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Analysis of Variance (ANOVA)



The reason for doing an ANOVA is to see if there is any difference between groups on some variable.

For example, you might have data on student performance in non-assessed tutorial exercises as well as their final grading. You are interested in seeing if tutorial performance is related to final grade. ANOVA allows you to break up the group according to the grade and then see if performance is different across these grades.

ANOVA is available for both parametric (score data) and non-parametric (ranking/ordering) data.


Types of ANOVA

One-way between groups

The example given above is called a one-way between groups model.

You are looking at the differences between the groups.

There is only one grouping (final grade) which you are using to define the groups.

This is the simplest version of ANOVA.

This type of ANOVA can also be used to compare variables between different groups - tutorial performance from different intakes.

One-way repeated measures

A one way repeated measures ANOVA is used when you have a single group on which you have measured something a few times.

For example, you may have a test of understanding of Classes. You give this test at the beginning of the topic, at the end of the topic and then at the end of the subject.

You would use a one-way repeated measures ANOVA to see if student performance on the test changed over time.

Two-way between groups

A two-way between groups ANOVA is used to look at complex groupings.

For example, the grades by tutorial analysis could be extended to see if overseas students performed differently to local students. What you would have from this form of ANOVA is:

The effect of final grade

The effect of overseas versus local

The interaction between final grade and overseas/local

Each of the main effects are one-way tests. The interaction effect is simply asking "is there any significant difference in performance when you take final grade and overseas/local acting together".

Two-way repeated measures

This version of ANOVA simple uses the repeated measures structure and includes an interaction effect.

In the example given for one-way between groups, you could add Gender and see if there was any joint effect of gender and time of testing - i.e. do males and females differ in the amount they remember/absorb over time.


Non-parametric and Parametric

ANOVA is available for score or interval data as parametric ANOVA. This is the type of ANOVA you do from the standard menu options in a statistical package.

The non-parametric version is usually found under the heading "Nonparametric test". It is used when you have rank or ordered data.

You cannot use parametric ANOVA when you data is below interval measurement.

Where you have categorical data you do not have an ANOVA method - you would have to use Chi-square which is about interaction rather than about differences between groups.

How its done

What ANOVA looks at is the way groups differ internally versus what the difference is between them. To take the above example:

  1. ANOVA calculates the mean for each of the final grading groups (HD, D, Cr, P, N) on the tutorial exercise figure - the Group Means.
  2. It calculates the mean for all the groups combined - the Overall Mean.
  3. Then it calculates, within each group, the total deviation of each individual's score from the Group Mean - Within Group Variation.
  4. Next, it calculates the deviation of each Group Mean from the Overall Mean - Between Group Variation.
  5. Finally, ANOVA produces the F statistic which is the ratio 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 package will tell you if the F ratio is significant or not.

All versions of ANOVA follow these basic principles but the sources of Variation get more complex as the number of groups and the interaction effects increase.



It is unlikely that you would do an analysis of variance by hand. Except for small data sets it is very time consuming.


Available Software


The ANOVA routines in SPSS are OK for simple one-way analyses. Anything more complicated gets difficult.

All statistical packages (SAS, Minitab etc.) provide for ANOVA.


Excel allows you to ANOVA from the Data Analysis Add-on. The instructions are not good.