In the Randomized block design, all of the sample sizes are the same \(b\), which is the blocks, the mathematical model associated with \(Y_{ij}\) is :\(Y_{ij}=\mu_j+\beta_i+\epsilon_{ij}\), the term \(\beta_i\) represents the effect of the \(i^{th}\) block. \[ \begin{array}{|cc|cccc ccc|} \hline &&&\text{treatment}&\text{levels}& & & Block&Block& Block\\ & & 1 & 2 & \cdots & k && Totals & Means & Effects \\ \hline &1& Y_{11} & Y_{12} & \cdots & Y_{1k} && T_{1.} & \overline Y_{1.}&\beta_1\\ &2& Y_{21} & Y_{22} & \cdots & Y_{2k} && T_{2.} & \overline Y_{2.}&\beta_2\\ &\vdots&\vdots &\vdots &\cdots&\vdots && \vdots & \vdots & \vdots\\ &b&Y_{b1} &Y_{b2} &\cdots&Y_{bk} && T_{b.} & \overline Y_{b.}&\beta_b\\ \text{Sample totals:}&&T_{. 1}&T_{. 2}&\cdots&T_{. k} & &T_{. .}\\ \text{Sample means:}&&\overline Y_{. 1}&\overline Y_{. 2}&\cdots&\overline Y_{. k} & & &\overline Y_{. .}\\ \text{True means:}&&\mu_1&\mu_2&\cdots&\mu_k\\ \hline \end{array} \]
The estimate for \(\beta_i\) is \(E(\beta_i)=E(Y_{ij}-\mu_j-\epsilon_{ij})=\overline Y_{i.}-\mu=\overline Y_{i.}-\overline Y_{..}\). The expression for \(SSE\) is:
\[\begin{align}
SSE&=\sum_{i=1}^{b}\sum_{j=1}^{k}(Y_{ij}-\overline Y_{.j})^2\\
&=\sum_{i=1}^{b}\sum_{j=1}^{k}[(Y_{ij}-\overline Y_{.j})+(\overline Y_{i.}-\overline Y_{..})-(\overline Y_{i.}-\overline Y_{..})]^2\\
&=\sum_{i=1}^{b}\sum_{j=1}^{k}[(\overline Y_{i.}-\overline Y_{..})+(Y_{ij}-\overline Y_{.j}-\overline Y_{i.}+\overline Y_{..})]^2\\
&=\sum_{i=1}^{b}\sum_{j=1}^{k}(\overline Y_{i.}-\overline Y_{..})^2+2\sum_{i=1}^{b}\sum_{j=1}^{k}(\overline Y_{i.}-\overline Y_{..})(Y_{ij}-\overline Y_{.j}-\overline Y_{i.}+\overline Y_{..})+\sum_{i=1}^{b}\sum_{j=1}^{k}(Y_{ij}-\overline Y_{.j}-\overline Y_{i.}+\overline Y_{..})^2\\
&=\sum_{i=1}^{b}\sum_{j=1}^{k}(\overline Y_{i.}-\overline Y_{..})^2+\sum_{i=1}^{b}\sum_{j=1}^{k}(Y_{ij}-\overline Y_{.j}-\overline Y_{i.}+\overline Y_{..})^2
\end{align}\]
The first term \(\displaystyle\sum_{i=1}^{b}\sum_{j=1}^{k}(\overline Y_{i.}-\overline Y_{..})^2\) is called the sum of squares of block (\(SSB\)), and the second term \(\displaystyle\sum_{i=1}^{b}\sum_{j=1}^{k}(Y_{ij}-\overline Y_{.j}-\overline Y_{i.}+\overline Y_{..})^2\) is the new sum of squares of random error (\(SSE\)).
The sum of squares of total (\(SSTOT\)) is \(\displaystyle\sum_{i=1}^{b}\sum_{j=1}^{k}(Y_{ij}-\overline Y_{..})^2\).
The sum of squares of treatment (\(SSTR\)) is \(\displaystyle\sum_{i=1}^{b}\sum_{j=1}^{k}(\overline Y_{.j}-\overline Y_{..})^2\).
Then \(SSTOT=SSTR+SSB+SSE\)