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Analysis of Variance (ANOVA)
Purpose 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 nonassessed 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 nonparametric (ranking/ordering) data.
Types of ANOVA Oneway between groups The example given above is called a oneway 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. Oneway 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 oneway repeated measures ANOVA to see if student performance on the test changed over time. Twoway between groups A twoway 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 oneway tests. The interaction effect is simply asking "is there any significant difference in performance when you take final grade and overseas/local acting together".
Twoway repeated measures This version of ANOVA simple uses the repeated measures structure and includes an interaction effect. In the example given for oneway 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.
Nonparametric 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 nonparametric 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 Chisquare 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:
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.
Comments 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 SPSS The ANOVA routines in SPSS are OK for simple oneway analyses. Anything more complicated gets difficult. All statistical packages (SAS, Minitab etc.) provide for ANOVA. Excel Excel allows you to ANOVA from the Data Analysis Addon. The instructions are not good. 