Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Section 13.1
Chapter Summary
Chapter 11 compared two means; this chapter compares three or more at once. The naive approach — running a t-test on every pair — is a trap: with 5 groups there are 10 pairwise tests, and each carries a 5% false-alarm rate, so the chance of at least one false “significant” finding balloons. One-way ANOVA (analysis of variance) solves this with a single test: H₀ says all population means are equal; H₁ says at least one differs.
The clever idea is in the name: analysis of variance. ANOVA compares two sources of variation — how much the group means differ from each other (between-groups variation) versus how much individuals differ within their own groups (within-groups variation). If the groups’ means are spread out far more than individuals vary inside groups, that’s evidence the populations really differ. The ratio of these, scaled by degrees of freedom, is the F-statistic; a large F (tiny P-value) rejects H₀. Realistic example: three teaching methods produce class averages of 78, 81, and 79 — but with within-class SD around 10, ANOVA gives P-value ≈ 0.55: the means differ less than students vary within each class, so there’s no significant evidence any method is better. Requirements matter here: roughly normal populations and equal variances across groups (check with boxplots — similar spreads and no extreme skew). And note the limit: rejecting H₀ only says at least one mean differs, not which ones — identifying the specific pairs needs follow-up (post-hoc) tests.
Study Guide
Key terms and definitions
- One-way ANOVA — tests equality of three or more population means using a single F-test. H₀: μ₁ = μ₂ = … = μ_k; H₁: at least one mean differs.
- F-statistic — F₀ = (mean square between) ÷ (mean square within); large values argue against H₀.
- Mean square between (MSB) — variation among the group means, scaled by df = k − 1.
- Mean square within (MSW, or MSE) — variation of individuals around their own group mean, scaled by df = N − k (N = total observations).
- Requirements: independent random samples; each population approximately normal; equal population variances (check with side-by-side boxplots — similar spreads).
- Post-hoc tests — follow-up pairwise comparisons (e.g., Tukey’s) used only after ANOVA rejects H₀, to find which means differ.
Essential formulas
- Sums of squares: SSTotal = SSBetween + SSWithin.
- Mean squares: MSB = SSBetween ÷ (k − 1); MSW = SSWithin ÷ (N − k).
- F-statistic: F₀ = MSB ÷ MSW, with df₁ = k − 1 and df₂ = N − k.
- P-value: area to the right of F₀ under the F-distribution. Reject H₀ if P-value ≤ α.
Step-by-step procedures
- Run a one-way ANOVA: (1) state H₀ (all means equal) and H₁; (2) check requirements — side-by-side boxplots for approximate normality and similar spreads; (3) compute (or let StatCrunch compute) the ANOVA table: SS, df, MS, F₀; (4) find the P-value from the F-distribution; (5) conclude: if significant, “at least one population mean differs” — then consider post-hoc tests to find which.
- Read an ANOVA table: locate the F-statistic and its P-value; the “Between” row is the group effect, the “Within/Error” row is background noise.
Common mistakes to avoid
- Running many pairwise t-tests instead of ANOVA — this inflates the overall Type I error rate.
- Concluding “all means differ” from a significant ANOVA — it only shows at least one differs.
- Ignoring the equal-variance requirement — wildly different spreads across groups invalidate the F-test.
- Forgetting that ANOVA needs quantitative response data; for categorical outcomes, use chi-square (Chapter 12).
Self-check: I can…
- [ ] Explain why multiple t-tests are a bad substitute for ANOVA.
- [ ] State the hypotheses and requirements for one-way ANOVA.
- [ ] Explain the logic: between-groups vs. within-groups variation.
- [ ] Read an ANOVA table (F-statistic, P-value) and state a conclusion.
- [ ] Explain what a significant ANOVA does and doesn’t tell you.
StatCrunch How-To
- One-way ANOVA: Stat > ANOVA > One Way — if each group’s data is in its own column, select the columns; if the data are stacked with a group label column, choose “Responses in one column, factor in another” and select both > Compute.
- Check requirements first: Graph > Boxplot, grouped by the factor column — look for roughly symmetric boxes and similar spreads across groups.
- Read the output: the ANOVA table gives DF, SS, MS, F-Stat, and P-value — the P-value is what you compare to α.
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