**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 (X^{2}), 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

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## Using chi-square tests

Chi-square tests are usually written using the symbol** X ^{2}**. 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}

- Xs
^{2}= 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

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## 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 X
^{2}(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 X
^{2}symbol for chi-square - Add a space on each side of the equal sign
- If X
^{2}is less than 0, you must include the leading zero and two significant figures after the decimal point - The X
^{2}tests report should be featured alongside its degrees of freedom, sample proportion, and p-value

## FAQs

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.

## Sources

^{1} Frimodig, Benjamin. “Chi Square (X^{2}) 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.