Why squaring $R$ gives explained variance? This may be a basic question, but I was wondering why an $R$ value in a regression model can simply be squared to give a figure of explained variance?
I understand that $R$ coefficient can give the strength of a relationship, but I don't understand how simply squaring this value gives a measure of explained variance. 
Any easy explanation of this?
Thanks very much for helping with this!
 A: Hand-wavingly, the correlation $R$ can be thought of as a measure of the angle between two vectors, the dependent vector $Y$ and the independent vector $X$.
If the angle between the vectors is $\theta$, the correlation $R$ is $\cos(\theta)$.
The part of $Y$ that is explained by $X$ is of length $\Vert Y\Vert\cos(\theta)$ and is parallel to $X$ (or the projection of $Y$ on $X$).  The part that is not explained is of length $\Vert Y\Vert\sin(\theta)$ and is orthogonal to $X$.  In terms of variances, we have
$$\sigma_Y^2 = \sigma_Y^2\cos^2(\theta) + \sigma_Y^2\sin^2(\theta)$$
where the first term on the right is the explained variance and the second the unexplained variance.  The fraction that is explained is thus $R^2$, not $R$.
A: You can do this a long way and show that the total variance of the dependent variable is a sum of the variance of predicted and the error variance. The ratio of the variance or predicted to the variance of dependent variable is called $R^2$, and it's between 0 and 1 in OLS. It happens so that when you have only one independent variable $\sqrt{R^2}=R$, the Pearson correlation coefficient. That's why you can say that squaring the correlation coefficient gives the explained variance, i.e. the portion predicted variance to the total variance.
