The Student's T-Distribution

When sampling from a normal population with known standard deviation $\sigma,$ the Central Limit Theorem gives us

$$Z = \dfrac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1).$$

In practice $\sigma$ is rarely known. We replace it with the sample standard deviation $s,$ producing the $t$-statistic

$$T = \dfrac{\bar{X} - \mu}{s/\sqrt{n}}.$$

This quantity does not follow a standard normal distribution. It follows the Student's $t$-distribution with $n-1$ degrees of freedom, written

$$T \sim t_{n-1}.$$

The $t$-distribution is symmetric about $0$ and bell-shaped, but has heavier tails than $N(0,1)$ -- reflecting the extra uncertainty from estimating $\sigma$ by $s.$ As $n \to \infty,$ the sample standard deviation stabilizes and $t_{n-1}$ converges to $N(0,1).$

For instance, a sample of size $n = 25$ with $\bar{x} = 105$ and $s = 10,$ tested against $\mu = 100,$ yields

$$t = \dfrac{105 - 100}{10/\sqrt{25}} = \dfrac{5}{2} = 2.5,$$

with $24$ degrees of freedom.