SECTION 14.1 Summary

• One-way analysis of variance (ANOVA) is used to compare several population means based on independent SRSs from each population. We assume that the populations are Normal and that, although they may have different means, they have the same standard deviation.
• To do an analysis of variance, first examine the data. Side-by-side boxplots give an overview. Examine Normal quantile plots (either for each group separately or for the residuals) to detect outliers or extreme deviations from Normality.
• In addition to Normality, the ANOVA model assumes equal population standard deviations. Compute the ratio of the largest to the smallest sample standard deviation. If this ratio is less than two and the Normality assumption is reasonable, ANOVA can be performed.
• If the data do not support equal standard deviations, consider transforming the response variable. This often makes the group standard deviations more nearly equal and makes the group distributions more Normal.
• The null hypothesis is that the population means are all equal. The alternative hypothesis is true if there are any differences among the population means.
• ANOVA is based on separating the total variation observed in the data into two parts: variation among group means and variation within groups. If the variation among groups is large relative to the variation within groups, we have evidence against the null hypothesis.
• An analysis of variance table organizes the ANOVA calculations. Degrees of freedom, sums of squares, and mean squares appear in the table. The statistic and its -value are used to test the null hypothesis.
• The ANOVA test shares the robustness of the two-sample test. It is relatively insensitive to moderate non-Normality and unequal variances, especially when the sample sizes are similar.