Degrees of freedom

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 $(r - 1)(c - 1)$.