Contingency tables and association
Big idea: A contingency table counts two categorical variables together.
We test whether variables are associated or independent.
Example: transport mode (bus/car/walk) vs punctuality (on-time/late).
| Concept | Meaning |
|---|---|
| Observed frequency | Actual count in each cell |
| Expected frequency | Count expected if variables are independent |
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How to find expected frequencies
Worked example
Table has row total 40, column total 30, grand total 120.
Find expected frequency for that cell.
Step by step
- Write formula first
- Substitute and simplify
- E=10
Final answer
Expected frequency is 10.
Mark-saving habit: Show formula before substitution to secure method credit.
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Chi-squared test statistic
Decision idea: Large chi-squared means observed counts differ from expected counts more than random chance would suggest.
Worked example
One cell has O=14, E=10.
Contribution to chi-squared?
Step by step
- Use formula for one cell
- Compute numerator: 16
- Contribution = 1.6
Final answer
This cell contributes 1.6 to total chi-squared.
Conditions and exam traps
Common mistakes
- Using percentages instead of counts
- Forgetting expected frequencies
- Not checking expected>=5
Correct method
- Use observed counts
- Compute every expected cell
- State conditions before conclusion
Exam Tips:
- State hypotheses in context.
- Use counts, not percentages.
- Write conclusion in words about association.