8.6 Carrying Out a Chi-Square Test for Homogeneity or Independence

3 min readjanuary 8, 2023

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

AP Statistics 📊

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Source: Aponia Data

Test, Test, Test!

Now that we have chosen the correct test, checked our necessary conditions, and written our hypotheses for our test, it is now time to actually carry out the test! As with our GOF test, the test will consist of two mathematical elements: the test statistic (χ2 statistic) and our p-value. 😳

Test Statistic

The next step is to calculate the test statistic, which in this case is the chi-squared statistic. This is done by comparing the observed frequencies in the contingency table to the expected frequencies, which are calculated based on the assumption that the null hypothesis is true. The formula for the chi-squared statistic is: 🪑
χ2 = ∑ (O - E)^2 / E,
where O is the observed frequency and E is the expected frequency. The sum is taken over all cells in the contingency table.
The formula for our χ2 value can also be found using the formula on the formula sheet given for the exam. A much easier way of finding the test statistic is to use our graphing calculator. 📱

Degrees of Freedom

Our degrees of freedom are found by taking the number of rows and subtracting 1 and multiplying by the number of columns minus 1. 🔐
(number of rows - 1) * (number of columns - 1)
Let's say that you have a contingency table with 3 rows and 4 columns. To find the degrees of freedom, you would first subtract 1 from the number of rows to get 3 - 1 = 2. Then, you would subtract 1 from the number of columns to get 4 - 1 = 3. Finally, you would multiply these two values together to get the degrees of freedom, which in this case would be 2 * 3 = 6.
Hence, the degrees of freedom for a contingency table with 3 rows and 4 columns would be 6.


Once you finally get your χ2 value, you calculate your p-value by finding the probability of getting that particular χ2 by random chance. As always, if our p is low, we reject the H0. 🅿️
As mentioned above, the best way of doing all of this together is using your graphing calculator device and performing the χ2 GOF test. Just be sure to write out your χ2 value and your p-value from your calculator output. 


Just as we concluded hypothesis tests in previous units, we must compare our p-value from our calculator to a given ɑ value. If it is less than our alpha, we conclude that we reject the H0 and have convincing evidence of the Ha. Otherwise, we fail to reject the null and do not have convincing evidence of the Ha. Remember two things: ❗
  1. Never “accept” anything!
  2. Include context!
"Since our p-value (~0) is less than 0.05, we reject the null hypothesis. We have convincing evidence that at least one of the proportions for how people rank on the happiness scale is incorrect."


  • First part -- "Since our (p value) is </> 0.05, we reject/fail to reject our null."
  • Second part:
    • Test for Independence -- "We have/do not have convincing evidence that there is an association between variable x and y in our intended population."
    • Test for Homogeneity -- "We have/do not have convincing evidence that the distribution of categorical variable x is different between population x and population y."
🎥  Watch: AP Stats Unit 8 - Chi Squared Tests
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