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Unformatted text preview: erences from all the raw scores to the grand mean. Each group has ni raw scores, so the total number of raw scores is Σni. Because the grand mean is part of the formula for SStotal, SStotal is not really a sum of Σni squares (even though it appears to be); it’s actually a sum of Σni – 1 squares. This is because of the algebraic magic of degrees of freedom—if we wrote out the grand mean in terms of the raw scores and then rearranged the formula for SStotal, one of the summands would disappear. df total = " n i # 1 The variability explainable by the group differences, SStreatment, is based on the differences 2 between the group means and the grand mean. Even though we add up ( M i " M ) for all !
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the subjects (so that it appears there are Σni squares being added), ( M i " M ) is the same for all the subjects in each group, so there are only k independent squares in the formula for SStreatment (one for each group). Moreover, because the grand mean is part of the !
formula, one of those squares disappears, so the final degrees of freedom is k – 1. Another !
way to see this is to realize that SStreatment is closely related to the variance of the group means (see the section on Relationship between two views of ANOVA). Since there are k group means, their variance has k – 1 degrees of freedom. df treatment = k " 1 The residual variability, SSresidual, is based on the differences between all raw scores and their respective group means. Once again, we appear to be adding Σni squares because !
that’s how many raw scores there are. However, because all the group means are part of the formula (and the group means depend on the raw scores themselves), one degree of...
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This document was uploaded on 02/25/2014 for the course PSYC 3101 at Colorado.
 Spring '08
 MARTICHUSKI
 Psychology

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