Early sample measures of variability. (English) Zbl 1059.62500

David, H. A.

Stat. Sci. 13, No. 4, 368-377 (1998).

Summary: This paper attempts a brief account of the history of sample measures of dispersion, with major emphasis on early developments. The statistics considered include standard deviation, mean deviation, median absolute deviation, mean difference, range, interquartile distance and linear functions of order statistics. The multiplicity of measures is seen to result from constant efforts to strike a balance between efficiency and ease of computation, with some recognition also of the desirability of robustness and theoretical convenience. Many individuals shaped this history, especially Gauss. The main contributors to our story are in chronological order, Lambert, Laplace, Gauss, Bienaymé, Abbe, Helmert and Galton

MSC:

62-03	History of statistics
01A55	History of mathematics in the 19th century

Keywords:

Measures of dispersion; standard deviation; mean deviation; median absolute deviation; mean difference; range; interquartile distance; order statistics; chi-squared distribution; Gauss; Laplace; Bienaymé; Abbe; Helmert

References:

[1]	Abbe, E. (1863). Ueber die Gesetzmässigkeit in der Vertheilung der Fehler bei Beobachtungsreihen. In Gesammelte Abhandlungen 2. Fischer, Jena.
[2]	Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. Wiley, New York. · Zbl 0850.62008
[3]	Balakrishnan, N. and Cohen, A. C. (1991). Order Statistics and Inference. Estimation Methods. Academic Press, New York. · Zbl 0732.62044
[4]	Bertrand, J. (1855). Méthode des Moindres Carrés. Mémoire sur la Combinaison des Observations, par Ch.-Fr. Gauss. MalletBachelier, Paris.
[5]	Bienay mé, I.-J. (1852). Sur la probabilité des erreurs d’aprés la méthode des moindres carrés. J. Math. Pures Appl. 17 33-78.
[6]	Daniell, P. J. (1920). Observations weighted according to order. Amer. J. Math. 42 222-236. JSTOR: · JFM 47.0491.04 · doi:10.2307/2370465
[7]	David, H. A. (1968). Gini’s mean difference rediscovered. Biometrika 55 573-575. · Zbl 0177.46501
[8]	David, H. A. (1981). Order Statistics, 2nd ed. Wiley, New York. · Zbl 0553.62046
[9]	David, H. A. (1992). Introduction to Mosteller (1946), ”On some useful ‘inefficient’ statistics.” In Breakthroughs in Statistics (S. Kotz and N. L. Johnson, eds.) 2 203-208. Springer, New York.
[10]	Daw, R. H. and Pearson, E. S. (1972). Studies in the history of probability and statistics. XXX. Abraham De Moivre’s 1733 derivation of the normal curve: a bibliographical note. Biometrika 59 677-680. JSTOR: · Zbl 0256.01011
[11]	Edwards, A. W. F. (1974). The history of likelihood. Internat. Statist. Rev. 42 9-15. JSTOR: · Zbl 0289.62006 · doi:10.2307/1402681
[12]	Eisenhart, C. (1961). Boscovich and the combination of observations. In Roger Joseph Boscovich (L. L. Why te, ed.) 200-212. Allen and Unwin, London. [Reprinted in Kendall and Plackett (1977) 88-100.]
[13]	Encke, J. F. (1834). Über die Methode der kleinsten Quadrate. Berliner Astronomisches Jahrbuch 1834 249-312.
[14]	Fisher, R. A. (1920). A mathematical examination of the methods of determining the accuracy of an observation by the mean error, and by the mean square error. Monthly Notices of the Roy al Astronomical Society 80 758-770. [Reprinted as Paper 2 in Fisher (1950).]
[15]	Fisher, R. A. (1934). Statistical Methods for Research Workers, 5th ed. Oliver and Boy d, Edinburgh. · JFM 60.1162.01
[16]	Fisher, R. A. (1939). ”Student.” Annals of Eugenics 9 1-9.
[17]	Fisher, R. A. (1950). Contributions to Mathematical Statistics. Wiley, New York. · Zbl 0040.36201
[18]	Galton, F. (1882). Report of the Anthropometric Committee. In Report of the 51st Meeting of the British Association for the Advancement of Science, 1881 245-260. John Murray, London.
[19]	Gauss, C. F. (1809). Theoria Motus Corporem Coelestium. [English Translation (1857) by C. H. Davis. Little, Brown, Boston.]
[20]	Gauss, C. F. (1816). Bestimmung der Genauigkeit der Beobachtungen. [Reprinted in (1880) Carl Friedrich Gauss Werke 4 109-117. Königliche Gesellschaft der Wissenschaften, Göttingen.]
[21]	Gauss, C. F. (1821, 1823, 1826). Theoria combinationis observationum erroribus minimis obnoxiae. [Reprinted in (1880) Carl Friedrich Gauss Werke 4 3-93. Königliche Gesellschaft der Wissenschaften, Göttingen. For English translation see Stewart (1995).]
[22]	Gini, C. (1912). Variabilit a e mutabilit a, contributo allo studio delle distribuzioni e delle relazione statistiche. Studi Economico-Giuredici dell’ Universit a di Cagliari 3(part 2) i-iii, 3-159.
[23]	Godwin, H. J. (1949). On the estimation of dispersion by linear sy stematic statistics. Biometrika 36 92-100. JSTOR: · Zbl 0033.19801 · doi:10.1093/biomet/36.1-2.92
[24]	Godwin, H. J. and Hartley, H. O. (1945). On the distribution of the estimate of the mean deviation obtained from samples from a normal population. Biometrika 32 254-265. JSTOR: · Zbl 0060.29902 · doi:10.1093/biomet/33.3.254
[25]	Govindarajulu, Z. (1966). Best linear estimates under sy mmetric censoring of the parameters of a double exponential population. J. Amer. Statist. Assoc. 61 248-258. [Correction (1976) 71 255.] JSTOR: · doi:10.2307/2283060
[26]	Hald, A. (1998). A History of Mathematical Statistics from 1750 to 1930. Wiley, New York. · Zbl 0979.01012
[27]	Harter, H. L. (1970). Order Statistics and Their Uses in Testing and Estimation 1, 2. U.S. Government Printing Office, Washington, D.C.
[28]	Harter, H. L. (1978). A Chronological Annotated Bibliography of Order Statistics 1. Pre-1950. U.S. Government Printing Office, Washington, DC. [Reprinted by American Sciences Press, Sy racuse, NY (1983)]. · Zbl 0515.62045
[29]	Hartley, H. O. (1942). The range in random samples. Biometrika 32 334-348. JSTOR: · Zbl 0063.01943 · doi:10.1093/biomet/32.3-4.334
[30]	Helmert, F. R. (1875). Ueber die Berechnung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler. Zeitschrift f ür Mathematik und physik 20 300- 303. Helmert, F. R. (1876a). Ueber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen. Zeitschrift f ür Mathematik und physik 21 192-218. Helmert, F. R. (1876b). Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Fehlers directer Beobachtungen gleicher Genauigkeit. Astronom. Nachr. 88 113-132. · JFM 07.0113.01
[31]	Hey de, C. C. and Seneta, E. (1977). I. J. Bienay mé: Statistical Theory Anticipated. Springer, Berlin.
[32]	Hoaglin, D. C., Mosteller, F. and Tukey, J. W., eds. (1983). Understanding Robust and Exploratory Data Analy sis. Wiley, New York. · Zbl 0599.62007
[33]	Johnson, N. L. and Kotz, S., eds. (1997). Leading Personalities in Statistical Sciences. Wiley, New York. · Zbl 0878.01018
[34]	Jordan, W. (1869). Ueber die Bestimmung der Genauigkeit mehrfach wiederholter Beobachtungen einer Unbekannten. Astronom. Nachr. 74 209-226. · JFM 02.0841.03
[35]	Kendall, M. G. (1971). Studies in the history of probability and statistics. XXVI. The work of Ernst Abbe. Biometrika 58 369- 373. JSTOR: · Zbl 0224.01023
[36]	Kendall, M. G. and Plackett, R. L., eds. (1977). Studies in the History of Statistics and Probability 2. Griffin, London. · Zbl 0389.01005
[37]	Kruskal, W. H. (1946). Helmert’s distribution. Amer. Math. Monthly 53 435-438. JSTOR: · Zbl 0063.03378 · doi:10.2307/2306241
[38]	Kruskal, W. H. and Stigler, S. M. (1997). Normative terminology: ”normal” in statistics and elsewhere. In Statistics and Public Policy (B. D. Spencer, ed.) 85-111. Clarendon, Oxford.
[39]	Lancaster, H. O. (1966). Forerunners of the Pearson 2. Austral. J. Statist. 8 117-126. · Zbl 0171.17301 · doi:10.1111/j.1467-842X.1966.tb00262.x
[40]	Laplace, P. S. (1799). Traité de Mécanique Celeste 3. Duprat, Paris.
[41]	Laplace, P. S. (1818). Théorie Analy tique des Probabilités,
[42]	Deusci eme Supplément. [Included in Laplace (1820).]
[43]	Laplace, P. S. (1820). Théorie Analy tique des Probabilités, 3rd ed. Courcier, Paris. [Reprinted as (1847) Oeuvres de Laplace 7. Imprimerie Roy ale, Paris.]
[44]	Lloy d, E. H. (1952). Least-squares estimation of location and scale parameters using order statistics. Biometrika 39 88- 95. JSTOR: · Zbl 0046.36604 · doi:10.1093/biomet/39.1-2.88
[45]	McKay, A. T. and Pearson, E. S. (1933). A note on the distribution of range in samples of n. Biometrika 25 415-420. · JFM 59.1172.04
[46]	Mosteller, F. (1946). On some useful ”inefficient” statistics. Ann. Math. Statist. 17 377-408. · Zbl 0063.04123 · doi:10.1214/aoms/1177730881
[47]	Ogawa, J. (1951). Contributions to the theory of sy stematic statistics, I. Osaka Math. J. 3 175-213. · Zbl 0044.34301
[48]	Pearson, E. S. (1932). The percentage limits for the distribution of range in samples from a normal population. Biometrika 24 404-417. · Zbl 0005.36801
[49]	Pearson, E. S. (1935). The application of statistical methods to industrial standardization and quality control. No. 600, British Standards Institution.
[50]	Pearson, E. S. (1945). The probability integral of the mean deviation. Editorial note. Biometrika 32 252-253. JSTOR: · Zbl 0060.29901 · doi:10.2307/2332304
[51]	Pearson, E. S. and Hartley, H. O. (1942). The probability integral of the range in samples of n observations from a normal population. Biometrika 32 301-310. JSTOR: · Zbl 0063.06142
[52]	Pearson, K. (1900). On the criterion that a given sy stem of deviations from the probable in the case of a correlated sy stem of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine, 5th Series 50 157-175. · JFM 31.0238.04
[53]	Pearson, K. (1920). On the probable errors of frequency constants, III. Biometrika 13 113-132.
[54]	Pearson, K. (1931). Historical note on the distribution of the standard deviations of samples of any size drawn from an indefinitely large normal parent population. Biometrika 23 416-418. · JFM 57.0634.02
[55]	Peters, C. A. F. (1856). Über die Bestimmung des wahrscheinlichen Fehlers einer Beobachtung aus den Abweichungen der Beobachtungen von ihrem arithmetischen Mittel. Astronom. Nachr. 44 29-32.
[56]	Quetelet, A. (1846). Lettres a S.A.R. le Duc Regnant de SaxeCobourg et Gotha, sur la Théorie des Probabilités, Appliquée aux Sciences Morales et Politiques. Hay ez, Bruxelles. [English translation (1849) by O. G. Downes. Lay ton, London.]
[57]	Sarhan, A. E. and Greenberg, B. G. (eds.) (1962). Contributions to Order Statistics. Wiley, New York. · Zbl 0102.35204
[58]	Seal, H. L. (1967). Studies in the history of probability and statistics. XV. The historical development of the Gauss linear model. Biometrika 54 1-24. JSTOR: · Zbl 0154.25103
[59]	Seneta, E. (1983). Modern probabilistic concepts in the work of E. Abbe and A. de Moivre. Math. Sci. 8 75-80. · Zbl 0528.60003
[60]	Shey nin, O. B. (1966). Origin of the theory of errors. Nature 211 1003-1004. · Zbl 0161.24301
[61]	Shey nin, O. B. (1971). J. H. Lambert’s work on probability. Arch. Hist. Exact Sci. 7 244-256. · Zbl 0263.01017 · doi:10.1007/BF00357218
[62]	Shey nin, O. B. (1973). R. J. Boscovich’s work on probability. Arch. Hist. Exact Sci. 9 306-324. · Zbl 0263.01018 · doi:10.1007/BF00348539
[63]	Shey nin, O. B. (1979). C.F. Gauss and the theory of errors. Arch. Hist. Exact Sci. 20 21-72. · Zbl 0424.01006 · doi:10.1007/BF00776066
[64]	Shey nin, O. (1995). Helmert’s work in the theory of errors. Arch. Hist. Exact Sci. 49 73-104. · Zbl 0831.01004 · doi:10.1007/BF00374700
[65]	Shey nin, O. (1996). The History of the Theory of Errors. HänselHohenhausen, Egelsbach, Germany.
[66]	Sprott, D. A. (1978). Gauss’s contributions to statistics. Historia Math. 5 183-203. · Zbl 0395.01004 · doi:10.1016/0315-0860(78)90049-6
[67]	Stewart, G. W. (1995). Theory of the Combination of Observations Least Subject to Errors. SIAM, Philadelphia. [Transla
[68]	tion of Gauss (1821, 1823, 1826).] Stigler, S. M. (1973a). Laplace, Fisher, and the discovery of the concept of sufficiency. Biometrika 60 439-445. [Reprinted in Kendall and Plackett (1977) 271-277.] Stigler, S. M. (1973b). Simon Newcomb, Percy Daniell, and the history of robust estimation, 1885-1920. J. Amer. Statist. Assoc. 68 872-879. [Reprinted in Kendall and Plackett (1977) 410-417.] JSTOR: · Zbl 0286.01011 · doi:10.2307/2284515
[69]	Stigler, S. M. (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Harvard Univ. Press. · Zbl 0656.62005
[70]	Student (1907). On the error of counting with a haemacy tometer. Biometrika 5 351-360.
[71]	Tippett, L. H. C. (1925). On the extreme individuals and the range of samples taken from a normal population. Biometrika 17 364-387. · JFM 51.0392.01
[72]	Trotter, H. F. (1957). Gauss’s Work (1803-1826) on the Theory of Least Squares, an English Translation. Princeton Univ.
[73]	Tukey, J. W. (1960). A survey of sampling from contaminated distributions. In Contributions to Probability and Statistics (I. Olkin, S. G. Ghury e, W. Hoeffding, W. G. Madow and H. B. Mann, eds.) 448-485. Stanford Univ. Press. · Zbl 0201.52803
[74]	von Andrae, C. G. (1869). Schreiben des Herrn Geheimen Etatsraths von Andrä an den Herausgeber. Astronom. Nachr. 74 283-284.
[75]	von Andrae, C. G. (1872). Ueber die Bestimmung des wahrscheinlichen Fehlers durch die gegebenen Differenzen von m gleich genauen Beobachtungen einer Unbekannten. Astronom. Nachr. 79 257-272. von Bortkiewicz, L. (1922a). Variationsbreite und mittlerer Fehler. Sitzungsberichte der Berliner Mathematischen Gesellschaft 21 3-11. von Bortkiewicz, L. (1922b). Die Variationsbreite beim Gauss’schen Fehlergesetz. Nordisk Statistisk Tidskrift 1 1138, 193-220.
[76]	von Mises, R. (1923). Über die Variationsbreite einer Beobachtungsreihe. Sitzungsberichte der Berliner Mathematischen Gesellschaft 22 3-8. [Reproduced in (1964) Selected Papers of Richard von Mises 2 129-134. Amer. Math. Soc., Providence, RI.] · JFM 49.0376.04