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William Sealy Gosset worked at Guinness as a chemist and brewer and developed methods of statistical inference for small samples. He published under the pseudonym "Student". His employment as a chemist is independently recorded by MacTutor; the Dictionary of Irish Biography identifies him as an applied statistician and brewer and explains the pseudonym policy. ^{[ref] [ref]}

The 1908 paper The Probable Error of a Mean appeared in Biometrika 6(1), pages 1–25, with the byline "Student". Its introduction identifies the uncertainty introduced by estimating a standard deviation from few observations and explicitly assumes a normal distribution. Sections II and III develop the relation between sample spread and the standardized sample mean. Its notation uses a sample spread with divisor $n$ and calls the ratio $z$; with that convention, the modern $T$ equals $\sqrt{n-1}z$. The modern exact independence argument is given under studentized normal sample mean. ^[ref]

Section VI describes a physical sampling check: 3,000 cards carrying existing paired measurements were shuffled, then divided into 750 groups of four. He compared observed frequencies of sample spread and standardized sample mean with theoretical frequencies. This was a check using finite, grouped, approximately normal distribution data; it did not prove exact calibration for arbitrary populations. ^[ref]

Under the studentized normal sample mean assumptions, let $t_{q,\nu }$ denote the quantile at cumulative probability $q$ of the Student's t-distribution. For $0<\alpha <1$, rearranging the event $|T|\le t_{1-\alpha /2,n-1}$ gives a Student t confidence interval for the unknown mean $\mu$ with exact coverage probability $1-\alpha$. ^[ref]

Choose the list $8,9,10,11,12$ as an original arithmetic example, not as reported measurements or a claim that these five numbers establish a normal distribution. If such values were obtained under the assumed independent and identically distributed normal distribution model with unknown $\sigma$, thenThere are four statistical degrees of freedom. For a 95% confidence interval, $t_{0.975,4}=2.776445105\dots$, soUsing the standard normal quantile $z_{0.975}=1.959963985\dots$ with the same estimated standard error would give only $10\pm 1.385903824\dots$. That underestimates the correct margin by about 29.4%. Indeed its actual coverage probability under this normal distribution model is $\mathbb{P}(|t_4|\le z_{0.975})\simeq 0.8784$, rather than 0.95.

The observations, the sample mean and the confidence interval all have different meanings. The bar estimates the population mean; it is not an interval intended to contain 95% of individual observations. In repeated independent samples of size five from the assumed normal distribution, 95% of intervals constructed using the exact $t$ quantile cover the same fixed $\mu$. This is not a 95% posterior probability that the fixed statistical parameter lies in this particular realized interval. Such a probability requires a separate Bayesian inference model.

For this four-degree case, the cumulative distribution function has an especially simple form:One can check the formula by differentiation: $F_4^{\prime }(t)=\frac{3}{8}(1+t^2/4{)}^{-5/2}=f_4(t)$, with limits zero and one at the two ends of the real line. Solving the increasing equation $F_4(t)=0.975$ by the bisection method gives the quantile used above.