Chi-Squared Tests

Chi-squared tests are performed on data that reveal the frequencies with which a variable of interest occurs in given categories. This is in contrast to tests on data that are interval or ratio measurements. The chi-squared tests are less powerful than many other methods of hypothesis testing, such as t-tests, ANOVAs, and regression analysis. Therefore, if a research question can be answered by a research design that leads to a more powerful analysis test, that is preferred.

There are two chi-squared tests that we will study: the Chi-Squared Test for Goodness of Fit and the Chi-Squared Test of Independence.

The Chi-Squared Test for Goodness of Fit is used when there is a set of observed or experimental data, and you wish to compare it to predicted or expected values. The expected values can be obtained from theoretical probability or prior research. For example, you may wish to test a die to see if it is fair or not. Theoretical probability predicts that, if the die is rolled sixty times, each number would appear ten times. The observed/experimental data would be the results from actually rolling the die sixty times and recording the frequency with which each number appeared. It should be clear that it would be unlikely that the observed data would match the predicted values exactly. But, if the null hypothesis it true and the die is fair, it would also be unlikely that the observed data would vary a great deal from the theoretically predicted frequencies, and a measure of the discrepancy between the observed and expected values is what the Chi-Squared Goodness of Fit test calculates as the chi-squared statistic.

The Chi-Squared Test of Independence does not compare observed data to predicted data for a single categorical factor, but instead considers the relationship between two factors. For example, you could use the test of independence to determine if one categorical factor, such as gender (male or female), was independent of another categorical factor, such as political group affiliation (democrat, republican, independent, green party, or other). The test will provide the statistical probability that any relationship that appears to exist between the factors (based on the observed data and expected values) is due to chance rather than due to an actual relationship between the variables.


Original work on this document was done by Central Virginia Governor's School students Richard Barnes, Kim Tibbs, and Ryan Nash (Class of '00). This document was updated by Central Virginia Governor's School students Matthew James and Kyle Nenninger (Class of '03).


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