The arithmetic of ANOVA calculations

In the preceding ANOVA calculation al was estimated in two different ways. If the null hypothesis were true, al could also be estimated in a third way by treating the data as one large sample. This would involve summing the squares of the deviations from the overall mean  

This method of estimating al is not used in the analysis because the estimate depends on both the within- and between-sample variations. However, there is an exact algebraic relationship between this total variation and the sources of variation which contribute to it. This, especially in more complicated ANOVA calculations, leads to a simplification of the arithmetic involved. The relationship between the sources of variation is illustrated by Table 3.4, which summarizes the sums of squares and degrees of freedom. It will be seen that the values for the total variation given in the last row of the table are the sums of the values in the first two rows for both the sum of squares and the degrees of freedom. This additive property holds for all the ANOVA calculations described in this book. 

Just as in the calculation of variance, there are formulae which simplify the calculation of the individual sums of squares. These formulae are summarized below  

One-way ANOVA tests for a significant difference between means when there are more than two samples involved. The formulae used are  

Between-samples Within-samples X n/n- TVN by subtraction h- by subtraction 

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