I have been trying to get my head around $R^2$ in a bit more details instead of just seeing it as a number.

So far I have looked at the process in the following manner:

If I knew very little about the $x$ and $y$ variables and was asked to make a prediction for $y$ for any $x$ I'd guess the mean of $y$ as my prediction. However if I could see the regression line I'd use this and this to give me a more accurate prediction.

Here's where my confusion starts…

I measure the difference between a $y$ point and the mean for $y$ and then square this. I repeat for every $y$ point and sum them this is total variance in $y$. Lets label this B.

I also measure the difference between every $y$ point and its predicted $y$ point on the line and again square this. I take the total of all of these and This is called explained variation mean. Lets label this A.

So 1 - (A/B) gives me my $R^2$. So A/B is some kind of ratio? But I don't get what it means by explained variation? How does the regression line explain these points?

Whats does it mean for the regression line to account for variance?

It seems the further the points are from the mean while being closer to the line the better?



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