Is the question about one population or several?
You will be able to: Distinguish chi-square homogeneity and independence by study design.
Is the question about one population or several?
A school records travel mode and club session for one random sample. Another study takes separate random samples from two schools and compares travel-mode distributions.
A useful starting point: How do you test a difference between two shares? →
Words and symbols before equations
- Independence test
- Examines association between two categorical variables in one sampled population.
- Homogeneity test
- Compares a categorical response distribution across populations or treatments.
- Expected count E
- Count predicted by the no-association or equal-distribution null model.
- Contingency table
- A two-way count table classifying each unit once.
What this picture assumes
Two rows A/B and three travel categories: Walk, Bus, Car. Fixed equal row and column margins; all expected cells equal the chosen count. Independent random sampling and 10% conditions from large populations are assumed. This Fall 2026 CED requires every expected count >5. Each 3D block is one synthetic person.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- χ²=10; df=2. Right-tail p=0.006738. At α=0.05, reject independence. The design determines the population interpretation.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
One random sample classified by two variables calls for an independence test: H₀ says the variables are independent in the population; Hₐ says they are associated.
Independent random samples from multiple populations, or an appropriate randomized experiment, call for homogeneity: H₀ says the response distributions are the same; Hₐ says at least one differs. Calculations can be identical even though the question and design differ.
Use actual counts, mutually exclusive categories and independent observational units. Check randomization and any 10% sampling conditions. The revised CED states all expected counts should be greater than 5; this unit follows that stricter wording. This threshold concerns expected, not observed, counts.
| Feature | Independence | Homogeneity |
|---|---|---|
| Data collection | One sample, two categorical variables | Separate samples or randomized treatments |
| Null claim | No population association | Same response distributions |
| Alternative | A population association | At least one distribution differs |
A worked example, step by step
Separate SRSs from three schools record whether students walk, bus or drive. Name the test and hypotheses.
- There are three separately sampled populations and one categorical response.
- Use chi-square homogeneity.
- H₀: the travel-mode distributions are the same across the three school populations; Hₐ: at least one differs.
- Check independent random samples, each 10% sampling condition and all expected cell counts greater than 5.
The data-collection design determines the interpretation; the table’s rectangular shape does not distinguish the tests.
Does a chi-square association establish a causal direction?
Compare with an explanation
No. The design must justify causation; association alone does not.
Predict. Change one thing. Explain.
Inspect the same two-way table under the two design descriptions. Explain which hypotheses change and why the calculation can remain the same.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
χ²=10; df=2. Right-tail p=0.006738. At α=0.05, reject independence. The design determines the population interpretation.
| Cell | Observed O | Expected E | (O−E)²/E |
|---|---|---|---|
| A / Walk | 15 | 10 | 2.5 |
| A / Bus | 10 | 10 | 0 |
| A / Car | 5 | 10 | 2.5 |
| B / Walk | 5 | 10 | 2.5 |
| B / Bus | 10 | 10 | 0 |
| B / Car | 15 | 10 | 2.5 |
Two rows A/B and three travel categories: Walk, Bus, Car. Fixed equal row and column margins; all expected cells equal the chosen count. Independent random sampling and 10% conditions from large populations are assumed. This Fall 2026 CED requires every expected count >5. Each 3D block is one synthetic person.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the probability values, reference groups, graph scales or model assumptions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionOne SRS of city residents records neighborhood and transport mode. State the test, hypotheses and count check.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Use chi-square independence for two categorical variables in one sampled population.
- 1 point: H₀: neighborhood and transport mode are independent in the city population.
- 1 point: Hₐ: they are associated.
- 1 point: Verify randomization, independence/10% as appropriate and every expected cell count >5 under this framework.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What distinguishes homogeneity from independence?
The collection design and population question.
RECALL 2What does homogeneity’s alternative claim?
At least one population/treatment response distribution differs.
RECALL 3Are percentages alone suitable input counts?
No; use the actual observed counts.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Is the question about one population or several?
- One sample, two categorical variables → independence.
- Multiple populations/treatments, one response → homogeneity.
- Use counts and check all expected counts >5 for this CED.
Remember: The data-collection design determines the interpretation; the table’s rectangular shape does not distinguish the tests.
Conditions: Two rows A/B and three travel categories: Walk, Bus, Car. Fixed equal row and column margins; all expected cells equal the chosen count. Independent random sampling and 10% conditions from large populations are assumed. This Fall 2026 CED requires every expected count >5. Each 3D block is one synthetic person.
Refresh Kid · AP Statistics Unit 3 · Objectives 3.14.B, 3.14.C, 3.14.D · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 3.14, objectives 3.14.B, 3.14.C, 3.14.D. Framework effective Fall 2026, checked September 17, 2026. Unit 3 includes inference for one and two population proportions, errors and power, and chi-square homogeneity/independence tests; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Inference requires a justified design and appropriate counts. Intervals use observed proportions; null tests use their reference-model proportions. The revised CED states chi-square expected counts should be greater than 5; this unit follows that wording even though some companion texts use at least 5. Normal and chi-square inference are approximate. The chi-square goodness-of-fit test is not included in this unit’s official scope.
The Organic Chemistry Tutor companion title and destination were located; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original categorical count grid uses self-hosted Three.js with its MIT license. Two rows and three columns organize synthetic people into categorical cells. Each stacked block represents one person. Camera rotation changes only the view; exact counts, expectations and contributions are always available in the 2D table. Inference curves and intervals remain 2D; use the exact table rather than apparent 3D size for comparisons. Complete labeled diagrams, count tables and explanations remain available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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