Pearson chi-squared goodness-of-fit test

For observed cell counts $O_j$ and null expected counts $E_j$, the statistic isWith $k$ specified null cell probabilities and sufficiently large expected counts, its null limit is ${\chi }_{k-1}^2$.

For an $r$ by $c$ contingency table, the statistichas asymptotically a chi-squared distribution with $(r-1)(c-1)$ degrees of freedom under independence.