# Bortkiewicz, Ladislaus Von

# Bortkiewicz, Ladislaus Von

Ladislaus von Bortkiewicz (first spelled Bortkewitsch as in the Russian transcription) was born of Polish descent in 1868 in St. Petersburg and studied at the university there. His first papers (1890 and 1891) were published in Russian. He continued his studies in Götingen under Lexis, where he wrote his doctoral thesis (1893). In 1895 he became *Privatdozent* in Strassburg and subject to the influence of Knapp, but he returned in 1899 to Russia, where he taught at the Alexandrowsky Lyceum in St. Petersburg. He became associate professor at the University of Berlin in 1901, and finally in 1920 he became full professor *ad personam* of economics and statistics. He remained in Berlin for thirty years, until his death in 1931. With rare exceptions he wrote in German. He was one of the few representatives of mathematical statistics in Germany and as such a lonely figure, highly respected but rarely understood.

Besides classical economics, the work of Bortkiewicz covered population statistics and theory, actuarial science, mathematical statistics, probability theory, mathematical economics, and physical statistics—fields separate in content but analogous in methodology. He contributed to the process of consolidating each of these disciplines and did classic work in mathematical statistics.

Many of his investigations dealt with mortality tables. In a stationary population, the birth rate equals the death rate and the expectation of life of a newborn equals the reciprocal of the common value of the two rates. For increasing populations it was believed that the expectation of life could be obtained from the observed birth and death rates. Bortkiewicz showed (1893), however, that a correct answer can be obtained only by the construction of a mortality table. He returned to this problem when dealing with different methods of comparing mortality rates (1904*b*; 1911). In an increasing population there are more infants and fewer old people than in a stationary one. The first influence raises, the second lowers, the general mortality. Bortkiewicz showed that the second influence prevails in general so that the growth of the population tends to decrease the mortality rate. The study of life tables led him to actuarial science (see 1903; 1929).

The work that made his name widely known was a brochure (1898) of sixty pages, Das *Gesetz der kleinen Zahlen* (“The Law of Small Numbers”). Poisson had shown in 1837 that besides the usual normal limit for Bernoulli’s distribution there is a second limit, requiring that the number (*n*) of observations increase and the probability *(p)* decrease so that the product *(np)* has a limiting value. In this distribution *n* and *p* enter only through their product λ = *np*, which is the expected number of happenings. (The Poisson limit is primarily useful as an approximation when λ is small.) Poisson’s important derivation remained practically unknown for sixty years; at least, its importance was not recognized. Bortkiewicz was the first to note the fact that events in a large population, with low frequency, can be fitted by a Poisson distribution even when the probability of an event varies somewhat between the strata of the population. This is what he called the law of small numbers—the name refers to small numbers of events (see also 1915*a*).

A striking example was the number of soldiers killed by horse kicks per year per Prussian army corps. Fourteen corps were examined, each for twenty years. For over half the corps–year combinations there were no deaths from horse kicks; for other combinations the number of deaths ranged up to four. Presumably the risk of lethal horse kicks varied over years and corps, yet the over-all distribution was remarkably well fitted by a Poisson distribution.

In this distribution the variance, that is, the square of the standard deviation, is equal to the expectation. The corresponding observed quotient should therefore be near unity. This is called “normal dispersion” in the Lexis theory. The law of small numbers says that rare events usually show normal dispersion; for a mathematical explanation of this fact consistent with Lexis’ theory, see Gosset (1919). Bortkiewicz computed tables of Poisson’s distribution and discussed estimation of its expectation by the sample mean. In addition he discussed errors of estimation in quantitative terms and used them as criteria for the validity of the theory. Thus Bortkiewicz created an important instrument for mathematical statistics and probability theory. However, the name he gave it was unfortunate because it implied a nonexistent contrast to the law of large numbers and led to much confusion and unnecessary argument. [For *a discussion of the law of large numbers see*Probability, *article on*formal probability; *see also the biography of*Poisson.] It would have been better to speak of “rare events.”

Many recent studies on the meaning of the different derivations and uses of Poisson’s formula are linked to Bortkiewicz’s discovery. The Poisson distribution has become the subject of important work that extends to statistically dependent events and varying probabilities. Large parts of operational research and queueing theory are based on the Poisson distribution.

Bortkiewicz also contributed to the theory of runs with the publication of his book *Die Iterationen* (1917). This work was motivated by an attack made by the psychologist Karl Marbe (1916–1919) on the easy assumption of independence in applications of probability theory, for example, to successive flips of a coin or to sequences of male and female births. Marbe believed that a run of male births leads to a heightened probability of a female birth, as nature tries to equalize or make uniform the sex ratio. Bortkiewicz showed, however, that Marbe’s mathematics was wrong and that a mathematically correct approach gives agreement between theoretical independence and observed sequences in cases of the kind discussed.

Bortkiewicz devoted a book (1913) to the statistical interpretation of radioactivity. He showed that regularities considered as physical laws could be expressed by existing theorems on mean values of stochastic processes.

He also showed (1922*a*) that the extreme values, which had been considered unsuitable for the analysis and the characterization of a distribution, are statistical variables depending upon initial distribution and sample size. He gave the exact distribution of the normal range and computed its mean for sample sizes up to 20. With primitive equipment, he reached good numerical results (Tippett 1925) and checked them by many observations. The statistical importance of this work is obscured by mathematical complexities incident to the normal distribution.

The study of dispersion was central to the thought of Bortkiewicz. He confirmed and extended (1904*b*) the ideas of his teacher, Lexis, and strengthened them by his derivation of the standard error of the coefficient of dispersion (1918). He defended the importance and originality of Lexis (1930). The generalization of these methods led to the modern analysis of variance.

From the start Bortkiewicz worked on political economy and, like Lexis, he shared none of the usual vulgar prejudices to Marx. According to Schumpeter ([1932] 1960, p. 303) “By far his most important achievement is his analysis of the theoretical framework of the Marxian system [(1906–1907; 1907)], much the best thing ever written on it and, incidentally, on its other critics. A similar masterpiece is his paper on the theories of rent of Rodbertus and Marx [(1910–1911)].”

Bortkiewicz succeeded in embedding in a mathematical form both Marx’s determination of the average profit rate for simple reproduction and Marx’s transformation, implicit therein, of value into price. According to Marx (*Das Kapital*, vol. 3), to yield the average profit rate the total surplus value is divided by the total capital, that is, the sum of constant and variable capital. In this solution, however, the input is measured in values and the output in prices. Bortkiewicz was the first of Marx’s many critics to see this inconsistency. He made the necessary modifications that rendered the Marxian scheme of surplus values and prices consistent. However, his dry presentation prevented the Marxists (except for Klimpt) from accepting his method.

His investigations on price index numbers (1923–1924) are noteworthy contributions to mathematical economics. Irving Fisher (1922) had developed many such numbers. Bortkiewicz brought clarity and order into this system of index numbers by stating the requirements that such a number must satisfy in order to fulfill its purpose.

Bortkiewicz argued vigorously for his views. In 1910 he wrote an article attacking Alfred Weber’s geometrical representation of the location of industries. His polemical article (1915*b*) against the Pearson school clarified his fundamental attitude —namely, that it is worthless to construct formulas to reproduce observations if these formulas have no theoretical meaning. In another polemic (1922*b*) against Pearson he insisted on Helmert’s priority in discovering the distribution of the mean square residual when individual errors are normal. Yet Bortkiewicz’s answer (1923) was quite mild when Keynes (1921, p. 403, note 2) wrote, “… Bortkiewicz does not get any less obscure as he goes on. The mathematical argument is right enough and often brilliant. But what it is all really about, and what it really amounts to, and what the premises are, it becomes increasingly perplexing to decide.”

In his article (1931*a*) on the disparity of income distributions Bortkiewicz used Pareto’s law. While Pareto had not been very clear about the role of his basic parameter α, Bortkiewicz established different measures of income concentration and showed that α is such a measure. Bortkiewicz’s work on concentration was published in ignorance of the prior work of Gini.

Bortkiewicz had a characteristic way of working. He presented each problem from all sides with extreme thoroughness and patience after an extensive study of the literature. This multiple foundation makes the solution unassailable, but the reader can trace no single line from premises to conclusion: the central line of thought is entwined with numerous sidelines and extensive polemics, especially on matters of scientific priority. He criticized with equal zeal and profundity important and insignificant mistakes, printing errors, and numerical miscalculations. A large part of his work appeared as reviews and critical analyses in remote journals. His writings stimulated numerous scientists in Germany, in the northern European countries and in Italy, but not in England. He did not create a school, perhaps because of his austere character and his poor teaching. He underestimated his own work and even doubted, wrongly, its practical significance. His cautious nature forbade him to strive for external honors. He was, from 1903, a member of both the International Statistical Institute, which then consisted mainly of administrative statisticians, and the Swedish Academy of Sciences. He maintained objectivity in the face of popular slogans as well as “untimely opinions.” He was a true scholar of the old school and his life was passed in enviable quietness.

Four of his contributions are decisive: the proof that the Poisson distribution corresponds to a statistical reality; the introduction of mathematical statistics into the study of radioactivity; the inception of the statistical theory of extreme values; and the lonely effort to construct a Marxian econometry.

E. J. Gumbel

*[Other relevant material may be found in*Distributions, Statistical, *article on*approximations to distributions; Economic thought, *articles on*socialist thought; Nonparametric statistics. *articles on*order statistics*and*runs; Queues; *and in the biographies of*Lexis*and*Pareto.]

## WORKS BY BORTKIEWICZ

1890 Smertnost’ i dolgovechnost’ muzhskago pravoslavnago naseleniia evropeiskoi Rossii (Mortality and Lifespan of the Male Russian Orthodox Population of European Russia). I. Akademiia Nauk, *Zapiski* 63: Supplement no. 8.

1891 Smertnost’ i dolgovechnost’ zhenskago pravoslavnogo naseleniia evropeiskoi Rossii (Mortality and Lifespan of the Female Russian Orthodox Population of European Russia). I. Akademiia Nauk, *Zapiski* 66: Supplement no. 3.

1893 *Die mittlere Lebensdauer: Die Methoden ihrer Bestimmung und ihr Verhältnis zur Sterblichkeitsmessung*. Jena (Germany): Fischer.

1898 *Das Gesetz der kleinen Zahlen*. Leipzig: Teubner.

1901 Anwendungen der Wahrscheinlichkeitsrechnung auf Statistik. Volume 1, pages 821–851 in *Encyklopädie der mathematischen Wissenschaften*. Leipzig: Teubner.

1903 Risicoprämie und Sparprämie bei Lebensversicherungen auf eine Person. *Assekuranz-Jahrbuch* 24, no. 2:3–16.

1904*a* Über die Methode der “Standard Population.” International Statistical Institute, *Bulletin* 14, no. 2: 417–437.

1904*b* Die Theorie der Bevölkerungsund Moralstatistik nach Lexis. *Jahrbücher für Nationalökonomie und Statistik* 82:230–254.

(1906–1907) 1952 Value and Price in the Marxian System. *International Economic Papers* 2:5–60. → First published in German.

(1907) 1949 On the Correction of Marx’s Fundamental Theoretical Construction in the Third Volume of *Capital.* Pages 197–221 in Eugen von Böhm-Bawerk, *Karl Marx and the Close of His System*. New York: Kelley. → First published in German.

1909–1911 Statistique. Part 1, volume 4, pages 453–490 in *Encyclopédie des sciences mathématiques*. Paris: Gauthier-Villars. → Substantially the same as Bortkiewicz 1901, with changes by the translator, F. Oltramare.

1910 Eine geometrische Fundierung der Lehre vom Standort der Industrien. *Archiv für Sozialwissenschaft und Sozialpolitik* 30:759–785.

1910–1911 Die Rodbertus’sche Grundrententheorie und die Marx’sche Lehre von der absoluten Grundrente. *Archiv für die Geschichte des Sozialismus und der Arbeiterbewegung* 1:1–40, 391–434.

1911 Die Sterbeziffer und der Frauenüberschuss in der stationären und der progressiven Bevölkerung. International Statistical Institute, *Bulletin* 19:63–141.

1913 *Die radioaktive Strahlung als Gegenstand wahrscheinlichkeitstheoretischer Untersuchungen*. Berlin: Springer.

1915*a* Über die Zeitfolge zufälliger Ereignisse. International Statistical Institute, *Bulletin* 20, no. 2:30–111.

1915*b* Realismus und Formalismus in der mathematischen Statistik. *Allgemeines statistisches Archiv 9:* 225–256.

1917 *Die Iterationen: Ein Beitrag zur Wahrscheinlichkeitstheorie*. Berlin: Springer.

1918 Der mittlere Fehler des zum Quadrat erhobenen Divergenzkoeffizienten. Deutsche Mathematiker-Vereinigung, *Jahresbericht* 27:71–126.

1919 *Bevölkerungswesen*. Leipzig: Teubner.

1922*a* Die Variationsbreite beim Gaussschen Fehlergesetz. *Nordisk statistisk tidskrift* 1:11–38, 193–220.

1922*b* Das Helmertsche Verteilungsgesetz für die Quadratsumme zufälliger Beobachtungsfehler. *Zeitschrift für angewandte Mathematik und Mechanik* 2:358–375.

1923 Wahrscheinlichkeit und statistische Forschung nach Keynes. *Nordisk statistisk tidskrift* 2:1–23.

1923–1924 Zweck und Struktur einer Preisindexzahl. *Nordisk statistisk tidskrift* 2:369–408; 3:208–251, 494–516.

1929 Korrelationskoeffizient und Sterblichkeitsindex. *Blätter für Versicherungs-Mathematik und verwandte Gebiete* 1:87–117.

1930 Lexis und Dormoy. *Nordic Statistical Journal* 2: 37–54.

1931*a* Die Disparitätsmasse der Einkommensstatistik. International Statistical Institute, *Bulletin* 25, no. 3: 189–291.

1931*b* The Relations Between Stability and Homogeneity. *Annals of Mathematical Statistics* 2:122.

## SUPPLEMENTARY BIBLIOGRAPHY

Anderson, Oskar 1931 Ladislaus von Bortkiewicz. *Zeitschrift für Nationalökonomie* 3:242–250.

Andersson, Thor 1931 Ladislaus von Bortkiewicz: 1868–1931 Nordic *Statistical Journal* 3:9–26. → Includes a bibliography.

Crathorne, A. R. 1928 The Law of Small Numbers. *American Mathematical Monthly* 35:169–175.

Fisher, Irving (1922) 1927 *The Making of Index Numbers: A Study of Their Varieties, Tests, and Reliability*. 3d ed., rev. Boston: Houghton Mifflin.

Freudenberg, Karl 1951 Die Grenzen für die Anwendbarkeit des Gesetzes der kleinen Zahlen. *Metron* 16: 285–310.

Gini, C. 1931 Observations … à la communication … du M. L. von Bortkiewicz. International Statistical Institute, *Bulletin* 25, no. 3:299–306.

Gosset, William S. (1919) 1943 An Explanation of Deviations From Poisson’s Law in Practice. Pages 65–69 in [William S. Gosset], *“Student’s” Collected Papers*. Cambridge Univ. Press.

Gumbel, E. J. 1931 L. von Bortkiewicz. *Deutsches statistisches Zentralblatt* 23, cols. 231–236.

Gumbel, E. J. 1937 Les centenaires. *Aktuárske védy* (Prague) 7:10–17.

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

KÜhne, Otto 1922 *Untersuchungen über die Wert und Preisrechnung des Marxschen Systems: Eine dogmenkritische Auseinandersetzung mit L. von Bortkiewicz*. Greifswald (Germany): Bamberg.

Lorenz, Charlotte 1951 *Forschungslehre der Sozialstatistik*. Volume 1: Allgemeine Grundlegung und Anleitung. Berlin: Duncker & Humblot.

Marbe, Karl 1916–1919 *Die Gleichförmigkeit in der Welt: Untersuchungen zur Philosophic und positiven Wissenschaft*. 2 vols. Munich: Beck.

Newbold, Ethel M. 1927 Practical Applications of the Statistics of Repeated Events, Particularly to Industrial Accidents. *Journal of the Royal Statistical Society* 90:487–535.

Schumacher, Hermann 1931 Ladislaus von Bortkiewicz *Allgemeines statistisches Archiv* 21:573–576.

Schumpeter, Joseph A. (1932) 1960 Ladislaus von Bortkiewicz: 1868–1931. Pages 302–305 in Joseph A. Schumpeter, *Ten Great Economists From Marx to Keynes*. New York: Oxford Univ. Press. → First published in Volume 42 of the *Economic Journal*.

Tippett, L. H. C. 1925 On the Extreme Individuals and the Range of Samples Taken From a Normal Population. *Biometrika* 17:364–387.

Weber, Erna 1935 *Einführung in die Variations- und Erblichkeits-statistik*. Munich: Lehmann.

Windsor, Charles P. 1947 Quotations: *Das Gesetz der kleinen Zahlen. Human Biology* 19:154–161.

Woytinsky, Wladimir S. 1961 *Stormy Passage; A Personal History Through Two Russian Revolutions to Democracy and Freedom: 1905-1960*. New York: Van-guard.

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