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Regression Sum of Squares Formula

Also known as explained sum, the model sum of squares or sum of squares dues to regression. It helps to represent how well a data that has been model has been modelled. It there is some variation in the modelled values to the total sum of squares, then that explained sum of squares formula is used. Helps measure how much variation there is in the data observed. 

\[\large SSR=\sum\left(\widehat{y}-\overline{y}\right)^{2}\]

Solved example

Question: Find the regression sum of square line for the data set {(1, 2), (2, 1), (4, 6), (5, 6)}?

Solution:
Given data set : {(1, 1), (2, 3), (4, 6), (5, 6)}

$\overline{x}=\frac{\left(1+2+4+6\right)}{4}=\frac{13}{4}=3.25$
$\overline{y}=\frac{\left(1+3+6+6\right)}{4}=\frac{16}{4}=4$

$SS_{xx}=\left(\left(1-3.25\right)^{2}+\left(2-3.25\right)^{2}+\left(4-3.25\right)^{2}+\left(6-3.25\right)^{2}\right)=14.75$
$SS_{yy}=\left(\left(1-4\right)^{2}+\left(3-4\right)^{2}+\left(6-4\right)^{2}+\left(5-4\right)^{2}\right)=15$
$SS_{xy}=\left(\left(1-3.25\right)\left(1-4\right)+\left(2-3.25\right)\left(3-4\right)+\left(4-3.25\right)\left(6-4\right)+\left(6-3.25\right)\left(5-4\right)\right)=12.25$

$b_{1}=\frac{12.25}{10}=1.22$
$b_{0}=4-12.22\times 2.25=0.035$
$\overline{y}=0.03\times 1.22=0.0366x$

1 =  0.0366
2 = 0.0732
4 = 0.1464
6 = 0.2196

Using the formula: $SSR=\sum\left(\widehat{y}-\overline{y}\right)^{2}$

$=\left(0.0366-4\right)^{2}+\left(0.0732-4\right)^{2}+\left(0.1464-4\right)^{2}+\left(0.2196-4\right)^{2}=60.26995492$

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