One-way ANOVA is a statistical test used to compare the means of three or more independent groups. It helps answer whether the differences among sample means are larger than we would expect from random variation alone. This matters because running many separate t-tests increases the chance of a false positive.
ANOVA gives one overall test of whether at least one group mean is different.
Key Facts
- One-way ANOVA tests H0: μ1 = μ2 = μ3 = ... = μk against the claim that at least one mean differs.
- Total variation is split into between-group variation and within-group variation: SST = SSB + SSW.
- Between-group mean square: MSB = SSB / (k - 1), where k is the number of groups.
- Within-group mean square: MSW = SSW / (N - k), where N is the total number of observations.
- The test statistic is F = MSB / MSW.
- A large F value suggests group means differ more than expected from within-group noise, leading to a small p-value.
Vocabulary
- One-way ANOVA
- A hypothesis test that compares the means of three or more independent groups using one categorical factor.
- Between-group variation
- The variation caused by differences between each group mean and the overall mean.
- Within-group variation
- The variation caused by differences between individual data values and their own group mean.
- F statistic
- A ratio that compares between-group variation to within-group variation in an ANOVA test.
- ANOVA table
- A table that organizes sources of variation, sums of squares, degrees of freedom, mean squares, the F statistic, and the p-value.
Common Mistakes to Avoid
- Running many t-tests instead of ANOVA, because this increases the probability of making a Type I error when comparing several groups.
- Concluding that all group means are different after a significant ANOVA, because ANOVA only shows that at least one mean differs and follow-up tests are needed.
- Using one-way ANOVA for paired or repeated measurements, because the standard one-way ANOVA assumes independent observations.
- Ignoring the within-group spread, because group means that look different may not be statistically different if the data inside groups vary a lot.
Practice Questions
- 1 Three teaching methods have sample means of 78, 84, and 90 with equal sample sizes of 10. If SSB = 720 and SSW = 1620, compute MSB, MSW, and F.
- 2 An ANOVA has k = 4 groups and N = 32 total observations. If SSB = 150 and SSW = 560, find the between-group degrees of freedom, within-group degrees of freedom, MSB, MSW, and F.
- 3 A researcher compares four fertilizers and gets a large F statistic with p = 0.003. Explain what this result says about the fertilizer means and why it does not identify which specific fertilizers differ.