The t Distribution and t Intervals

The t distribution is simply a bell curve with fatter tails, used in place of the normal when you've had to estimate the spread from your sample. It's also called Student's t, after the pen name of William Gosset, who worked it out for small batches at the Guinness brewery. The formula $\bar x \pm t^*\frac{s}{\sqrt n}$ is the interval from the last lesson with two swaps: $s$ (your sample's standard deviation) in place of $\sigma$, and $t^*$ (a slightly bigger multiplier from a t table) in place of $z^*$. The degrees of freedom, $n - 1$, just tell you which row of the table to read. Nothing else changes.

Why fatter tails? Think about what can go wrong with a small sample. By bad luck, $s$ might come out smaller than the true $\sigma$, and then your interval would be too narrow and miss more often than it should. The t distribution builds in that risk by spreading out a little more than the normal, so $t^*$ is bigger than $z^*$ and the interval is wider.

The t distribution looks like the standard normal but with heavier tails, because $s$ varies from sample to sample. As $df$ grows it approaches the normal: by $df = 30$ they're close, and $t^*$ approaches $z^*$. That makes sense too: with lots of data, $s$ is a very good estimate of $\sigma$, so there's little extra uncertainty left to pay for.

Assumptions: a random sample, and either a roughly normal population or a large enough $n$ (about 30 or more, less if the data are not too skewed).

In practice σ is almost never known, so this is the interval actually used for means.

A common mistake. It's tempting to use $df = n$, since $n$ is the number you have in front of you. Use $n - 1$. Computing $s$ means first estimating the mean from the same data, which uses up one degree of freedom, the same reason $s$ divides by $n - 1$.

Clinical measurements, materials testing and any study with a modest sample size and unknown spread use t intervals; Student's t was invented at the Guinness brewery to work with small batches.