Dixie Blake

2023-03-31

Tell about the meaning of Sxx and Sxy in simple linear regression,, especially the meaning of those formulas

popunjati7z8a

In simple linear regression, Sxx and Sxy are mathematical terms that are used to calculate the slope and intercept of the regression line.
Sxx represents the sum of squares of the deviations of the predictor variable (x) from its mean (x̄). It measures the total variation or spread in the x-values. The formula for Sxx is:
$Sxx={\sum }_{i=1}^{n}\left({x}_{i}-\overline{x}{\right)}^{2}$
where n is the number of observations, xi is the value of x for the ith observation, and x̄ is the mean of x.
Sxy represents the sum of cross-products of the deviations of both the predictor variable (x) and the response variable (y) from their respective means (x̄ and ȳ). It measures the combined variation between x and y. The formula for Sxy is:
$Sxy={\sum }_{i=1}^{n}\left({x}_{i}-\overline{x}\right)\left({y}_{i}-\overline{y}\right)$
where n is the number of observations, xi and yi are the values of x and y for the ith observation, and x̄ and ȳ are the means of x and y.
These terms are used in the calculations of the slope (β1) and intercept (β0) of the regression line in simple linear regression. The slope is given by:
$\beta 1=\frac{Sxy}{Sxx}$
And the intercept is given by:
$\beta 0=\overline{y}-\beta 1\overline{x}$
In summary, Sxx represents the variation in the predictor variable, while Sxy represents the combined variation between the predictor variable and the response variable. These terms are essential in determining the relationship between the variables and fitting the regression line to the data.

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