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AP Statistics Chi-Square Test Explained | Step-by-Step 2025 Guide

••6 min read

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:

  1. Goodness-of-Fit (GOF): Does observed data match a theoretical distribution?
  2. Homogeneity: Are two or more groups distributed the same way across categories?
  3. 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?

  1. Hypotheses:
    • H₀: The die is fair.
    • Hₐ: The die is not fair.
  2. Expected counts:
    60 ÷ 6 = 10 each.
  3. 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.
  4. Degrees of freedom: df = 6 – 1 = 5.
  5. p-value: Using calculator → p ≈ 0.42.
  6. 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

  1. Hypotheses:
    • H₀: Gender and platform preference are independent.
    • Hₐ: They are not independent.
  2. Expected counts:
    Example for Male-Netflix: E=(75)(65)150=32.5E = \frac{(75)(65)}{150} = 32.5
  3. χ² calculation: Sum all (O – E)² / E.
  4. Degrees of freedom: (rows–1)(columns–1) = (2–1)(3–1) = 2.
  5. p-value: Calculator/Desmos → p ≈ 0.21.
  6. 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.

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