Chi-Square Tests and Two-Way Tables
A free Statistics and Data Analysis lesson from the “Inference and Conclusions” unit, with a worked example and practice problems including step-by-step solutions.
Chi-square tests use counts in categories. For two-way tables, they help decide whether two categorical variables show evidence of association. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Recognize chi-square test situations
- Compare observed and expected counts
- Interpret evidence of association
Worked example
Problem. A chi-square test is used on a two-way table of grade level and club participation. What kind of relationship is being tested?
- Worked Example: First identify exactly what the question is asking: A chi-square test is used on a two-way table of grade level and club participation. What kind of relationship is being tested?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Two-way tables contain counts for categorical variables.
- Chi-square tests can check association between those variables.
Answer: association between categorical variables
Practice problems
1. Practice case A: What does a chi-square test on a two-way table check?
Choices: a normal z-score · association between categorical variables · a difference in two means · a regression slope
Show solution
- Warm-up: First identify exactly what the question is asking: What does a chi-square test on a two-way table check?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Two-way tables organize categorical counts.
- Chi-square tests check whether the categories are associated.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: association between categorical variables
2. Practice case B: Which study question matches a two-way-table chi-square test?
Choices: slope of height versus weight · z-score for one person · grade level by club membership counts · average score for one class
Show solution
- Warm-up: First identify exactly what the question is asking: Which study question matches a two-way-table chi-square test?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Chi-square uses counts in categories.
- Grade level and club membership are categorical.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: grade level by club membership counts
3. Practice case C: Expected counts in a chi-square test describe:
Choices: the exact observed counts · sample means · confidence levels · counts predicted if there were no association
Show solution
- Warm-up: First identify exactly what the question is asking: Expected counts in a chi-square test describe:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Expected counts come from the no-association model.
- Observed counts are compared to them.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: counts predicted if there were no association
4. Practice case D: If observed counts are far from expected counts, what does that suggest?
Choices: association · no variables · a smaller sample size only · a guaranteed causal effect
Show solution
- Warm-up: First identify exactly what the question is asking: If observed counts are far from expected counts, what does that suggest?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Big observed-versus-expected differences are evidence against independence.
- That suggests association.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: association
5. Practice case E: What should a student not claim from a chi-square association test alone?
Choices: the conclusion should mention context · one category caused the other · there is evidence of association · the variables are categorical
Show solution
- Warm-up: First identify exactly what the question is asking: What should a student not claim from a chi-square association test alone?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Association is not automatically causation.
- Study design controls causal claims.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: one category caused the other
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