9
$\begingroup$

A recent question, related question, and cited source, recently made me aware that the $N-1$ correction for sample estimates of population variance is referred to as Bessel's correction. Bessel was dead by 1846 (wikipedia citation) and the t-test was published in 1908 (wikipedia citation). For some reason, I had always assumed that the contribution of Gosset (aka Student) in formulating the t-test was the use of $N-1$ in the calculation of $s^2$. Now it seems this contribution clearly belongs to Bessel. In this vein, I ask what was Gosset's contribution in formulating the t-test?

$\endgroup$
7
$\begingroup$

E. L. Lehmann addressed this question in an introduction to a reprint of Gosset's 1908 article in Breakthroughs in Statistics, Volume II--Methodology and Distribution (Samuel Kotz & Norman L. Johnson, eds., 1992).

Lehmann first describes the state of the art in Gosset's time: it amounted to a "z test" where the estimated standard deviation was treated as if it were a constant. Then he discusses Gosset's contribution:

However, if the sample size $n$ is small, $S^2$ will be subject to considerable variation. It was the effect of this variation that concerned Student, the pseudonym of W. S. Gosset... . He pointed out that if the form of the distribution of the $X$'s is known, this variation can be taken into account, since for any given $n$ the distribution of $t$ is then determined exactly. He proposed to work out this distribution for the case in which the $X$'s are normal.

This in fact is what Gosset did, albeit without mathematical rigor: he derived some properties of the distribution of $t$ for the normal case, matched them to properties of known distributions, and correctly guessed its distribution--acknowledging that this was less than rigorous. To support his guess, he conducted a Monte-Carlo simulation using samples of four from a dataset.

Gosset wrote pseudonymously because his employer (the Guinness brewery) apparently felt that this improved understanding of small-sample variation was a bit of an advantage in the business : it would have led to improved quality control procedures.

$\endgroup$
  • $\begingroup$ Thank you. My answer got chopped up it seems. Besides, yours is better in nearly all respects. $\endgroup$ – russellpierce Mar 3 '13 at 22:24
  • 3
    $\begingroup$ Monte Carlo in 1908 - men were men in those days... $\endgroup$ – Corone Mar 4 '13 at 7:06
  • $\begingroup$ @Corone That is very observant: he probably carried out all calculations with pencil and paper. $\endgroup$ – whuber Mar 4 '13 at 15:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.