最小二乗法

${\displaystyle {\Delta }^2={\displaystyle \sum _{i=1}^n\Delta i^2}={\displaystyle \sum _{i=1}^n{\left\{y_i-\left(a{x}_i+b\right)\right\}}^2}}$

${\displaystyle {\sigma }_{xy}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i{y}_i-\overline{x}\overline{y}}$ (参照：共分散)

${\displaystyle {\sigma }_x^2=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2-{\overline{x}}^2}$ (参照：分散)

${\displaystyle ={\displaystyle \sum _{i=1}^n\left\{y_i^2-2\left(a{x}_i+b\right)y_i+{\left(a{x}_i+b\right)}^2\right\}}}$

${\displaystyle ={\displaystyle \sum _{i=1}^n\left\{y_i^2-2a{x}_i{y}_i-2b{y}_i+a^2{x}_i^2+2ab{x}_i+b^2\right\}}}$

${\displaystyle ={\displaystyle \sum _{i=1}^n{y}_i^2}+{\displaystyle \sum _{i=1}^n\left(-2a{x}_i{y}_i\right)}+{\displaystyle \sum _{i=1}^n\left(-2b{y}_i\right)}}$ ${\displaystyle +{\displaystyle \sum _{i=1}^n{a}^2{x}_i^2}}$ ${\displaystyle +{\displaystyle \sum _{i=1}^{n}2ab{x}_i}}$ ${\displaystyle +{\displaystyle \sum _{i=1}^n{b}^2}}$

${\displaystyle =A-2aB-2bC+a^{2}D+2abE+n{b}^2}$

${\displaystyle =n{b}^2+2\left(Ea-C\right)b+D{a}^2-2Ba+A}$

${\displaystyle =n{\left(b+\frac{Ea-C}{n}\right)}^2-\frac{1}{n}{\left(Ea-C\right)}^2}$ ${\displaystyle +D{a}^2}$ ${\displaystyle -2Ba+A}$

${\displaystyle =n{\left(b+\frac{Ea-C}{n}\right)}^2}$ ${\displaystyle +\frac{1}{n}\left(-E^2{a}^2+2CEa-C^2+nD{a}^2-2nBa+nA\right)}$

${\displaystyle =n{\left(b+\frac{Ea-C}{n}\right)}^2}$ ${\displaystyle +\frac{1}{n}\left\{\left(nD-E^2\right)a^2+2\left(CE-nB\right)a+nA-C^2\right\}}$

${\displaystyle {\displaystyle =\frac{{\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i{y}_i}-\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i}\right)\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{y}_i}\right)}{{\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2}-{\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i}\right)}^2}}}$

${\displaystyle \overline{x}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i}$ ， ${\displaystyle \overline{y}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{y}_i}$ (参照：平均)　･･････(1)

${\displaystyle {\sigma }_{xy}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i{y}_i-\overline{x}\overline{y}}$ (参照：共分散)　･･････(2)

${\displaystyle {\sigma }_x^2=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2-{\overline{x}}^2}$ (参照：分散)　･･････(3)

${\displaystyle =\frac{{\displaystyle \frac{1}{n^2}}\left({\displaystyle {\displaystyle \sum }_{i=1}^n{x}_i^2{\displaystyle \sum }_{i=1}^n{y}_i-{\displaystyle \sum }_{i=1}^n{x}_i{y}_i{\displaystyle \sum }_{i=1}^n{x}_i}\right)}{{\displaystyle \frac{1}{n^2}}\left\{{\displaystyle n{\displaystyle \sum }_{i=1}^n{x}_i^2-{\left({\displaystyle \sum }_{i=1}^n{x}_i\right)}^2}\right\}}}$

${\displaystyle =\frac{\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2}\right)\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{y}_i}\right)-\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i{y}_i}\right)\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i}\right)}{{\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2-{\left(\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i\right)}^2}}}$

(1)より

${\displaystyle =\frac{\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2}\right)\overline{y}-\left({\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i{y}_i}\right)\overline{x}}{{\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2-{\overline{x}}^2}}}$

${\displaystyle =\frac{{\displaystyle \left(\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2\right)\overline{y}-{\overline{x}}^2\overline{y}+{\overline{x}}^2\overline{y}-\left(\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i{y}_i\right)\overline{x}}}{{\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2-{\overline{x}}^2}}}$

${\displaystyle =\frac{{\displaystyle \left\{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2-{\overline{x}}^2\right\}\overline{y}-\overline{x}\left\{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i{y}_i-\overline{x}\overline{y}\right\}}}{{\displaystyle \frac{1}{n}{\displaystyle \sum }_{i=1}^n{x}_i^2-{\overline{x}}^2}}}$