水準Lと繰返しRが有る標本平均周りの平方和の式展開

水準と繰返しが有る標本平均周りの平方和の式展開(分解あり)

L:水準, R:繰返し $$\begin{array}{rclrc}\sum _{n=1}^N(x_n-\overline{x}{)}^2& =& \sum _{l=1}^L\sum _{r=1}^R(x_{lr}-\overline{x}{)}^2& & \dots L:\mathrm{水}\mathrm{準},R:\mathrm{繰}\mathrm{返}\mathrm{し},N=LR\\ & =& \sum _{l=1}^L\sum _{r=1}^R(\overline{x_l}-\overline{x}{)}^2& +\sum _{l=1}^L\sum _{r=1}^R(x_{lr}-\overline{x_l}{)}^2& \dots \sum _{l=1}^L\sum _{r=1}^R(\overline{x_l}-\overline{x}{)}^2:\mathrm{各}\mathrm{水}\mathrm{準}\mathrm{平}\mathrm{均}\mathrm{と}\mathrm{全}\mathrm{体}\mathrm{平}\mathrm{均}\mathrm{の}\mathrm{差}\mathrm{の}\mathrm{平}\mathrm{方}\mathrm{和},\sum _{l=1}^L\sum _{r=1}^R(x_{lr}-\overline{x_l}{)}^2:\mathrm{各}\mathrm{値}\mathrm{と}\mathrm{各}\mathrm{水}\mathrm{準}\mathrm{と}\mathrm{の}\mathrm{差}\mathrm{の}\mathrm{平}\mathrm{方}\mathrm{和}\\ & =& R\sum _{l=1}^L({\overline{x_l}}^2-2\overline{x_l}\overline{x}+{\overline{x}}^2)& +\sum _{l=1}^L\sum _{r=1}^R(x_{lr}^2-2x_{lr}\overline{x_l}+{\overline{x_l}}^2)& \dots (a-b{)}^2=a^2-2ab+b^2\\ & =& R\sum _{l=1}^L{\overline{x_l}}^2-R\sum _{l=1}^{L}2\overline{x_l}\overline{x}+R\sum _{l=1}^L{\overline{x}}^2& +\sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-\sum _{l=1}^L\sum _{r=1}^{R}2x_{lr}\overline{x_l}+\sum _{l=1}^L\sum _{r=1}^R{\overline{x_l}}^2& \dots \sum _{k=1}^K(x_k+y_k)=\sum _{k=1}^K{x}_k+\sum _{k=1}^K{y}_k\\ & =& R\sum _{l=1}^L{\overline{x_l}}^2-2R\overline{x}\sum _{l=1}^L\overline{x_l}+R{\overline{x}}^2\sum _{l=1}^{L}1& +\sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-2\sum _{l=1}^L(\overline{x_l}\sum _{r=1}^R{x}_{lr})+\sum _{l=1}^L({\overline{x_l}}^2\sum _{r=1}^{R}1)& \dots \sum _{k=1}^{K}C{x}_k=C\sum _{k=1}^K{x}_k\\ & =& R\sum _{l=1}^L{\overline{x_l}}^2-2R\overline{x}L\overline{x}+R{\overline{x}}^{2}L& +\sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-2\sum _{l=1}^L(\overline{x_l}R\overline{x_l})+\sum _{l=1}^L{\overline{x_l}}^{2}R& \dots \sum _{k=1}^{K}1=K,\overline{x}:=\frac{\sum _{k=1}^K{x}_k}{K}\to \sum _{k=1}^K{x}_k=K\overline{x}\\ & =& R\sum _{l=1}^L{\overline{x_l}}^2-2LR{\overline{x}}^2+LR{\overline{x}}^2& +\sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-2R\sum _{l=1}^L{\overline{x_l}}^2+R\sum _{l=1}^L{\overline{x_l}}^2\\ & =& R\sum _{l=1}^L{\overline{x_l}}^2-LR{\overline{x}}^2& +\sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-R\sum _{l=1}^L{\overline{x_l}}^2\\ & =& -LR{\overline{x}}^2& +\sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2\\ & =& \sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-LR{\overline{x}}^2\\ & =& \sum _{n=1}^N{x}_n^2-N{\overline{x}}^2& \dots LR=N\end{array}$$

水準と繰返しが有る標本平均周りの平方和の式展開(分解なし)

$$\begin{array}{rcl}\sum _{n=1}^N(x_n-\overline{x}{)}^2& =& \sum _{l=1}^L\sum _{r=1}^R(x_{lr}-\overline{x}{)}^2\dots L:\mathrm{水}\mathrm{準},R:\mathrm{繰}\mathrm{返}\mathrm{し},N=LR\\ & =& \sum _{l=1}^L\sum _{r=1}^R(x_{lr}^2-2x_{lr}\overline{x}+{\overline{x}}^2)\\ & =& \sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-\sum _{l=1}^L\sum _{r=1}^{R}2x_{lr}\overline{x}+\sum _{l=1}^L\sum _{r=1}^R{\overline{x}}^2\\ & =& \sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-2\overline{x}\sum _{l=1}^L\sum _{r=1}^R{x}_{lr}+{\overline{x}}^2\sum _{l=1}^L\sum _{r=1}^{R}1\\ & =& \sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-2\overline{x}LR\overline{x}+{\overline{x}}^{2}LR\\ & =& \sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-2LR{\overline{x}}^2+LR{\overline{x}}^2\\ & =& \sum _{l=1}^L\sum _{r=1}^R{x}_{lr}^2-LR{\overline{x}}^2\\ & =& \sum _{n=1}^N{x}_n^2-N{\overline{x}}^2& \dots LR=N\\ & =& \sum _{n=1}^N{x}_n^2-N(\frac{\sum _{n=1}^N{x}_n}{N}{)}^2\\ & =& \sum _{n=1}^N{x}_n^2-\frac{1}{N}(\sum _{n=1}^N{x}_n{)}^2\\ & =& \sum _{n=1}^N{x}_n^2-\frac{1}{N}(N\overline{x}{)}^2\\ & =& \sum _{n=1}^N{x}_n^2-\frac{1}{N}{N}^2{\overline{x}}^2\\ & =& \sum _{n=1}^N{x}_n^2-N{\overline{x}}^2\end{array}$$

各水準平均と全体平均の差の平方和と各値と各水準との差の平方和に分解できること

以上より分解有りと無しで同じ値になることから，$\sum _{n=1}^N(x_n-\overline{x}{)}^2=\sum _{l=1}^L\sum _{r=1}^R(x_{lr}-\overline{x}{)}^2$は$\sum _{l=1}^L\sum _{r=1}^R(\overline{x_l}-\overline{x}{)}^2$と$\sum _{l=1}^L\sum _{r=1}^R(x_{lr}-\overline{x_l}{)}^2$に分解できることがわかる．