The Relationship Between SSW, SSB, SST

Introduces the fundamental ANOVA identity SST = SSB + SSW. Students learn to compute SST directly from data, to verify the decomposition, and to recover any one of the three sums of squares from the other two.

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Tutorial

The ANOVA Identity

In a one-factor study, the total variability of the observations splits exactly into the variability between groups and the variability within groups. This is the fundamental identity of ANOVA:

\text{SST} = \text{SSB} + \text{SSW}

Recall the meaning of each term:

SST (total sum of squares) measures how all observations vary around the grand mean.
SSB (between-groups sum of squares) measures how the group means vary around the grand mean.
SSW (within-groups sum of squares) measures how observations vary around their own group's mean.

Every observation's squared deviation from the grand mean breaks cleanly into a between-group piece and a within-group piece — no leftover, no overlap.

For instance, if a one-factor study produces $\text{SSB} = 18$ and $\text{SSW} = 42,$ then

\text{SST} = 18 + 42 = 60.