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

$$R^2$$ 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.

• I think because $R^2$ is nicely differentiable and more easy to deal with instead of $R$. Aug 4 at 18:08
• @Allan What does "more easy to deal with" mean? Also: Welcome, J. Park! Aug 4 at 18:31
• @Alexis, I mean that is differentiable, so if use want to use $R^2$ a an objective function is more easily to compute the derivatives and apply numerical methods, for example, gradient descent. Aug 4 at 18:39
• Variances add; standard deviations don't.
– whuber
Aug 4 at 18:57
• @Bernhard Right--that's one of the implications.
– whuber
Aug 4 at 19:38

$$R^2$$ 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 $$R^2$$.