About the ANOVA Test

Another e-lesson on the t-test demonstrated how to compare differences of means between two groups, such as comparing outcomes between control and treatment groups in an experimental study. The t-test is a useful tool for comparing the means of two groups; however, the t-test is not good in situations calling for the comparison of three or more groups. It can only compare one group's mean to a known distribution or compare the means of two groups. With three or more groups, the t-test is not an effective statistical tool. On a practical level, using the t-test to compare many means is a cumbersome process in terms of the calculations involved. On a statistical level, using the t-test to compare multiple means can lead to biased results.

Yet there are many kinds of questions in which we might want to compare the means of several different groups at once. For example, in evaluating the effects of a particular social program, we might want to compare the mean outcomes of several different program sites. Or we might be interested in examining the relative performance of different members of a corporate sales team in terms of their monthly or annual sales records. Alternatively, in an organization with several different sales managers, we might ask whether some sales managers get more out of their sales staff than others.

With questions such as these, the preferred statistical tool is the ANOVA, (Analysis Of Variance. There are some similarities between the t-test and ANOVA. Like the t-test, ANOVA is used to test hypotheses about differences in the average values of some outcome between two groups; however, while the t-test can be used to compare two means or one mean against a known distribution, ANOVA can be used to examine differences among the means of several different groups at once. More generally, ANOVA is a statistical technique for assessing how nominal independent variables influence a continuous dependent variable.

This module describes and explains the one-way ANOVA, a statistical tool that is used to compare multiple groups of observations, all of which are independent but may have a different mean for each group. A test of importance for many kinds of questions is whether or not all the averages of a set of groups are equal. There is another form of ANOVA that examines how two explanatory variables affect an outcome variable; however, this application is not discussed in this module.


Analysis of Variance methods have in common a set of two assumptions:

  1. The standard deviations (SD) of the populations for all groups are equal - this is sometimes referred to as an assumption of the homogeneity of variance. Again, we can represent this assumption for groups 1 through n as
  2. Mathematical Markup

  3. The samples are randomly selected from the population.