# Analysis of Variance (ANOVA): Definition and Examples

Analysis of Variance, or ANOVA for short, is a statistical test that looks for significant differences between means on a particular measure. 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.

### Key Takeaways: Analysis of Variance (ANOVA)

• Researchers conduct an ANOVA when they are interested in determining whether two groups differ significantly on a particular measure or test.
• There are four basic types of ANOVA models: one-way between groups, one-way repeated measures, two-way between groups, and two-way repeated measures.
• Statistical software programs can be used to make conducting an ANOVA easier and more efficient.

## ANOVA Models

There are four types of basic ANOVA models (although it is also possible to conduct more complex ANOVA tests as well). 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. The example above, of education level among different sports teams, would be an example of this type of model. It is called a one-way ANOVA because there is only one variable (type of sport played) that is being used to divide participants into different groups.

## One-way repeated measures ANOVA

If you are interested in assessing a single group at more than one time point, you should use a one-way repeated measures ANOVA. 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. Conducting a one-way repeated measures ANOVA would allow you to find out whether the students’ test scores changed significantly from the beginning to the end of the course.

## Two-way between groups ANOVA

Imagine now that you have two different ways in which you want to group your participants (or, in statistical terms, you have two different independent variables). For example, imagine you were interested in testing whether test scores differed between student athletes and non-athletes, as well as for freshmen versus seniors. In this case, you would conduct a two-way between groups ANOVA. You would have three effects from this ANOVA—two main effects and an interaction effect. The main effects are the effect of being an athlete and the effect of class year. The interaction effect looks at the impact of both being an athlete and class year. Each of the main effects is a one-way test. The interaction effect is simply asking if the two main effects impact each other: for example, if student athletes scored differently than non-athletes did, but this was only the case when studying freshmen, there would be an interaction between class year and being an athlete.

## Two-way repeated measures ANOVA

If you want to look at how different groups change across time, you can use a two-way repeated measures ANOVA. Imagine you’re interested in looking at how test scores change across time (as in the example above for a one-way repeated measures ANOVA). However, this time you’re also interested in assessing gender as well. For example, do males and females improve their test scores at the same rate, or is there a gender difference? A two-way repeated measures ANOVA can be used to answer these types of questions.

## 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

1. 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.
2. The overall mean is then calculated for all of the groups combined.
3. Within each group, the total deviation of each individual’s score from the group mean is calculated. This tells us whether the individuals in the group tend to have similar scores or whether there is a lot of variability between different people in the same group. Statisticians call this within group variation.
4. Next, how much each group mean deviates from the overall mean is calculated. This is called between group variation.
5. Finally, an F statistic is calculated, which is the ratio of between group variation to the within group variation.

If there is significantly greater between group variation than within group variation (in other words, when the F statistic is larger), then it is likely that the difference between the groups is statistically significant. Statistical software can be used to calculate the F statistic and determine whether it is significant or not.

All types of ANOVA follow the basic principles outlined above. However, as the number of groups and the interaction effects increase, the sources of variation will become more complex.

## Performing an ANOVA

Because conducting an ANOVA by hand is a time-consuming process, most researchers use statistical software programs when they are interested in conducting an ANOVA. SPSS can be used to conduct ANOVAs, as can R, a free software program. In Excel, you can do an ANOVA by using the Data Analysis Add-on. SAS, STATA, Minitab, and other statistical software programs that are equipped for handling bigger and more complex data sets can also be used to perform an ANOVA.

## References

Monash University. Analysis of Variance (ANOVA). http://www.csse.monash.edu.au/~smarkham/resources/anova.htm

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