Abstract
The accuracy of an estimate of a normally distributed quantity is judged by reference to its variance, or rather, to an estimate of the variance based on the available sample. In 1908 “Student” examined the ratio of the mean to the standard deviation of a sample.1 The distribution at which he arrived was obtained in a more rigorous manner in 1925 by R.A. Fisher,2 who at the same time showed how to extend the application of the distribution beyond the problem of the significance of means, which had been its original object, and applied it to examine regression coefficients and other quantities obtained by least squares, testing not only the deviation of a statistic from a hypothetical value but also the difference between two statistics.
Presented at the meeting of the American Mathematical Society at Berkeley, April 11,1931.
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Notes
Biometrika, vol. 6 (1908), p. 1.
Applications of Student’s Distribution, Metron, vol. 5 (1925), p. 90.
V. Romanovsky, On the criteria that two given samples belong to the same normal population (on the different coefficients of racial likeness), Metron, vol. 7 (1928), no. 3, pp. 3-46; K. Pearson, On the coefficient of racial likeness, Biometrika, vol. 18 (1926), pp. 105-118.
“Mean Value” is used in the sense of mathematical expectation; the variance of a quantity whose mean value is zero is defined as the expectation of its squares; the covariance of two such quantities is the expectation of their product. Thus the correlation of the two in a hypothetical infinite population is the ratio of their covariance to the geometric mean of the variances.
Metron, loc. cit., and Statistical Methods for Research Workers, Oliver and Boyd, third edition (1928).
This geometrical interpretation of T shows its affinity with the multiple correlation coefficient, whose interpretation as the cosine of an angle of a random line with a V p enabled R.A. Fisher to obtain its exact distribution (Phil. Trans., vol. 213B, 1924, p. 91; and Proc. Roy. Soc., vol. 121A, 1928, p. 654). The omitted steps in Fisher’s argument may be supplied with the help of generalized polar coordinates as in the text. Other examples of the use of these coordinates in statistics have been given by the author in The Distribution of Correlation Ratios Calculated from Random Data, Proc. Nat. Acad. Sci., vol. 11 (1925), p. 657, and in The Physical State of Protoplasm, Koninklijke Akademie van Wetenschappen te Amsterdam, verhandlingen, vol. 25 (1928), no. 5, pp. 28-31.
Tracts for Computers, no. 7 (1921).
On certain expansions in series of polynomials of incomplete B-functions (in English), Recueil Math, de la Soc. de Moscou, vol. 33 (1926), pp. 207-229.
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Hotelling, H. (1992). The Generalization of Student’s Ratio. In: Kotz, S., Johnson, N.L. (eds) Breakthroughs in Statistics. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0919-5_4
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