Chi-square and F Ratio: Which should be used when?

  • Richard L. Gorsuch (UniMult Inc.)
  • Curtis Lehmann (Azusa Pacific University)


Approximations for Chi-square and F distributions can both be computed to provide a p-value, or probability of Type I error, to evaluate statistical significance. Although Chi-square has been used traditionally for tests of count data and nominal or categorical criterion variables (such as contingency tables) and F ratios for tests of non-nominal or continuous criterion variables (such as regression and analysis of variance), we demonstrate that either statistic can be applied in both situations. We used data simulation studies to examine when one statistic may be more accurate than the other for estimating Type I error rates across different types of analysis (count data/contingencies, dichotomous, and non-nominal) and across sample sizes (Ns) ranging from 20 to 160 (using 25,000 replications for simulating p-value derived from either Chi-squares or F-ratios). Our results showed that those derived from F ratios were generally closer to nominal Type I error rates than those derived from Chi-squares. The p-values derived from F ratios were more consistent for contingency table count data than those derived from Chi-squares. The smaller than 100 the N was, the more discrepant p-values derived from Chi-squares were from the nominal p-value. Only when the N was greater than 80 did the p-values from Chi-square tests become as accurate as those derived from F ratios in reproducing the nominal p-values. Thus, there was no evidence of any need for special treatment of dichotomous dependent variables. The most accurate and/or consistent p's were derived from F ratios. We conclude that Chi-square should be replaced generally with the F ratio as the statistic of choice and that the Chi-square test should only be taught as history.

Keywords: Chi-square, F ratio, teaching statistics, simulation study

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Published on
23 Aug 2018
Peer Reviewed