t-distribution

The $t$-distribution accounts for the extra uncertainty of estimating the population standard deviation with the sample standard deviation $s$. Its exact shape depends on the degrees of freedom: fewer degrees of freedom give fatter tails, and as they grow the curve approaches the standard normal. For example, a one-sample $t$ test with $n = 15$ uses the $t$ model with $15 - 1 = 14$ degrees of freedom. The test statistic is $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$, where $\bar{x}$ is the sample mean and $\mu_0$ the hypothesized mean.