Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A chi-square test compares observed counts with the expected counts you would see if H₀ were true.
- The calculations are identical for the test for homogeneity (one variable, several populations or treatments) and the test for independence (two variables, one population). Only the hypotheses, the randomization condition and the conclusion wording differ.
- Large χ² means the data are far from H₀, so only the right tail counts as evidence against H₀.
- Interpret the p-value assuming H₀ is true, in context. Decide by comparing the p-value with α.
Key relationships
| Quantity | How to find it | Notes |
|---|---|---|
| Expected count | (row total × column total) ÷ table total | Keep decimals; rows and columns add to the observed totals |
| Component | (observed − expected)² ÷ expected | Never negative; one per cell |
| χ² statistic | sum of the components over every cell | Use counts, not percentages; leave out totals |
| Degrees of freedom | (rows − 1)(columns − 1) | Category rows and columns only |
| p-value | P(χ² ≥ observed value) for that df | Table gives a range; technology gives a value |
| Decision | p-value ≤ α: reject H₀; p-value > α: fail to reject H₀ | Quote both numbers |
Assumptions and conditions
- Randomization: independent random samples or a randomized experiment (homogeneity); one random sample (independence).
- 10% condition: n ≤ 10% of N for each sample taken without replacement; not needed for a randomized experiment.
- Expected counts: all expected counts greater than 5.
- When H₀ is true and these hold, χ² follows approximately a chi-square distribution with (r − 1)(c − 1) degrees of freedom.
Conclusion templates
- Homogeneity: “Because the p-value of … is less than α = …, we reject H₀. There is convincing statistical evidence that the distribution of [variable] differs across [populations].”
- Independence: “Because the p-value of … is greater than α = …, we fail to reject H₀. There is not convincing statistical evidence of an association between [variable 1] and [variable 2] in [population].”
Mistakes to avoid
- Percentages in the formula instead of counts.
- Rounding expected counts to whole numbers.
- df = r × c or (r × c) − 1 instead of (r − 1)(c − 1).
- Left tail or doubled p-value. It is always the right tail.
- Checking observed counts for the “greater than 5” condition instead of expected counts.
- “Accept H₀” or “this proves the variables are independent”.
- A conclusion with no population, or homogeneity wording for an independence test.
Quick self-check
- A two-way table has 150 individuals. A cell’s row total is 60 and its column total is 45. What is its expected count? (60 × 45 ÷ 150 = 18)
- What are the degrees of freedom for a table with 3 rows and 4 columns of categories? ((3 − 1)(4 − 1) = 6)
- A cell has observed count 12 and expected count 20. What is its component of χ²? ((12 − 20)² ÷ 20 = 3.2)
Next: practice questions.