Student's t distribution

The t distribution is the symmetric, heavier-tailed sampling distribution of a standardised mean when σ is estimated by the sample SD.

T = \frac{\xbar - \mu}{s/\sqrt{n}} \sim t_{n-1}

Where it comes from (normal population)

$\xbar$: sample mean
$s$: sample standard deviation (divisor $n - 1$ )
$n - 1$: degrees of freedom (df)

\E(T) = 0 \;(\nu > 1), \qquad \Var(T) = \frac{\nu}{\nu - 2} \;(\nu > 2)

Mean and variance

$\nu$: degrees of freedom

When to use

Critical values and p-values for means when the population SD is unknown: one-sample, two-sample and paired t procedures, regression coefficients ( $\nu = n - p$ ). Welch's procedures use a non-integer df, which the engine handles exactly.

How to interpret

Heavier tails than the normal, so critical values are larger: extra uncertainty from estimating σ.
As df grows, $t$ approaches $N(0,1)$ : $t_{0.975}$ is 2.776 (df 4), 2.262 (df 9), 2.045 (df 29), 1.960 (normal).

Common mistakes

Using $z = 1.96$ with a small sample and an estimated SD.
Wrong df: a one-sample t uses $n - 1$ , a pooled two-sample t uses $n_1 + n_2 - 2$ .
Using the one-tailed critical value for a two-sided interval (or the reverse).

Example

With df $= 9$ , $t = 2.1$ : upper tail $P(T > 2.1) = 0.0326$ , so the two-sided p-value is $0.0651$ . The critical value for a 95% two-sided interval is $t_{0.975, 9} = 2.2622$ .