~Coefficient of Determination~

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~Coefficient of determination = r². This is defined as the proportion of the total variation made by using the regression line to predict the observed y-values, instead of predicting the y-values by their average in each case. (so, that's why it's refered to as the proportion of the total variation in the observed y-values that is EXPLAINED by the regression line). The closer r² is to 1, the more useful the regression equation is for prediction.

~Technically, r² = 1 - [∑(y-y_L)²]/[∑(y-y_A)²], where y_L is the y-value predicted from the regression line & y_A is the average of the observed
y-values.

~In words, r² = [Explained Variation] divided by [Total Variation]

~The following will give you the motivation for this definition

~Since y-y_A=(y_L-y_A)+(y-y_L), we can square both sides, to get:

(y-y_A)² = [(y_L-y_A) + (y-y_L)]², or

(y-y_A)² = [(y_L-y_A)]² +2[(y_L-y_A)][(y-y_L)] + [(y-y_L)]²

~Using Sigma (Sum) properties, we get:
∑[(y-y_A)²] = ∑[(y_L-y_A)]² + 2∑[(y_L-y_A)][(y-y_L)] + ∑[(y-y_L)]²

~The middle term on the right side can be shown to = 0 in the derivation process of the "least squares line" (using the two equations in a & b).
(see link on Correlation & Regression)

~We then have, ∑[(y-y_A)²] = ∑[(y_L-y_A)]² + ∑[(y-y_L)]²

which are the expressions that give us

~Total Variation = Explained Variation + Unexplained Variation, or

~Explained variation = Total Variaton - Unexplained variation

~Dividing both sides by the total variation, we get:

~[Explained Variaton]/[total Variation]=

1-[Unexplained Variation]/[Total Variation] or r²
(equation for r² given initially)

~The Unexplained Variation, ∑[(y-y_L)]², can be due to other variables of our problem not considered.