Degrees of Freedom vs T-Distribution

Degrees of freedom count the independent pieces of information left after a statistic uses some up to estimate other quantities. For a one-sample $t$ procedure it is $n - 1$, because the sample mean has already been estimated from the $n$ values. For example, a sample of $n = 20$ has $20 - 1 = 19$ degrees of freedom for the $t$-distribution. For a chi-square test of a two-way table with $r$ rows and $c$ columns, the degrees of freedom are $(r - 1)(c - 1)$.

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.