion have also been considered.

Lexis proposed a statistic, based on the above quadratic forms, to describe the extent to which a given series is homogeneous, supernormal, or sub-normal. The statistic, called the Lexis quotient, is

a monotone increasing function of another statistic, which might be used alternatively. A. A. Chuprov showed later (1922) that in the case of statistical homogeneity, E(L) = 1 and the variance of L is approximately 2/(n – 1).

A further elaboration of this leads to significance tests. A first step in that direction is made by adding to and subtracting from the expected value of L its standard error, obtaining 1 If L lies within the boundaries thus calculated, one may conclude with confidence of approximately two out of three that the statistical mass is homogeneous; if L is significantly larger than 1, one may conclude that the series was drawn from a Lexis universe; and if L is significantly smaller than 1, one may conclude that the series is Poisson.

The relevance and connections of the Lexis series and Lexis’ L to the analysis of variance and the chi-square distribution were later shown by many authors. The formal connection between L and the X2 statistic of Pearson was elaborated by R. A. Fisher. Fisher showed that in the case of a 2 x n classification the x2 statistic is just nL. [SeeCounted data; see also R. A. Fisher 1928; Gebelein & Heite 1951.]

The relation of the L-statistic to the F-statistic is very direct, and we may say that Lexis anticipated the F-statistic (see Coolidge 1921; Rietz 1932; Geiringer 1942 a; 1942 b; Gini 1956; Herdan 1966). Whereas in L one compares the variance between the samples to the variance of all n x m observations, that is,

in the F-statistic one compares the variance between the samples to the variance within the samples, that is,


if and only if

This concurrence is based on the previously stated equality

Although the asymptotic distribution of the F-statistic as m → ∞ was established by R. A. Fisher (1925, p. 97) and by W. G. Cochran (1934, p. 178) and generalized by M. G. Madow (1940), the same distribution was established for Lexis’ L as early as 1876 by F. R. Helmert, using the method of characteristic functions.

Bortkiewicz extended the application of Lexis’ theory of dispersion, and Chuprov (1922) extended Lexis’ theory and gave it the most comprehensive treatment. Others influenced by Lexis were J. von Kries, H. Westergaard, and F. Y. Edgeworth, the only Anglo-American scholar closely familiar with statistical work on the Continent at that time (see Keynes 1921).


Lexis’ contributions to economic theory were less appreciated than were his contributions to statistics; many economists, including Schumpeter, largely ignored them. Such a negative assessment of Lexis’ work proves to be not entirely justified after closer examination of the reasons that led him to criticize certain aspects of the mathematical and “pure” economics of his contemporaries. His main contribution was a valid criticism of the work done at his time, particularly that of the Austrian school and the Lausanne school. His criticism was informed in part by the outlook of the historical school, which was prevalent in Germany, and accordingly he believed that it was necessary to incorporate in any theory of value and demand the element of time as well as the phenomenon of the recurrence of wants.

Lexis accepted Gossen’s analysis of human behavior because Gossen appreciated all the shortcomings of any such theory. The criticisms Lexis made of the Austrian school seem contradictory but are only superficially so: he deplored the lack of mathematics in the work of some authors, especially Carl Menger, and found fault with the application of inappropriate mathematics in the work of others, especially Auspitz and Lieben.

Lexis regarded the concept of utility as being rather vague, since utility cannot be measured. He argued that to say that the utility of a good (set of goods) is equal to, larger than, or less than the utility of some other good permits a partial or complete preordering of utilities. Complementarity and substitution effects imply, however, subadditivity and superadditivity of utilities, which render futile any attempt to aggregate utilities and demand correspondences. Lexis questioned the convexity and continuity assumptions of preference orderings.

The controversy then raging over how to determine total utility given the marginal utility correspondences (a controversy between E. von Bohm-Bawerk and F. von Wieser) was correctly interpreted by Lexis and led him to a discussion of Gossen’s other laws, most notably the equalization of marginal utilities. Any such theorem, he believed, must be hedged by a number of qualifications; it is particularly important to consider the time element connected with demand and consumption. Want and satisfaction are both felt and exercised over time. At one and the same time only a limited set of wants can be satisfied. One can eat, drink, sleep, and work, but these activities are to some extent mutually exclusive. Thus, the individual has to decide what sequence to follow in satisfying his set of wants. This sequence will be determined, according to Lexis, by the intensity of wants and by their periodicity, the most fundamental rhythms being the day, the year, and one lifetime. The demand of an individual will be classified and exercised accordingly. Intensive wants will be satisfied first, on a daily, yearly, or other basis, depending on the periodicity of recurrence of wants. Other, less intensive wants will be satisfied after full satisfaction of the intensive wants has been achieved. The implications are far-reaching but perhaps misleading; they have not been accepted by subsequent economists. Individual demand correspondences have to be defined by the period to which they relate, which in turn requires a reformulation of the theory of demand and implies the necessity of defining the demand for (consumption of) each good at different times as quantitatively different. This entails no theoretical difficulties in a general mathematical equilibrium analysis, but it does prevent the theory from having any operational value.

The concept of preferential ordering over time induced Lexis to observe that certain more intensive wants will be saturated whereas other, less intensive wants will be satisfied partially, implying satisfaction of zero marginal utility for the first set of wants and some positive marginal utility for the second. Lexis supported this conclusion by referring to economic reality. However, his statement about the equalization of marginal utilities turns out ultimately to be incorrect if we allow for errors of judgment, evaluations of uncertainties and risks, and diversity of attributes of each good for any individual.

Thus, Lexis’ skepticism about the potential of the marginal utility theory in economics was based on the difficulty of measuring utility, the existence of subadditivities and superadditivities in utility correspondences, and the impossibility of aggregating individual preferences. The introduction of the time element into the theory of value and demand adds interesting arguments to general equilibrium analysis which imply, according to Lexis, obvious refutations of the equalization of marginal utilities. As a consequence of his skepticism, Lexis turned, in the rest of his economic work, to a rather dry description of economic events, which failed to be attractive to more speculative minds.

Klaus-Peter Heiss


1870 Die franzosischen Ausfuhrprdmien im Zusammenhange mit der Tarifgeschichte und Handelsentwicklung Frankreichs seit der Restauration. Bonn: Marcus.

1875 Einleitung in die Theorie der Bevolkerungsstatistik. Strassburg: Trübner.

1876 Das Geschlechtsverhältniss der Geborenen und die Wahrscheinlichkeitsrechnung.Jahrbücher für Natiönalokonomie und Statistik 27:209-245.

1877 Zur Theorie der Massenerscheinungen in der menschlichen Gesellschaft. Freiburg irn Breisgau: Wagner.

(1879a) 1942 Uber die Theorie der Stabilitdt statistischer Reihen (The Theory of the Stability of Statistical Series). Minneapolis, Minn: WPA. → First published in Volume 32 of Jahrbiicher fur Nationalokonomie und Statistik.

1879 b Gewerkvereine und Unternehmerverbande in Frankreich. Verein fur Socialpolitik, Berlin, Schriften 17:1-280.

1881 Erorterungen iiber die Wdhrungsfrage. Leipzig: Duncker ' Humblot.

(1882 a) 1890 Die volkswirthschaftliche Konsumtion. Volume 1, pages 685–722 in Handbuch der politischen Oekonomie. 3d ed. Edited by Gustav Schönberg. Tübingen: Laupp.

(1882 b) 1891 Handel. Volume 2, pages 811–938 in Handbuch der politischen Oekonomie. 3d ed. Edited by Gustav Schonberg. Tübingen: Laupp.

1886 Über die Wahrscheinlichkeitsrechnung und deren Anwendung auf die Statistik.Jahrbücher für Nation-alokonomie und Statistik 47:433-450.

1891 Bevölkerungswesen, II: Bevölkerungswechsel, 1: Allgemeine Theorie des Bevölkerungswechsels. Volume 2, pages 456–463 in Handwörterbuch der Staatswissenschaften. Jena: Fischer.

1893 Die deutschen Universitdten: Fur die Universitdtsäusstellung in Chicago 1893. 2 vols. Berlin: Asher.

(1895a) 1896 The Present Monetary Situation. American Economic Association, Economic Studies, Vol. 1, No. 4. New York: Macmillan. → First published as Der gegenwärtige Stand der Währungsfrage.

1895 b Grenznutzen. Volume 1, pages 422–432 in Hand-wörterbuch der Staatswissenschaften: Supplement-band. Jena: Fischer.

1901 Die neuen franzosischen Universitdten. Munich: Academischer Verlag.

1902 LEXIS, WILHELM (editor)Die Reform des höheren Schulwesens in Preussen. Halle: Waisenhaus.

1903 Abhandlungen zur Theorie der Bevölkerungs- und Moralstatistik. Jena: Fischer. →Contains reprints of 1876 and 1879a.

1904 a LEXIS, WILHELM (editor) Das Unterrichtswesen im Deutschen Reich. 4 vols. Berlin: Asher.

1904b A General View of the History and Organisation of Public Education in the German Empire. Berlin: Asher.

1906 a Das Wesen der Kultur. Pages 1–53 in Die allgemeinen Grundlagen der Kultur der Gegenwart. Die Kultur der Gegenwart, vol. 1, part 1. Berlin: Teubner.

1906 b Das Handelswesen. 2 vols. Sammlung Göschen, Vols. 296-297. Berlin: Gruyter. → Volume 1: Das Handelspersonal und der Warenhandel. Volume 2: Die Effektenborse und die innere Handelspolitik.

1908 Systematisierung, Richtungen und Methoden der Volkswirtschaftslehre. Volume 1, pages 1:1-45 in Die Entwicklung der deutschen Volkswirtschaftslehre im neunzehnten Jahrhundert. Leipzig: Duncker ' Hum-blot.

(1910) 1926 Allgemeine Volkswirtschaftslehre. 3d ed., rev. Die Kultur der Gegenwart, vol. 2, part 10, section 1. Berlin and Leipzig: Teubner.(1914) 1929 Das Kredit- und Bankwesen. 2d ed. Samm-lung Goschen, Vol. 733. Berlin: Gruyter.


Bauer, Rainald K. 1955 Die Lexissche Dispersionstheorie in ihren Beziehungen zur modernen statistischen Methodenlehre, insbesondere zur Streuungsanalyse (Analysis of Variance). Mitteilungsblatt fur mathematische Statistik und ihre Anwendungsgebiete 7: 25-45.

BienaymÉ, Jules (1855) 1876 Sur un principe que M. Poisson avait cru decouvrir et qu’il avait appelé loi des grands nombres.Journal de la Sociéeté de Statistique de Paris 17:199-204.

Bortkiewicz, Ladislaus VON 1901 Über den Präcisiongrad des Diver genzkoeffizienten. Verband der Osterreichischen und Ungarischen Versicherungs-Techniker,Mitteilungen 5:1-3.

Bortkiewicz, Ladislaus VON 1909-1911 Statistique. Part 1, Volume 4, pages 453–490 in Encyclopédie des sciences mathematique s. Paris: Gauthier-Villars.

Bortkiewicz, Ladislaus VON 1915 Wilhelm Lexis [Obituary]. International Statistical Institute,Bulletin 20, no.1:328–332.

Bortkiewicz, Ladislaus VON 1917 Wahrscheihlichkeitstheoretische Untersuchungen iiber die Knabenquote bei Zwillingsgeburten. Berliner Mathematische Gesellschaft,Sitzungsberichte 17:8-14.

Bortkiewicz, Ladislaus VON 1918 Der mittlere Fehler des zum Quadrat erhobenen Divergenzkoëfhzienten. Deutsche Mathematiker-Vereinigung,Jahresbericht 27: 71-126.

Bortkiewicz, Ladislaus VON 1930 Lexis und Dormoy. Nordic Statistical Journal 2:37-54.

Bortkiewicz, Ladislaus VON 1931 The Relations Between Stability and Homogeneity.Annals of Mathematical Statistics 2:1-22.

Campbell, Robert 1859 On the Probability of Uniformity in Statistical Tables. Philosophical Magazine 18:359-368.

Chuprov, Aleksandr A. 1905 Die Aufgaben der Theorie der Statistik.Jahrbuch fur Gesetzgebung, Verwaltung und Volkswirtschaft im Deutschen Reich 29: 421-480. The author℉s name is given its German transliteration, Tschuprow.

Chuprov, Aleksandr A. 1922 1st die normale Stabilitat empirisch nachweisbar?Nordisk statistisk tidskrift 1:369-393. → The author℉s name is given its German transliteration, Tschuprow.

COCHRAN, W. G. 1934 The Distribution of Quadratic Forms in a Normal System, With Applications to the Analysis of Covariance. Cambridge Philosophical Society,Proceedings 30:178-191.

Coolidge, Julian L. 1921 The Dispersion of Observations. American Mathematical Society,Bulletin 27: 439-442.

Cournot, Antoine Augustin 1843 Exposition de la theorie des chances et des probabilites. Paris: Hachette.

Dormoy, Émile 1874 Théorié mathématique des paris de courses. Paris: Gauthier-Villars. → Also published in Volume 3 of Journal des actuaires francafs.

Dormoy, É;MILE 1878 Theorie mathématique des assurances sur la vie. 2 vols. Paris: Gauthier-Villars.

Edgeworth, F. Y. 1885 Methods of Statistics. Pages 181–217 in Royal Statistical Society, London,Jubilee Volume. London: Stanford.

Fisher, Arne 1915 The Mathematical Theory of Probabilities and Its Application to Frequency Curves and Statistical Methods. Vol. 1. London: Macmillan.

Fisher, R. A. 1925 Applications of “Student’s” Distribution.Metron 5, no. 3:90-104.

FISHER, R. A. 1928 On a Distribution Yielding the Error Functions of Several Well Known Statistics. Volume 2, pages 805–813 in International Congress of Mathematicians (New Series), Second, Toronto, 1924,Proceedings. Univ. of Toronto Press.

Gebelein, Hand; and Heite, H.-J. 1951 Statistische Urteilsbildung. Berlin: Springer.

Geirestger, Hilda 1942a A New Explanation of Non-normal Dispersion in the Lexis Theory. Econometrica 10:53-60.

Geiringer, Hilda 1942 b Observations on Analysis of Variance Theory. Annals of Mathematical Statistics 13:350-369.

GINI, C. 1956 Generalisations et applications de la theorie de la dispersion.Metron 18, no. 1/2:1-75.

Helmert, F. R. 1876 Ueber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und iiber einige damit im Zusammenhange stehenden Fragen. Zeitschrift fur Mathematik und Physik 21:192-218.

Herdan, G. 1966 The Advanced Theory of Language as Choice and Chance. New York: Springer.

Keynes, John Maynard (1921) 1952 A Treatise on Probability. London: Macmillan. → A paperback edition was published in 1962 by Harper.

KRIES, JOHANNES VON (1886) 1927 Die Principien der Wahrscheinlichkeitsrechnung: Fine logische Untersuchung. 2d ed. Töbingen: Mohr.

Madow, William G. 1940 Limiting Distributions of Quadratic and Bilinear Forms.Annals of Mathematical Statistics 11:125-146.

Pólya, GEORG 1919 Anschauliche und elementare Darstellung der Lexisschen Dispersionstheorie.Zeitschrift fur schweizerische Statistik und Volkswirtschaft 55: 121-140.

Rietz, H. L. 1932 On the Lexis Theory and the Analysis of Variance. American Mathematical Society, Bulletin 38:731-735.

Schumpeter, Joseph A. (1954) 1960 History of Economic Analysis. Edited by E. B. Schumpeter. New York: Oxford Univ. Press.

Von Mises, Richard 1932 Théorie des probabilites: Fondements et applications. Paris, Universite de, Institut Henri Poincare,Annales 3:137-190.