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The chi-square test of independence is a statistical test used to determine whether there is a significant relationship between two categorical variables. The test involves comparing the observed frequency distribution (i.e., what you see in the sample) with the expected frequency distribution (i.e., what you expect to see if the two variables are unrelated).
To calculate the chi-square test statistic, you sum the squared differences between the observed and expected frequencies divided by the expected frequencies. If the test statistic is greater than the critical value, the null hypothesis (that there is no association between the variables) is rejected, and it is concluded that there is a significant association between the variables.
Below is a video of an example chi-square test of independence that I recorded for my online applied statistics class. I primarily work with criminal justice and criminology students, so the example below focuses on death penalty opinions by gender. It is pulled from Statistics for Criminology and Criminal Justice (3rd edition) by Jacinta M. Gau (pages 221-232 and 235).
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