Why squaring RR 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!

Answer

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 θ, the correlation R is cos(θ).
The part of Y that is explained by X is of length ||Y||cos(θ) and is parallel to X (or the projection of Y on X). The part that is not explained is of length ||Y||sin(θ) and is orthogonal to X. In terms of variances, we have
σ2Y=σ2Ycos2(θ)+σ2Ysin2(θ)
where the first term on the right is the explained variance and the second the unexplained variance. The fraction that is explained is thus R2, not R.

Attribution
Source : Link , Question Author : David , Answer Author : Dilip Sarwate

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