Comparing means of small samples when the variance itself is estimated from the data.The Student's t-test is a statistical hypothesis test used to determine whether the means of two groups are significantly different from each other. It is specifically designed for small sample sizes, where the central limit theorem may not guarantee a normal distribution of sample means, and where the population variance is unknown and must be estimated from the data. The test accounts for the extra uncertainty introduced by this estimation, making it a cornerstone of classical statistics.
The test was developed by William Sealy Gosset while working at the Guinness Brewery in Dublin, and published in 1908 under the pseudonym 'Student' because his employer forbade employees from publishing research.
The test calculates a t-statistic, which is the difference between the two sample means divided by the standard error of that difference. The standard error is computed from the sample variances and sample sizes. The t-statistic is then compared to a t-distribution, which has heavier tails than the normal distribution, reflecting the additional uncertainty from estimating the variance. The degrees of freedom of the t-distribution depend on the sample sizes and whether the variances are assumed equal.
It is used in virtually every field that compares averages — from clinical trials testing a new drug against a placebo, to A/B testing in web design, to comparing the mean yield of two agricultural plots.
Gosset's pseudonym 'Student' was so well-known that the test is still called 'Student's t-test' today, even though his real identity was revealed shortly after his death.