Chi-square tests are used to test hypotheses on the distribution of observations. Various types of chi-square tests are used to determine if your data is significantly different from your expectations.
This blog will discuss the types of chi-square tests.
Definition: Chi-square tests
The chi-square test, usually written as (X2), is a statistical test that helps you test your hypotheses by determining if your data differs from your expectations.1
There are two types of chi-square tests:
- Chi-square goodness of fit test
- Chi-square test of independence
Using chi-square tests
Chi-square tests are usually written using the symbol X2. They are usually used to test statistics that don’t follow the expectations of normal distribution.2
In contrast, parametric tests cannot test hypotheses regarding categorical variables. Instead, they may involve categorical variables as independent variables. Categorical variables are nominal or ordinal variables that represent sets such as species and races.1
You can use them when:
- You wish to test a hypothesis on a single or more categorical variables
- You randomly selected your sample from the population
- You anticipate at least five observations in each set or group combinations
Hypotheses testing of frequency distributions
Two types of Pearson’s chi-square tests exist that determine if the detected frequency dispersal of categorical variables differs notably from the anticipated frequency distribution in the hypothesis. A frequency distribution aims to describe the distribution of observations between various groupings and is usually displayed on a frequency distribution table.3
Frequency distribution tables usually display the number of observations in individual groupings. Contingency frequency distribution tables are perfect where there are two categorical variables since they showcase the number of observations in each group combination.1
Example of a frequency distribution table & contingency frequency distribution table
You can use chi-square tests of independence to determine if the observed frequencies differ notably from the anticipated frequencies if the handedness is not related to skin color.
The chi-square formula
The two chi-square tests have the same formula:2
- Xs2 = chi-square test
- Ʃ = sum (take the sum of)
- Ο = observed frequency
- Ε = anticipated frequency
The types of chi-square tests
There are two primary types of Pearson’s chi-square tests: the goodness of fit and the test of independence.4
This chi-square test applies when you have a single categorical variable.
The test of independence applies when you have multiple categorical variables. This chi-square test helps you determine if two variables are correlated.4
Another type of chi-square test is the test of homogeneity. These chi-square tests are similar to the test of independence, as they determine if two populations hail from the same distribution.1
There is also McNemar’s test that applies the chi-square tests statistics. It examines if the variables’ proportions are equal.
Other types of chi-square tests that are not in Pearson’s category are:1
- Test of a single variance
- Likelihood ratio chi-square test
Conducting a chi-square test correctly
The procedure usually differs depending on the types of chi-square tests.
However, the standard steps are:3
- Construct a table showing the observed and anticipated frequencies
- Calculate X2 (chi-square value) using the formula
- Determine the critical chi-square value using a statistical table or software
- Compare the chi-square and chi-square critical value
- Choose whether to reject the null hypothesis
Reporting chi-square tests
Chi-square test reports should appear in the final results section. Follow the rules below when reporting chi-square tests in APA:2
- A reference or formula is unnecessary
- Use the X2 symbol for chi-square
- Add a space on each side of the equal sign
- If X2 is less than 0, you must include the leading zero and two significant figures after the decimal point
- The X2 tests report should be featured alongside its degrees of freedom, sample proportion, and p-value
The two main types of chi-square tests are:
- the goodness of fit
- the test of independence
The test of independence applies when you have several categorical variables. This chi-square test helps you determine if two variables are correlated.
This chi-square test applies when you have a single categorical variable. It tests if the frequency distribution of the variable varies notably from your anticipations noted in the hypothesis.
Pearson’s chi-square tests are statistical tests used to determine if statistical data is notably different from the expectations in the hypothesis.
1 Frimodig, Benjamin. “Chi Square (X2) Test Statistic.” Simply Psychology. February 08, 2023. https://www.simplypsychology.org/chi-square.html.
2 Zach. “4 Examples of Using Chi-Square Tests in Real Life.” Statology. August 25, 2021. https://www.statology.org/chi-square-test-real-life-examples/.
3 JMP Statistical Discovery LLC. “The Chi-Square Test.” Accessed March 13, 2023. https://www.jmp.com/en_us/statistics-knowledge-portal/chi-square-test.html.
4 Biswal, Avijeet. “What is a Chi-Square Test? Formula, Examples & Application.” Simpli Learn. February 17, 2023. https://www.simplilearn.com/tutorials/statistics-tutorial/chi-square-test.