ANOVA Identity
The ANOVA identity decomposes each total deviation into a treatment part and a within-treatment part:
$$ y_{ij} - \bar{y}_{\cdot\cdot}
\left(\bar{y}{i\cdot} - \bar{y}{\cdot\cdot}\right) + \left(y_{ij} - \bar{y}_{i\cdot}\right). $$
Squaring and summing over all observations leads to the variance decomposition
$$ SST = SSTreat + SSE. $$This is the core identity behind one-way ANOVA.