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Explained sum of squares: Difference between revisions

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:<math>
\sum_{i=1}^n (y_{i}-\bar{y})^2=\sum_{i=1}^n (y_i - \hat{y}_{i}_i)^2+\sum_{i=1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i=1}^n 2(\hat{y}_{i}_i-\bar{y})(y_i - \hat{y}_i).
</math>
 
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:<math>
\begin{align}
\sum_{i=1}^n 2(\hat{y}_{i}_i-\bar{y})(y_{i}y_i-\hat{y}_i) & = \sum_{i=1}^{n} 2((\bar{y}-\hat{b}\bar{x}+\hat{b}x_{i}x_i)-\bar{y})(y_{i}y_i-\hat{y}_{i}_i) \\
& = \sum_{i=1}^{n}2((\bar{y}+\hat{b}(x_{i}-\bar{x}))-\bar{y})(y_{i}-\hat{y}_{i}) \\
& = \sum_{i=1}^{n}2(\hat{b}(x_{i}-\bar{x}))(y_{i}-\hat{y}_{i}) \\
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Again [[simple linear regression]] gives<ref name=Mendenhall/>
 
:<math>\hat{b}= \left(frac{\sum_{i=1}^{n} (x_{i}x_i-\bar{x})(y_{i}y_i-\bar{y})\right)/\left(}{\sum_{i=1}^{n} (x_{i}x_i-\bar{x})^2\right)}, </math>
 
:<math>
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