Degrees of Freedom vs Chi-Square Test

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

A chi-square test works with counts in categories rather than means, checking goodness of fit, independence, or homogeneity. It adds up the squared gaps between observed and expected counts, each scaled by the expected count, into one statistic. The formula is $\chi^2 = \sum \frac{(O - E)^2}{E}$, where $O$ is an observed count and $E$ an expected count. For example, rolling a die 60 times and comparing each face's tally to the expected 10 tests whether the die is fair, and a large $\chi^2$ gives a small p-value.