Analysis of Variance

In the following, analysis of variance (ANOVA or AOV) is illustrated for a case where there are  groups or regions, and , observations in the  group or region.  The model that underlies analysis of variance assumes that each observation has several components

The means are defined as follows

With this model for an individual observation in mind, the total variability of the data can be decomposed into two components:

where  is the “total sum of squares if the data (the sum of the squared deviation of each observation in each group about the grand mean),  is the “among (or between) groups sum of squares” (the sum of the squared deviations of each group mean about the grand mean), and  is the “within-groups sum of squares” (the sum of the deviatins within each group about the group mean).

The among-groups and within-groups sum of squares are adjusted for the appropriate degrees of freedom to produce the following variance-like quantities:

The test statistic, , provides a measure of the size of the among-groups variability relative to the within-groups variability

 may be compared to the  distribution with  degrees of freedom