Analysis of Variance (ANOVA)

Ultimate Frisbee
( CC BY 2.0) by CasparGirl

Analysis of Variance, or ANOVA for short, is a statistical test that looks for significant differences between means. For example, say you are interested in studying the education level of athletes in a community, so you survey people on various teams. You start to wonder, however, if the education level is different among the different teams. You could use an ANOVA to determine if the mean education level is different among the softball team versus the rugby team versus the Ultimate Frisbee team.

ANOVA Models

There are four types of ANOVA models. Following are descriptions and examples of each.

One-way between groups ANOVA

A one-way between groups ANOVA is used when you want to test the difference between two or more groups. This is the simplest version of ANOVA. The example of education level among different sports teams above would be an example of this type of model. There is only one grouping (type of sport played) that you are using to define the groups.

One-way repeated measures ANOVA

A one-way repeated measures ANOVA is used when you have a single group on which you have measured something more than one time. For example, if you wanted to test students’ understanding of a subject, you could administer the same test at the beginning of the course, in the middle of the course, and at the end of the course. You would then use a one-way repeated measures ANOVA to see if students’ performance on the test changed over time.

Two-way between groups ANOVA

A two-way between groups ANOVA is used to look at complex groupings. For example, the students’ grades in the previous example could be extended to see if students abroad performed differently to local students. So you would have three effects from this ANOVA: the effect of the final grade, the effect of abroad versus local, and the interaction between the final grade and overseas/local. Each of the main effects is a one-way test. The interaction effect is simply asking if there is any significant difference in performance when you test the final grade and overseas/local acting together.

Two-way repeated measures ANOVA

Two-way repeated measures ANOVA uses the repeated measures structure but also includes an interaction effect. Using the same example of one-way repeated measures (test grades before and after a course), you could add gender to see if there is any joint effect of gender and time of testing. That is, do males and females differ in the amount of information they remember over time?

Assumptions of ANOVA

The following assumptions exist when you perform an analysis of variance:

  • The expected values of the errors are zero.
  • The variances of all errors are equal to each other.
  • The errors are independent from one another.
  • The errors are normally distributed.

How an ANOVA is Done

  • The mean is calculated for each of your groups. Using the example of education and sports teams from the introduction in the first paragraph above, the mean education level is calculated for each sports team.
  • The overall mean is then calculated for all of the groups combined.
  • Within each group, the total deviation of each individual’s score from the group mean is calculated. This is called within group variation.
  • Next, the deviation of each group mean from the overall mean is calculated. This is call between group variation.
  • Finally, an F statistic is calculated, which is the ratio of between group variation to the within group variation.

If the between group variation is significantly greater than the within group variation, then it is likely that there is a statistically significant difference between the groups. The statistical software that you use will tell you if the F statistic is significant or not.

All versions of ANOVA follow the basic principles outlined above, but as the number of groups and the interaction effects increase, the sources of variation will get more complex.

Performing an ANOVA

It is very unlikely that you would do an ANOVA by hand. Unless you have a very small data set, the process would be very time-consuming. All statistical software programs provide for ANOVA. SPSS is okay for simple one-way analyses, however, anything more complicated becomes difficult. Excel also allows you to do ANOVA from the Data Analysis Add-on, however the instructions are not very good. SAS, STATA, Minitab, and other statistical software programs that are equipped for handling bigger and more complex data sets are all better for performing an ANOVA.


Monash University. Analysis of Variance (ANOVA).