# What was Student's (Gosset's) contribution in formulating the t-test?

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?

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.

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