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The coefficient of determination, usually symbolized by $R^2$, is the proportion of the total response variance explained by a regression model. Can also be used for various pseudo R-squared proposed, for instance for logistic regression (and other models.)

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Relationship between $R^2$ and correlation coefficient

This is true that $SS_{tot}$ will change ... but you forgot the fact that the regression sum of of squares will change as well. So let's consider the simple regression model and denote the Correlatio …
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