Assume that (Y₁, Y₂) follows a bivariate distribution with mean vector (μ₁, μ₂) and covariance matrix with entries σ₁², σ₂², σ₁₂, and let ρ=Cor(Y₁,Y₂).

If we collect i.i.d. observations (Yᵢ₁, Yᵢ₂) from this bivariate distribution, then the expected squared perpendicular distance of each such point from the 45 degree line in 2d plane is given by

E[(Y₁-Y₂)²] =

E[((Y₁-μ₁) - (Y₂-μ₂) + μ₁-μ₂)²] =

E[((Y₁-μ₁) - (Y₂-μ₂))²] + (μ₁-μ₂)² =

(μ₁-μ₂)²+σ₁²+σ₂²-2σ₁₂ =

(μ₁-μ₂)²+(σ₁-σ₂)²+2(1-ρ)σ₁σ₂