# Chi-Square Tests (X2) – Types and Examples

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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.

## Chi-Square Tests – In a Nutshell

• Pearson’s chi-square tests are used to determine if statistical data is notably different from the expectations in the hypothesis.1
• There are two types of chi-square tests:
1. The goodness of fit test
2. The test of independence
• Chi-square tests vary in their purposes.

## 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
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## 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

The frequency of toilet visits by men at different ages during 24 hours.

 Men’s age (yrs) Frequency 18 - 28 5 29 - 39 8 50 - 60 11 61 and above 16

Contingency table of the preferred hand of a sample of African Americans and White Americans.

 Sample Right-handed Left-handed White Americans 240 32 African Americans 292 29

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

#### Chi-square goodness of fit test

This chi-square test applies when you have a single categorical variable.

Example:

The hypothesis explaining the anticipations of equal proportions

• Null (HO): The men will visit the toilet in equal proportions
• Alternative (HA): Men of different ages will visit the toilet in different proportions

Anticipations of varying proportions

• Null (Ho): The men visit the toilet in the same proportion as the average over the past twenty-four hours
• Alternative (HA): The men visit the toilet in different proportions from the average over the past twenty-four hours

#### Chi-square test of independence

The test of independence applies when you have multiple categorical variables. This chi-square test helps you determine if two variables are correlated.4

Example:

• Null (Ho): The proportion of left-handed individuals is the same for white and African Americans
• Alternative (HA): The proportions of left-handed individuals vary depending on their skin color.

#### Additional types of chi-square test

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.

Example:

A sample of 500 kids is offered two flavors of soda and asked if they like how each taste.

Contingency table of soda flavor preference:

 Soda flavor Like Dislike Strawberry 37 22 Passion 8 13
• Null (Ho): The proportion of kids that like strawberry soda is the same as that of kids that like passion soda.
• Alternative (HA): The proportion of kids that like strawberry soda differs from that of kids that like passion soda.

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

1. Construct a table showing the observed and anticipated frequencies
2. Calculate X2 (chi-square value) using the formula
3. Determine the critical chi-square value using a statistical table or software
4. Compare the chi-square and chi-square critical value
5. Choose whether to reject the null hypothesis

Example:

There was no significant correlation between the handedness and skin color, X2 (1, n =346) = 0.55, p = .505

## 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

## FAQs

#### What are the two main types of chi-square tests?

The two main types of chi-square tests are:

• the goodness of fit
• the test of independence

#### What is the chi-square 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.

#### What is the chi-square goodness of fit test?

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.

#### What are chi-square tests?

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 (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.

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