# What is the meaning of 'Sxx' and 'Sxy' in simple

Margie Marx 2021-12-14 Answered
What is the meaning of Sxx and Sxy in simple linear regression? I know the formula but what is the meaning of those formulas?
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Cheryl King

${S}_{xx}$ 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 . Both of these are often rearranged into equivalent (different) forms when shown in textbooks.

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psor32

To add to the answer from above,
${S}_{xx}=\sum {x}^{2}-\frac{{\left(\sum x\right)}^{2}}{n}=\sum {x}^{2}-n{\stackrel{―}{x}}^{2}$
Intuitively, ${S}_{xy}$ is the result when you replace one of the x's with a y.
${S}_{xy}=\sum xy-\frac{\sum x\sum y}{n}=\sum xy-n\stackrel{―}{x}\stackrel{―}{y}$
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:
$r=\frac{\sum xy-n\stackrel{―}{x}\stackrel{―}{y}}{\sqrt{\left(\sum {x}^{2}-n{\stackrel{―}{x}}^{2}\right)\left(\sum {y}^{2}-n{\stackrel{―}{y}}^{2}\right)}}=\frac{{S}_{xy}}{\sqrt{{S}_{xx}{S}_{yy}}}$
or to find the gradient of the best-fit line $y=a+bx:$

The pragmatic importance of this is that if you are doing a long question about linear regression, calculating  at the beginning can save you a lot of work.

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