The main purpose of analysis of variance is to test the: 

The correct answer is homogeneity of means for several populations.  Key Points

• The main purpose of analysis of variance (ANOVA) is indeed to test the homogeneity of means for several populations or groups. ANOVA is a statistical technique that allows us to compare the means of two or more groups to determine if they are significantly different from each other. It helps in understanding whether there are any significant differences in the population means across the different groups or treatments being compared.
• The null hypothesis in ANOVA assumes that the means of all the populations or groups are equal. The alternative hypothesis suggests that at least one of the group means is different from the others. By analyzing the variability within and between the groups, ANOVA provides a statistical test to assess if the observed differences in the means are likely due to random variation or if they represent true differences.

In summary, ANOVA is specifically designed to determine if there are significant differences in means across multiple populations or groups, thereby assessing the homogeneity or equality of means.