Student's t distribution

The story

When you standardise a sample mean using the sample standard deviation $S$ instead of the true $\sigma$ , the extra uncertainty in $S$ makes extreme values more likely. William Gosset, writing as Student, worked out that for Normal data $(\bar{X} - \mu)/(S/\sqrt{n})$ follows a $t$ distribution with $\nu = n - 1$ degrees of freedom. Formally, $t$ is $Z/\sqrt{V/\nu}$ with $Z$ standard Normal and $V$ an independent $\text{Chi-squared}(\nu)$ .

Small $\nu$ means heavy tails: at $\nu = 1$ the mean does not even exist, and the variance $\nu/(\nu - 2)$ is finite only when $\nu > 2$ . As $\nu$ grows, $S$ becomes a reliable estimate of $\sigma$ and the $t$ distribution closes in on the standard Normal; by $\nu = 30$ the two are hard to tell apart by eye.

Formulas

Support

x \in \mathbb{R}

PDF

f(x) = \frac{\Gamma\!\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\, \Gamma\!\left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}

CDF

F(x) = 1 - \tfrac{1}{2} I_{\nu/(\nu+x^2)}\!\left(\tfrac{\nu}{2}, \tfrac{1}{2}\right) \quad \text{for } x \ge 0

Expectation

\mathbb{E}[X] = \begin{cases} 0 & \text{for } \nu > 1 \\ \text{undefined} & \text{otherwise} \end{cases}

Variance

\operatorname{Var}(X) = \begin{cases} \dfrac{\nu}{\nu - 2} & \text{for } \nu > 2 \\ \infty & \text{for } 1 < \nu \le 2 \end{cases}

Mode

0

Parameters

Parameter	What it controls
$\nu$	The degrees of freedom. Small $\nu$ gives heavy tails and frequent extreme values; large $\nu$ makes the curve almost exactly standard Normal.

Where it shows up

One-sample and two-sample t-tests on small samples.
Confidence intervals for a mean when $\sigma$ is unknown.
Testing regression coefficients in linear models.
Robust modelling of noisy data with occasional outliers, such as financial returns.

Try this

Shade from $-1.96$ to $1.96$ at $\nu = 30$ and you get about 0.94, a little under the Normal's 0.95. Now set $\nu = 1$ : the same interval holds only about 0.70, because so much probability has moved into the tails.
Set $\nu = 1$ , switch to Sample means and increase $n$ . The histogram of averages refuses to narrow: the mean of $n$ Cauchy draws is itself Cauchy, so the Central Limit Theorem does not apply when the variance is not finite.
Set $\nu = 3$ and press Draw 1000 several times, clearing in between. The theoretical variance is 3, but the sample variance jumps around wildly as rare, huge draws come and go. Repeat at $\nu = 30$ and see how much more stable it is.

Related distributions

$t$ with $\nu$ degrees of freedom tends to the standard Normal as $\nu \to \infty$ .
It is built as $Z/\sqrt{V/\nu}$ , where $Z$ is standard Normal and $V$ is an independent Chi-squared with $\nu$ degrees of freedom.
With $\nu = 1$ it is the Cauchy distribution, which has no mean and no variance.