Studet's n-Free t-Statistic

D. F. Andrews
University of Toronto
November 29, 2000

Abstract

Student's t-statistic is unnecessarily complicated. Any monotonic function of the statistic will share the same statistical properties. Here we advocate a much simpler form of the statistic. It is easier to compute. There are no square roots, no factors of (n-1), in fact there is no n. Because the critical values are less dependent on n, no table is required for n > 20.

Studet's n-free t-statistic

W.L.O.G. the statistic is defined to test the hypothesis E[x] = 0. The statistic is simply defined by

[Graphics:studetgr1.gif]

where SUM denotes a sum of the values of the variable x. For those rare instances where one sided tests are appropriate, the statisitic may be multiplied by the sign of SUM[x]. The statistic is much simpler to write, compute and calibrate than the version commonly taught:

[Graphics:studetgr2.gif]

The t2- tables

Critical values of t2 may be easily computed from the incomplete beta function I[1-t2/n, (n-1)/2, 1/2]. Here we focus on the common upper 0.05 point. The following table gives the p-values associated with constant critical values for the simplified statistic and Student's t-statistic.

The p-values associated with t2 = [Graphics:studetgr6.gif]

are very close to the nominal 0.05. The critical values of the simple statistic are relatively less dependent on the sample size, N.

Acknowledgements

Professor S. Stigler has kindly provided supporting evidence for the use of the simple statistic by noting that it (as with many other useful methods) was anticipated and indeed used by Laplace (1823). See Stigler (1986) pages 153, 155.

References

Laplace, P. S. (1823). De l'action de la lune sur l'atmosphere. Annales de Chimie et de Physique. 24: 280-294
Stigler, S. M. (1986). The History of Satistics. Harvard University Press. Cambridge.

Studet's n-Free t-Statistic

D. F. Andrews University of Toronto November 29, 2000

Abstract