t-distribution

The t-distribution, developed by William Gosset writing under the pen name "Student," describes the sampling distribution of the statistic t = (x̄ − μ) / (s / √n), where s is the sample standard deviation. Because s is itself an estimate that varies from sample to sample, this statistic carries more uncertainty than the z statistic, which uses the known population standard deviation σ. The t-distribution accounts for that extra uncertainty.

Visually the t-distribution is symmetric and centered at zero like the standard normal curve, but flatter in the middle and heavier in the tails. That shape means critical values are larger: a 95% confidence interval built with a t-distribution is wider than one built with a z, which is the correct penalty for not knowing σ. The distribution's single parameter is degrees of freedom, equal to n − 1 for a one-sample procedure. As degrees of freedom increase, the tails thin and the t-distribution converges on the standard normal — by roughly 30 degrees of freedom the two are close, and in the limit they coincide.

Practically, you use t rather than z whenever the population standard deviation is unknown, which covers nearly all real data analysis. The procedures include one-sample t intervals and tests for a mean, two-sample t procedures for comparing means, matched-pairs t tests, and t tests on the slope of a regression line. The conditions to check are random sampling, independence, and either an approximately normal population or a large enough sample for the Central Limit Theorem to apply.

AP Statistics tests the t-distribution heavily in its inference units, including choosing correct degrees of freedom and reading t tables. The FE Mechanical exam covers it under probability and statistics, especially hypothesis testing and the choice between a z-test and a t-test, and quantitative sections of the GRE draw on the underlying ideas of standard deviation and sampling variability.