The chi-square (χ2) test statistic is used in several settings, but the appropriate test depends on the question being asked and how the data were collected. This lesson compares the goodness-of-fit, independence, and homogeneity tests so you can identify which one applies.
Use a goodness-of-fit test to determine whether the distribution of one qualitative variable in a population matches a specified distribution. The data typically come from one survey question or one experiment conducted on a single population.
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What does a goodness-of-fit test assess?
It assesses whether the categorical distribution of one population matches a specified distribution. Its null hypothesis is that the population fits that distribution.
What does a chi-square test for independence assess?
It assesses whether two qualitative variables are independent or related within one population. The null hypothesis states that the variables are independent.
What does a chi-square test for homogeneity assess?
It assesses whether two or more populations have the same distribution for one qualitative variable. Its null hypothesis states that the populations follow the same distribution.
How do independence and homogeneity tests differ?
Independence concerns the relationship between two categorical variables in one population. Homogeneity compares the distribution of one categorical variable across different populations.
What two questions help identify the appropriate chi-square test?
Ask how many populations are involved and how many categorical variables are being studied. Then consider whether the goal is to compare with a target distribution, assess a relationship, or compare distributions across populations.
Quiz questions
Which chi-square test determines whether one population's categorical distribution matches a specified distribution?Answer: The goodness-of-fit test.
A survey records each person's age group and preferred news source in one population. Which test assesses whether these variables are associated?Answer: The test for independence.
Researchers compare preferred payment methods among separate samples from two cities. Which test assesses whether the distributions are the same?Answer: The test for homogeneity.
What design difference separates a test for independence from a test for homogeneity?Answer: Independence uses two variables in one population; homogeneity compares one variable across populations.
What can a chi-square goodness-of-fit result establish about category differences?Answer: It can indicate that at least one category differs, but not identify which one.
Based on Introductory Statistics 2e by Barbara Illowsky, Susan Dean (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.