${\displaystyle \mathbb {E} (\mathbb {E} (Y|X))=\mathbb {E} (Y)}$

${\displaystyle b_{0}={\overline {y}}-b_{1}{\overline {x}}}$

${\displaystyle b_{1}=r\cdot {\frac {s_{y}}{s_{x}}}}$

Degrees of freedom: ${\displaystyle (I-1)(J-1)}$

SS-Interaction: ${\displaystyle K\sum _{i=1}^{I}\sum _{j=1}^{J}({\overline {y}}_{ij\cdot }-{\overline {y}}_{i\cdot \cdot }-{\overline {y}}_{\cdot j\cdot }+{\overline {y}}_{\cdot \cdot \cdot })^{2}}$

Degrees of freedom: ${\displaystyle n_{\cdot }-IJ}$

SS-Within: ${\displaystyle \sum _{i=1}^{I}\sum _{j=1}^{J}\sum _{k=1}^{K}(y_{ijk}-{\overline {y}}_{ij\cdot })^{2}}$

${\displaystyle s_{\mathrm {pooled} }={\sqrt {\frac {\sum _{i=1}^{I}(n_{i}-1)s_{i}^{2}}{n_{\cdot }-I}}}={\sqrt {\frac {(4-1)3.559^{2}+(5-1)3.240^{2}+(4-1)3.464^{2}}{13-3}}}={\sqrt {11.59877}}\approx 3.406}$