Suppose r_xy is the correlation coefficient between two variables X and Y
where s.d.(X) = s.d.(Y). If θ is the angle between the two regression lines of Y on X and X on Y then:

Question

Question

This question was previously asked in

SSC CGL Tier-II ( JSO ) 2016 Official Paper ( Held On : 2 Dec 2016 )

Answer 1

Formula

If θ be the angle between the regression lines then,

Tan θ = [(1 - r²)/r] × (σ_x × σ_y)/(σ_x² + σ_y²)

Explanation

According to question

⇒ σ_x = σ_y

⇒ Tanθ = [(1 - r²/r] × (σ_x²/(2σ_x)

⇒ Tanθ = [(1 - r²/r] × 1/2

⇒ Tan θ = (1 - r²/2r

⇒ Tanθ = Sinθ/cosθ

⇒ Tanθ = Sinθ × cosθ

⇒ Sinθ = Tanθ /secθ

⇒ Sinθ = Tanθ/√(1 + tan²θ)

⇒ Sinθ = [(1 - r²/r]/√(1 + (1 - r²/2r)²)

⇒ Sinθ = (1 - r²)/√[(1 - r²)² + 4r²]

∴ \(sin \theta=\sqrt{\dfrac{1-r^2_{xy}}{1+r^2_{xy}}}\)