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Analysis Variance (ANOVA), Degrees of Freedom, Sum of Squares, Calculation…

- - - - the null hypothesis states that the treatment conditions have no effect on performance. That is, the population means for the three conditions are all the same. In general, there is no treatment effect.
    - - the treatment conditions are not all the same; that is, there is a real treatment effect
- - - - Remember that calculating variance is simply a method for measuring how big the differences are for a set of numbers. When you see the term variance, you can automatically translate it into the term differences. Thus, the between-treatments variance simply measures how much difference exists between the treatment conditions. There are two possible explanations for these between-treatment differences:
        
        The differences between treatments are not caused by any treatment effect but are simply the naturally occurring, random and unsystematic differences that exist between one sample and another. That is, the differences are the result of sampling error.
        
        The differences between treatments have been caused by the treatment effects. For example, if treatments really do affect performance, then scores in one treatment should be systematically different from scores in another condition.
    - - Inside each treatment condition, we have a set of individuals who all receive exactly the same treatment; that is, the researcher does not do anything that would cause these individuals to have different scores.the differences that exist within a treatment represent random and unsystematic differences that occur when there are no treatment effects causing the scores to be different. Thus, the within-treatments variance provides a measure of how big the differences are when H_0 is true.
- - - - To make this formula consistent with the ANOVA notation, we substitute the letter G in place of EX and obtain