t-distribution

The $t$-distribution is not one curve but a family, indexed by the degrees of freedom. It is the distribution of $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$ (x-bar minus mu, over s divided by root n) when the data come from a normal population. Swapping the fixed $\sigma$ (sigma) for the sample standard deviation $s$, which itself changes from sample to sample, is what puts the extra weight in the tails. Every member is symmetric about 0, and the family closes on the standard normal as the degrees of freedom grow.

The numbers make that convergence concrete. For a 95 percent interval the t-table gives $t^* = 2.145$ at 14 degrees of freedom, $2.042$ at 30, $1.984$ at 100 and $1.962$ at 1000, against $z^* = 1.960$ for the normal. The gap is 8.6 percent of the critical value at 14 degrees of freedom and about 0.1 percent at 1000.

"The sample is small so use $t$, and large so use $z$" is the wrong rule, and it is the one most students carry in. The trigger is whether $\sigma$ is known, not how big $n$ is. With 500 observations and a standard deviation estimated from them, the correct model is $t$ on 499 degrees of freedom, which happens to sit very close to the normal. Knowing $\sigma$ with $n = 8$ would put you back on $z$.

The heavier tails change verdicts, not just widths. A statistic of 2.00 read on $t$ with 14 degrees of freedom has a two-sided p-value of 0.0653, against the 0.0455 the normal returns for the same 2.00, so at $\alpha = 0.05$ one model rejects and the other does not.

Those tails cover the uncertainty in $s$ and nothing else. They do not repair a skewed population or a stray outlier, which is why a $t$ procedure still asks you to look at the shape of the sample first. The $t$-distribution enters the course at topic 4.2 of Unit 4, Inference for Quantitative Data: Means.