Student's t-test

Common types

• One-sample t-test: compares a sample mean to a known or hypothesized population mean (for example, testing if the average weight differs from 70 kg).
• Independent two-sample t-test: compares means of two independent groups. Variants include the pooled t-test (assumes equal variances) and Welch's t-test (does not assume equal variances).
• Paired (dependent) t-test: compares means of matched or repeated measurements (for example, before-and-after measurements on the same subjects).

Key assumptions

• Observations are independent within and between groups (unless the design is paired).
• The underlying distribution of the measurement is approximately normal, especially for small samples.
• For the pooled two-sample t-test, group variances are assumed equal; Welch's t-test relaxes this.
• The statistic is robust to modest departures from normality when sample sizes are moderate or large.

Formally, the one-sample t statistic is t = (x̄ − μ)/ (s/√n), where x̄ is the sample mean, μ the null mean, s the sample standard deviation and n the sample size. Two-sample formulas replace μ with the other sample mean and adjust the denominator to reflect pooled or separate variances; degrees of freedom differ between pooled and Welch's methods. Results are interpreted via two-sided or one-sided tests, p-values and confidence intervals. Effect-size measures such as Cohen's d complement p-values by quantifying practical magnitude.

History, uses and limitations

The test was published under the pen name "Student" by William Sealy Gosset in the early 20th century while working at a brewery. It remains a standard inferential tool for experiments and observational studies. Limitations include sensitivity to heavy-tailed distributions, small sample sizes, and nonindependence; alternatives include nonparametric tests (e.g., Wilcoxon), permutation tests, or bootstrap methods when assumptions are doubtful. In reporting, include test type, assumptions checked, test statistic, degrees of freedom, p-value, confidence interval and an effect-size estimate.

Practical implementations are available in most statistical software and programming languages; choose the variant (pooled vs Welch, paired vs independent) to match the study design and always examine data distribution and variance patterns before drawing conclusions.