Why do we use $R^2$ instead of $R$ in linear regression?

Question

$R^2$ equals the "amount of variance explained by the model".

However, we rarely use variance in descriptive statistics. We say a sample's weight is 78 ± 13 kg, which is $\bar x$ ± $\sigma$ (stdev), not $\bar x$ ± $\sigma^2$ (variance). This is because the standard deviation is in the same scale as the variable and is easier to understand than variance.

Thus my question: why don't we use "amount of standard deviation explained by the model"? Wouldn't you get it with $\sqrt{R^2}$? In my opinion, this would be much more intuitive to understand.

Answer 1

$R^2$ has several equivalent definitions. One is the squared correlation between the $x$ and $y$ variables in a simple linear regression. One is the squared correlation between the true values of $y$ and the predicted values; this has an advantage of working for multiple linear regression.

The definition that seems best-motivated to me involves the decomposition of the total sum of squares (SST) into the regression sum of squares (SSReg) and the residual sum of squares (SSRes). This $1-\frac{SSRes}{SST}$ expression can be considered on its own without thinking about the above two squared correlations. I think of that as a comparison of the performance of the fitted model to a naïve model that always predicts $\bar y$, no matter the feature values. Perhaps we could write $C=1-\frac{SSRes}{SST}$. However, it turns out that $C$ is equal to the above two squared correlations, so it has been given the name $R^2$.