Student's t-distribution is a continuous probability distribution that resembles the normal — bell-shaped, symmetric — but with heavier tails. It depends on a parameter called degrees of freedom (df).

When to use it: inference about a population mean when (1) population standard deviation $\sigma$ is unknown (estimated from sample as $s$ ), AND (2) sample size $n$ is small.

The t-statistic: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ follows a t-distribution with $n - 1$ degrees of freedom.

Properties: as $df \to \infty$ , t-distribution converges to standard normal $N(0, 1)$ . For $df < 30$ , heavy tails meaningfully widen confidence intervals — you "pay" for not knowing $\sigma$ .

History: developed by William Gosset at Guinness Brewery (publishing under "Student" because Guinness banned employee publications). Underlies t-tests (one-sample, two-sample, paired) and CIs for means with unknown variance.

Student's t-distribution

The t-distribution is bell-shaped like the normal but with heavier tails. Used for inference about means when sample size is small or σ is unknown.