χ² goodness-of-fit and independence tests
Compare observed counts with expected frequencies using the right degrees of freedom and defensible class sizes.
Section Paper 4 · Further Probability & Statistics · Topics 4.1–4.5
Exam scope and assessment notes
Mapped to Cambridge International AS & A Level Further Mathematics 9231. These original explanations and self-checks are independent study support, not Cambridge questions, marks or grades. The syllabus assumes prior knowledge of the relevant 9709 Mathematics content; check each component's prerequisites with your teacher. Full-paper assessment and human subject-specialist review remain outstanding.
Before you begin
- Cambridge expects learners to know the whole Cambridge International A Level Mathematics (9709) content. Paper 1 further pure mathematics draws on 9709 Papers 1 and 3. Confirm your exam year and component route with your school.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What makes a Further Mathematics solution convincing?
What you will learn
- Fit a theoretical distribution under a stated hypothesis
- Carry out a χ² goodness-of-fit test
- Test independence in a contingency table and combine small expected cells
Goodness-of-fit compares observed and expected counts
Under H₀, calculate each expected frequency Eᵢ from the proposed distribution. Use χ²=Σ(Oᵢ−Eᵢ)²/Eᵢ; combine adjacent classes when an expected count is below 5.
For k categories and p estimated parameters, degrees of freedom are k−1−p. If parameters are fully specified in advance, p=0. A large statistic indicates greater disagreement with H₀.
Independence uses a contingency table
For a cell, E=(row total×column total)/grand total. Sum the χ² contributions over all cells; degrees of freedom are (rows−1)(columns−1). Combine rows or columns when appropriate so expected frequencies meet the syllabus condition.
State the conclusion in context and distinguish evidence of association from a causal claim. Yates’ correction is not required for this syllabus.
Pause and explain
Which habit best checks your work on χ² goodness-of-fit and independence tests?
Worked example
A die is rolled 60 times with observed counts 12, 9, 8, 11, 10, 10. Test goodness of fit to a fair die by calculating χ².
Show the worked solution
- Under a fair-die model, each expected count is 60/6=10.
- χ²=Σ(O−E)²/E=(4+1+4+1+0+0)/10=1.
- No parameters were estimated, so degrees of freedom are 6−1=5.
Answer χ²=1 with 5 degrees of freedom; compare with the chosen critical value.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A model predicts category probabilities 0.2, 0.3 and 0.5 in 100 trials. The observed counts are 18, 32 and 50. Calculate χ².
Give me a hint
Find expected counts first, then sum each squared difference divided by its expectation.
Compare my reasoning
- Expected counts are 20, 30 and 50.
- Contributions are 4/20, 4/30 and 0/50.
- Add the contributions.
χ²=0.2+0.1333…≈0.333.
Look for these in your work
- Expected frequencies correct
- Each contribution uses expected denominator
- Total calculated
A contingency table has 3 rows and 2 columns. Find its degrees of freedom for a χ² independence test.
Give me a hint
Use (rows−1)(columns−1).
Compare my reasoning
- There are 3−1=2 independent row contrasts.
- There are 2−1=1 independent column contrast.
- Multiply the two values.
df=2.
Look for these in your work
- Correct formula
- Row and column factors
- Product stated
One expected cell frequency is 3 in a goodness-of-fit test. What should you check before calculating the final statistic, and how does combining categories affect degrees of freedom?
Give me a hint
The syllabus requires expected frequencies of at least 5 in each retained class.
Compare my reasoning
- Combine adjacent, substantively compatible classes where appropriate.
- Recalculate the combined observed and expected totals.
- Use the resulting number of classes, minus one and any estimated parameters, for degrees of freedom.
Combine suitable adjacent classes so each expected count is at least 5, then recalculate the statistic and degrees of freedom.
Look for these in your work
- Expected-count condition applied
- Counts recomputed
- Degrees of freedom updated
Common mistakes
- Using observed rather than expected counts in the denominator
- Forgetting parameters estimated from the data when finding degrees of freedom
- Claiming that an association test proves causation
What can you explain now?
In a goodness-of-fit χ² statistic, what is the denominator for a category's contribution?
Compare with the explanation
Its expected frequency E
The contribution is (O−E)²/E.
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Make it stick
How do expected frequencies, parameter estimation and table dimensions determine a χ² test?
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