r gives you the strength of the correlation which I guess is an empirical sort of measure: weak, strong or neither while R^2 give you a quantitative result such as R^2 = 0.75 means the 75% of the variation is y is accounted for by the variations in x. I an having a hard time wrapping my head around that concept.
Assuming a simple linear model using Pearson’s r, it seems to me that r ~ s(residuals) but if that’s the case then what is the interpretation of variance s^2(residuals) and why would that give me something along the lines of var(y|var(x)) = R^2 * var(x) or the variance of y given the variance of x = R^2 * variance of x. If R^2 = 0.75, is it a reasonable interpretation that 25% of the variance [and is that s^2(residuals)?] in y is independent of x?
Lastly, let’s say I’m x=net calories eaten vs y=weight gain and I get R^2 = 0.6. What is the interpretation of that? 40% of weight gain/loss is independent of calories? Or is there no real-life interpretation?