Bailee Richards

2022-11-26

Find the meaning of 'Sxx' and 'Sxy' in simple linear regression

CobeBedererKDs

Beginner2022-11-27Added 6 answers

${S}_{xx}$ is the sum of the squares of the difference between each x and the mean x value.

${S}_{xy}$ is sum of the product of the difference between x its means and the difference between y and its mean.

So ${S}_{xx}=\mathrm{\Sigma}(x-\overline{x})(x-\overline{x})$ and ${S}_{xy}=\mathrm{\Sigma}(x-\overline{x})(y-\overline{y})$. Both of these are often rearranged into equivalent (different) forms when shown in textbooks.

${S}_{xy}$ is sum of the product of the difference between x its means and the difference between y and its mean.

So ${S}_{xx}=\mathrm{\Sigma}(x-\overline{x})(x-\overline{x})$ and ${S}_{xy}=\mathrm{\Sigma}(x-\overline{x})(y-\overline{y})$. Both of these are often rearranged into equivalent (different) forms when shown in textbooks.

Reece Black

Beginner2022-11-28Added 2 answers

${S}_{xx}=\sum {x}^{2}-\frac{(\sum x{)}^{2}}{n}=\sum {x}^{2}-n{\overline{x}}^{2}$

Intuitively, ${S}_{xy}$ is the result when you replace one of the x's with a y.

${S}_{xy}=\sum xy-\frac{\sum x\sum y}{n}=\sum xy-n\overline{x}\overline{y}$

Also, just for your information, the good thing about this notation is that it simplifies other parts of linear regression.

For example, the product-moment correlation coefficient:

$r=\frac{\sum xy-n\overline{x}\overline{y}}{\sqrt{(\sum {x}^{2}-n{\overline{x}}^{2})(\sum {y}^{2}-n{\overline{y}}^{2})}}=\frac{{S}_{xy}}{\sqrt{{S}_{xx}{S}_{yy}}}$

or to find the gradient of the best-fit line y=a+bx:

$y-\overline{y}=b(x-\overline{x}),\text{where}b=\frac{{S}_{xy}}{{S}_{xx}}$

Intuitively, ${S}_{xy}$ is the result when you replace one of the x's with a y.

${S}_{xy}=\sum xy-\frac{\sum x\sum y}{n}=\sum xy-n\overline{x}\overline{y}$

Also, just for your information, the good thing about this notation is that it simplifies other parts of linear regression.

For example, the product-moment correlation coefficient:

$r=\frac{\sum xy-n\overline{x}\overline{y}}{\sqrt{(\sum {x}^{2}-n{\overline{x}}^{2})(\sum {y}^{2}-n{\overline{y}}^{2})}}=\frac{{S}_{xy}}{\sqrt{{S}_{xx}{S}_{yy}}}$

or to find the gradient of the best-fit line y=a+bx:

$y-\overline{y}=b(x-\overline{x}),\text{where}b=\frac{{S}_{xy}}{{S}_{xx}}$

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