$$ \displaystyle \sum^n_{i = 1} (y_i - \bar{y})^2 = ( \displaystyle \sum^n_{i = 1} (y_i - \bar{y})^2 - \displaystyle \sum^n_{i = 1} (y_i - \hat{y}_i)^2 ) + \displaystyle \sum^n_{i = 1} (y_i - \hat{y}_i)^2$$
My book says the LHS is the total variance, the first component of the RHS is the explained variance and the second component is the unexplained variance. A model of regression is good if the unexplained variance is much smaller than the total variance. (From working with excel, I believe the coefficient of determination is the explained variance divided by the total variance, but that is not much more than an educated guess).
So I want to see if I got this straight:
The first component of the RHS is explained because that is basically the total variance minus what you'd expect to be true according to your regression model. The second component of the RHS is unexplained because that is just the imperfection of your regression model.
My second question is, why is $ \displaystyle \sum^n_{i = 1} (y_i - \bar{y})^2 $ in this? Is this just because it would be easiest, or is there a deeper reason?