<Dl> <Dd> R S S = ∑ i = 1 n (y i − f (x i)) 2 = ∑ i = 1 n (y i − (a x i + b)) 2 = ∑ i = 1 n (y i − a x i − y _̄ + a x _̄) 2 (\ displaystyle RSS = \ sum _ (i = 1) ^ (n) (y_ (i) - f (x_ (i))) ^ (2) = \ sum _ (i = 1) ^ (n) (y_ (i) - (ax_ (i) + b)) ^ (2) = \ sum _ (i = 1) ^ (n) (y_ (i) - ax_ (i) - (\ bar (y)) + a (\ bar (x))) ^ (2)) <Dl> <Dd> = ∑ i = 1 n (a (x _̄ − x i) − (y _̄ − y i)) 2 = a 2 S x x − 2 a S x y + S y y = S y y − a S x y = S y y (1 − S x y 2 S x x S y y) (\ displaystyle = \ sum _ (i = 1) ^ (n) (a ((\ bar (x)) - x_ (i)) - ((\ bar (y)) - y_ (i))) ^ (2) = a ^ (2) S_ (xx) - 2aS_ (xy) + S_ (yy) = S_ (yy) - aS_ (xy) = S_ (yy) (1 - (\ frac (S_ (xy) ^ (2)) (S_ (xx) S_ (yy))))) </Dd> </Dl> </Dd> </Dl> <Dd> R S S = ∑ i = 1 n (y i − f (x i)) 2 = ∑ i = 1 n (y i − (a x i + b)) 2 = ∑ i = 1 n (y i − a x i − y _̄ + a x _̄) 2 (\ displaystyle RSS = \ sum _ (i = 1) ^ (n) (y_ (i) - f (x_ (i))) ^ (2) = \ sum _ (i = 1) ^ (n) (y_ (i) - (ax_ (i) + b)) ^ (2) = \ sum _ (i = 1) ^ (n) (y_ (i) - ax_ (i) - (\ bar (y)) + a (\ bar (x))) ^ (2)) <Dl> <Dd> = ∑ i = 1 n (a (x _̄ − x i) − (y _̄ − y i)) 2 = a 2 S x x − 2 a S x y + S y y = S y y − a S x y = S y y (1 − S x y 2 S x x S y y) (\ displaystyle = \ sum _ (i = 1) ^ (n) (a ((\ bar (x)) - x_ (i)) - ((\ bar (y)) - y_ (i))) ^ (2) = a ^ (2) S_ (xx) - 2aS_ (xy) + S_ (yy) = S_ (yy) - aS_ (xy) = S_ (yy) (1 - (\ frac (S_ (xy) ^ (2)) (S_ (xx) S_ (yy))))) </Dd> </Dl> </Dd> <Dl> <Dd> = ∑ i = 1 n (a (x _̄ − x i) − (y _̄ − y i)) 2 = a 2 S x x − 2 a S x y + S y y = S y y − a S x y = S y y (1 − S x y 2 S x x S y y) (\ displaystyle = \ sum _ (i = 1) ^ (n) (a ((\ bar (x)) - x_ (i)) - ((\ bar (y)) - y_ (i))) ^ (2) = a ^ (2) S_ (xx) - 2aS_ (xy) + S_ (yy) = S_ (yy) - aS_ (xy) = S_ (yy) (1 - (\ frac (S_ (xy) ^ (2)) (S_ (xx) S_ (yy))))) </Dd> </Dl> <Dd> = ∑ i = 1 n (a (x _̄ − x i) − (y _̄ − y i)) 2 = a 2 S x x − 2 a S x y + S y y = S y y − a S x y = S y y (1 − S x y 2 S x x S y y) (\ displaystyle = \ sum _ (i = 1) ^ (n) (a ((\ bar (x)) - x_ (i)) - ((\ bar (y)) - y_ (i))) ^ (2) = a ^ (2) S_ (xx) - 2aS_ (xy) + S_ (yy) = S_ (yy) - aS_ (xy) = S_ (yy) (1 - (\ frac (S_ (xy) ^ (2)) (S_ (xx) S_ (yy))))) </Dd>

Residual sum of squares vs residual standard error