Introduction: Why Chi-Square Matters in AP Statistics
The chi-square (χ²) test is one of the most intimidating topics for AP Statistics students — but it doesn’t have to be. Whether you’re tackling Multiple Choice Questions (MCQs) or Free Response Questions (FRQs), chi-square appears often in inference problems involving categorical data.
By the end of this guide, you’ll know:
- When to use chi-square tests.
- How to calculate expected counts, χ² statistics, and p-values.
- How to write full AP FRQ responses.
- How RevisionDojo’s chi-square practice banks, calculator tutorials, and step-by-step guides will help you crush this topic.
Step 1: What Is a Chi-Square Test?
Chi-square tests compare observed counts with expected counts to see if there’s a significant difference.
There are three types on the AP Exam:
- Goodness-of-Fit (GOF): Does observed data match a theoretical distribution?
- Homogeneity: Are two or more groups distributed the same way across categories?
- Independence: Are two categorical variables related?
👉 RevisionDojo provides flowcharts to help you pick the right test.
Step 2: Conditions for Chi-Square
Before running the test, always check:
- Random: Data comes from a random sample or experiment.
- Independence: Observations are independent.
- Large Sample Size: Expected counts ≥ 5 for all cells.
👉 If conditions fail, mention it in your FRQ response. RevisionDojo has condition-check drills for practice.
Step 3: The Chi-Square Formula
χ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}
Where:
- O = observed count
- E = expected count
👉 χ² is always positive, since it measures squared differences.
Step 4: How to Calculate Expected Counts
- Goodness-of-Fit: E=n⋅pE = n \cdot p (sample size × expected proportion).
- Homogeneity/Independence: E=(row total)(column total)grand totalE = \frac{(\text{row total})(\text{column total})}{\text{grand total}}
👉 RevisionDojo worksheets give step-by-step expected count practice.
Step 5: Example – Goodness-of-Fit Test
Question: A die is rolled 60 times. Observed counts: 5, 8, 12, 10, 15, 10. Is the die fair?
- Hypotheses:
- H₀: The die is fair.
- Hₐ: The die is not fair.
- Expected counts:
60 ÷ 6 = 10 each. - Calculate χ²: χ2=(5−10)210+(8−10)210+...+(10−10)210\chi^2 = \frac{(5-10)^2}{10} + \frac{(8-10)^2}{10} + ... + \frac{(10-10)^2}{10} = 5.
- Degrees of freedom: df = 6 – 1 = 5.
- p-value: Using calculator → p ≈ 0.42.
- Conclusion: Fail to reject H₀. No evidence die is unfair.
👉 RevisionDojo problem sets walk you through chi-square GOF examples step by step.
Step 6: Example – Test of Independence
Question: Is there an association between gender and preferred streaming platform?
Netflix Hulu Disney+ Total Male 30 20 25 75 Female 35 25 15 75 Total 65 45 40 150
- Hypotheses:
- H₀: Gender and platform preference are independent.
- Hₐ: They are not independent.
- Expected counts:
Example for Male-Netflix: E=(75)(65)150=32.5E = \frac{(75)(65)}{150} = 32.5 - χ² calculation: Sum all (O – E)² / E.
- Degrees of freedom: (rows–1)(columns–1) = (2–1)(3–1) = 2.
- p-value: Calculator/Desmos → p ≈ 0.21.
- Conclusion: Fail to reject H₀. No significant relationship.
👉 RevisionDojo has independence test worksheets with full solutions.
Step 7: Chi-Square on the Calculator (TI-84 + Desmos)
- TI-84:
STAT→EDIT→ enter table.STAT TESTS→ χ²-Test.- Gives χ², df, p-value.
- Desmos:
- Enter observed + expected.
- Use formula directly.
👉 RevisionDojo provides step-by-step calculator walkthroughs.
Step 8: Interpreting Chi-Square Results
Always write conclusions in context:
❌ Bad: “Reject H₀, p < 0.05.”
✅ Good: “Since p < 0.05, we reject H₀ and conclude there is convincing evidence that streaming preference depends on gender.”
👉 RevisionDojo FRQ banks emphasize full-sentence interpretations.
Step 9: Common Mistakes Students Make
- ❌ Forgetting to check conditions.
- ❌ Mixing up independence vs homogeneity.
- ❌ Using wrong formula for expected counts.
- ❌ Not stating hypotheses in context.
👉 RevisionDojo drills help students avoid these traps.
Step 10: Why Chi-Square Is Tested on AP Stats
College Board loves chi-square because it tests:
- Understanding of categorical data.
- Ability to apply formulas.
- Writing conclusions in context.
In other words: math + interpretation.
Step 11: The RevisionDojo Advantage
RevisionDojo makes chi-square easy with:
- Goodness-of-fit, homogeneity, and independence worksheets.
- Calculator + Desmos tutorials.
- FRQ practice with sample high-scoring responses.
- Condition-check drills.
👉 Check out RevisionDojo’s Chi-Square Practice Resources here.
Frequently Asked Questions (FAQs)
Q: What are the three chi-square tests on the AP Exam?
A: Goodness-of-fit, homogeneity, independence.
Q: How do I know when to use chi-square?
A: Use chi-square when data is categorical counts.
Q: Do I need to calculate by hand?
A: You may show the setup, but use the calculator for χ² and p-values.
Q: What if expected counts are less than 5?
A: Mention conditions not met. AP graders reward this awareness.
Q: How does RevisionDojo help with chi-square?
A: With step-by-step practice, calculator tutorials, and FRQ writing guides.
Final Thoughts
The chi-square test is one of the most conceptually important inference tests on the AP Statistics exam. If you can:
- Choose the right test.
- Calculate expected counts.
- Use your calculator effectively.
- Write strong conclusions in context.
…you’ll turn chi-square from a weakness into a point-scoring strength.
With RevisionDojo’s chi-square practice sets, calculator walkthroughs, and FRQ guides, you’ll be fully prepared to tackle any χ² problem on the 2025 AP Statistics exam.
