Statistical Science

“Student” and small-sample theory

E. L. Lehmann

Full-text: Open access


.The paper discusses the contributions Student W. S. Gosset made to the three stages in which small-sample methodology was established in the period 1908–1933: (i) the distributions of the test-statistics under the assumption of normality, (ii) the robustness of these distributions against nonnormality, (iii) the optimal choice of test statistics. The conclusions are based on a careful reading of the correspondence of Gosset with Fisher and E. S. Pearson.

Article information

Statist. Sci. Volume 14, Number 4 (1999), 418-426.

First available in Project Euclid: 24 December 2001

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Lehmann, E. L. “Student” and small-sample theory. Statist. Sci. 14 (1999), no. 4, 418--426. doi:10.1214/ss/1009212520.

Export citation


  • AIRY, G. B. 1879. On the Algebraic and Numerical Theory of Errors of Observations and the Combination of Observations. 3rd Ed. MacMillan, London. Z.
  • BOX, J. F. 1978. R. A. Fisher: The Life of a Scientist. Wiley, New York. Z.
  • BOX, J. F. 1981. Gosset, Fisher and the t-distribution. Amer. Statist. 35 61 66. Z.
  • EISENHART, C. 1979. On the transition from ``Student's'' z to ``Student's'' t. Amer. Statist. 33 6 10. Z.
  • FISHER, R. A. 1915. Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika 10 507 521. Z.
  • FISHER, R. A. 1921. On the ``probable error'' of a coefficient of correlation deduced from a small sample. Metron 1 3 32. Z.
  • FISHER, R. A. 1922a. On the mathematical foundations of theoretical statistics. Philos. Trans. Roy. Soc. London Ser. A 222 309 368. Z.
  • FISHER, R. A. 1922b. The goodness of fit of regression formulae, and the distribution of regression coefficients. J. Roy. Statist. Soc. 85 597 612. Z.
  • FISHER, R. A. 1924a. The distribution of the partial correlation coefficient. Metron 3 329 332. Z.
  • FISHER, R. A. 1924b. On a distribution yielding the error functions of several well known statistics. In Proc. Internat. Congress of Mathematics, Toronto 2 805 813. Z.
  • FISHER, R. A. 1925a. Applications of `Student's' distribution. Metron 5 90 104. Z.
  • FISHER, R. A. 1925b. Statistical Methods for Research Workers, Z. 2nd ed. 1928 Oliver and Boyd, Edinburgh. Z.
  • FISHER, R. A. 1928. The general sampling distribution of the multiple correlation coefficient. Proc. Roy. Soc. London Ser. A 121 654 673. Z.
  • FISHER, R. A. 1929. Statistics and biological research. Nature 124 266 267. Z.
  • FISHER, R. A. 1939. Student. Ann. Engenics 9 1 9. Z.
  • FISHER, R. A. 1971 1974. Collected Papers of R. A. Fisher Z. J. H. Bennett, ed.. Univ. Adelaide. Z.
  • GOSSET, W. S. 1970. Letters from W. S. Gosset to R. A. Fisher, 1915 1936 With Summaries by R. A. Fisher and a Foreword by L. McMullen. Printed for private circulation. Z. Z
  • HALD, A. 1998. A History of Mathematical Statistics from. 1750 1930. Wiley, New York. Z.
  • JEVONS, W. S. 1874, 1877. The Principles of Science. McMillan, London. Z.
  • LAIRD, N. M. 1989. A conversation with F. N. David. Statist. Sci. 4 235 246. Z.
  • LEHMANN, E. L. 1993. The Bertrand Borel debate and the origins of the Neyman Pearson theory. In Statistics and Z Probability: A Raghu Raj Bahadur Festschrift J. K. Ghosh, S. K. Mitra, K. R. Parthasarathy and B. L. S. Prakasa Rao,. eds.. Wiley Eastern Limited, New Delhi. Z.
  • MCMULLEN, L. 1939. `Student' as a man. Biometrika 30 205 210. Z.
  • MERRIMAN, M. 1884. A Textbook on the Method of Least Squares. Wiley, New York. Z.
  • MOSTELLER, R. and TUKEY, J. W. 1977. Data Analysis and Regression. Addison-Wesley, Reading, MA. Z.
  • NEYMAN, J. 1929. Methodes nouvelles de verification des hy´ ´ potheses. In Comptes Rendus du I Congres de Mathematiciens des Pays Slaves, Warsaw 355 366. ´ Z.
  • NEYMAN, J. 1967. A Selection of Early Statistical Papers of J. Neyman. Univ. California Press, Berkeley. Z.
  • NEYMAN, J. 1977. Frequentist probability and frequentist statistics. Synthese 36 97 131. Z.
  • NEYMAN, J. and PEARSON, E. S. 1928. On the use and interpretation of certain test criteria for purposes of statistical inference. Biometrika 20A 175 240 and 263 294. Z.
  • NEYMAN, J. and PEARSON, E. S. 1933. On the problem of the most efficient tests of statistical hypotheses. Philos. Trans. Roy. Soc. Ser. A 231 289 337. Z.
  • NEYMAN, J. and PEARSON, E. S. 1967. Joint Statistical Papers of J. Neyman and E. S. Pearson. Univ. California Press, Berkeley. Z.
  • PEARSON, E. S. 1929a. Review of R. A. Fisher: Statistical Methods for Research Workers, 2nd ed. Nature. Z.
  • PEARSON, E. S. 1931. The analysis of variance in cases of nonnormal variation. Biometrika 23 114 133. Z.
  • PEARSON, E. S. 1939. `Student' as a statistician. Biometrika 30 210 250. Z.
  • PEARSON, E. S. 1966a. The Selected Papers of E. S. Pearson. Univ. California Press, Berkeley. Z.
  • PEARSON, E. S. 1967. Some reflexions on continuity in the development of mathematical statistics, 1885 1920. Biometrika 54 341 355. Z.
  • PEARSON, E. S. 1968. Some early correspondence between W. S Gosset, R. A. Fisher and Karl Pearson with notes and comments. Biometrika 55 445 457. Z.
  • PEARSON, E. S. 1990. `Student' A Statistical Biography of Z William Sealy Gosset R. L. Plackett and G. A. Barnard,. eds.. Clarendon Press, Oxford. Z.
  • REID, C. 1982. Neyman From Life. Springer, New York. Z. STUDENT 1908a. The probable error of a mean. Biometrika 6 1 25.Z. STUDENT 1908b. Probable error of a correlation coefficient. Biometrika 6 302 310. Z. STUDENT 1917. Tables for estimating the probability that the mean of a unique sample of observations lies between and any given distance of the mean of the population from which the sample is drawn. Biometrika 6 414 417. Z. STUDENT 1925. New table for testing the significance of observations. Metron 5 105 108. Z. STUDENT 1929. Statistics in biological research. Nature 124 93. Z. STUDENT'S COLLECTED PAPERS 1942. E. S. Pearson and J. Wishart, eds. Cambridge Univ. Press. Z.
  • TANKARD, J. W., JR. 1984. The Statistical Pioneers. Schenkman, Cambridge. Z.
  • TIPPETT, L. H. C. 1927. Random Sampling Numbers. Cambridge Univ. Press. Z.
  • WELCH, B. L. 1958. `Student' and small sample theory. J. Amer. Statist. Assoc. 53 777 788.