Chi-Square Test

Chi-square is a statistical analysis that tests for goodness of fit or independence against the null hypothesis in two samples of data.

An example might be the expected number of students within a group of 100 who would attain an A, B, C, or failing grade (F). The expected frequency of an A grade may be 25, for B 25, for C 25, and 25 students might be expected to fail. This forms the statistical reasoning for the null hypothesis, that there will be an equal spread of grades across the four categories. When the sample of grades is collected and analyzed using the chi-square test, the expected frequencies (E) would be compared against the actual outcome (O). If there is a large difference and hence squared difference (X2), the researcher can reject the null hypothesis, whereas a small or zero difference would mean the null hypothesis is true.

Suppose that the null hypothesis in an educational psychology study is that study time does not influence the attained grade. The researcher would observe how the frequency of A, B, C, and F grades were distributed in terms of hours of study. If study time does influence the proportions of students getting higher grades, the affirmative hypothesis could be that more students putting in longer study hours would achieve A and B grades, and the null hypothesis could be rejected. The greater the squared difference in these longer study hour categories, the more confident the researcher can be in affirming that study time influences grade achievement. The researcher would consult standard chi-square tables to see whether the observed squared differences in these high grade categories was statistically significant, that is, unlikely to have occurred by random chance.

See also Statistics.



Daniel, Wayne W., and Chad Lee Cross. Biostatistics: A Foundation for Analysis in the Health Sciences, 10th ed. New York: Wiley, 2013.


Columbia CNMTL. “QMSS e-Lessons: About the ChiSquare Test.” (accessed July 15, 2015).

McDonald, John H. “Chi-square Test of Independence.” Handbook of Biological Statistics. (accessed July 15, 2015).


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