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Most statistics you meet at school assume you have plenty of data. A brewer testing a new batch of barley, or a doctor trying a drug on ten patients, does not. William Sealy Gosset worked out how to draw honest conclusions from a handful of measurements. This post covers who he was, what problem he solved, a worked t-test on the real data he used in 1908, common mistakes, and what you can learn from how he worked.
Who was William Gosset?
According to his MacTutor biography at the University of St Andrews, Gosset was born in Canterbury on 13 June 1876 and died in Beaconsfield on 16 October 1937. He studied chemistry and mathematics at New College, Oxford, and took first-class degrees in both. In 1899 he joined Arthur Guinness Son and Company as a chemist at its Dublin brewery.
In 1906–07 he spent a year at Karl Pearson's laboratory at University College London. MacTutor notes that he published under the name "Student", which is why his test is called Student's t-test, and that he found the form of the t distribution by combining mathematics with experiments on random numbers, an early use of what is now called the Monte Carlo method. From 1922 he built up a small statistics department at the brewery, and at the end of 1935 he moved to London to run Guinness's new brewery there.
What problem did Student's t-test solve?
Gosset's paper, The Probable Error of a Mean, appeared in Biometrika in March 1908. In its introduction, which you can read in a typeset copy of the paper, he explains the usual method of the time: take the sample's standard deviation s, divide by √n, and use the normal curve. But as the number of experiments falls, he wrote, the standard deviation found from the sample "becomes itself subject to an increasing error", until judgements reached this way "may become altogether misleading".
He also noted that some chemical, many biological and most agricultural experiments can only be repeated a few times, so this class of problem had been almost outside the range of statistical inquiry. His paper supplied tables for small samples. The distribution behind them has heavier tails than the normal curve: with little data, you must allow for larger surprises.
How do you run a t-test? A worked example with Gosset's own data
Gosset illustrated his method with figures from a 1904 study by A. R. Cushny and A. R. Peebles, giving the extra hours of sleep ten patients gained from two forms of the drug hyoscyamine. The difference between the two drugs for each patient was:
1.2, 2.4, 1.3, 1.3, 0, 1.0, 1.8, 0.8, 4.6, 1.4
Step 1, the mean. The sum is 15.8, so the mean is 15.8 ÷ 10 = 1.58 hours.
Step 2, the standard deviation. Today we divide the sum of squared deviations by n − 1 = 9, which gives s ≈ 1.23. (Gosset's table lists 1.17 because he divided by n; the conclusion is the same.)
Step 3, the standard error. s ÷ √n = 1.23 ÷ 3.162 ≈ 0.389.
Step 4, the t value. t = 1.58 ÷ 0.389 ≈ 4.06.
Step 5, compare. With 9 degrees of freedom, the NIST table of t critical values gives 2.262 for a two-sided test at the 5% level, compared with 1.960 from the normal curve. Our 4.06 is well beyond 2.262, so the difference between the drugs is very unlikely to be chance. Gosset put the odds at about 666 to one that the second drug was the better sleeping draught.
Notice the gap between 2.262 and 1.960. With only ten readings the bar is higher. With one degree of freedom the same table gives 12.706.
What mistakes do students make with small samples?
Using the normal curve for tiny samples. With few readings it overstates your confidence. Use t with n − 1 degrees of freedom.
Mixing up n and n − 1. Be clear which divisor your formula and your table assume.
Ignoring pairing. Gosset tested the differences for each patient, not the two columns separately. He noted the low spread of the differences was probably because both drugs acted similarly on the same patient.
Treating "significant" as "large". A t-test tells you a difference is unlikely to be chance, not that it matters in practice.
The t-test is not in the JEE Main syllabus (the 2024 statistics unit stops at mean, variance and standard deviation), so treat this as enrichment and check the current bulletin. The mean and standard deviation steps are exactly the ones you practise in mean and variance problems. MacTutor notes Gosset discussed statistical problems with Ronald Fisher, whose work you can read about in Ronald Fisher and the design of experiments.
What can a student learn from how Gosset worked?
Gosset's statistics grew straight out of his daily work: MacTutor quotes a colleague describing the "intimate connection" between his research and the practical problems he faced at the brewery. He also checked his maths against simulated data before trusting it. When you learn a formula, test it on a small example where you can work out the answer another way.
If you get stuck on the mean, standard deviation or standard error steps, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning.
Recompute Gosset's example yourself on paper, then check what happens to t if the patient who gained 4.6 hours is removed.
