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Question Number 19 by user1 last updated on 25/Jan/15

Let θ be the angle between the regression

line of y on x and the regression line of

x on y . Then prove that

\tan θ = {\frac{(1 - r^{2})}{r} \times \frac{σ_{x} \times σ_{y}}{(σ_{x}^{2} + σ_{y}^{2})}}

Answered by user1 last updated on 30/Oct/14

The equation of the line of regression

of x on y is

(x - \bar{x}) = r \times \frac{σ_{x}}{σ_{y}} (y - \bar{y}) \dots (i)

And the equation of the line of regression

of y on x is

(y - \bar{y}) = r \times \frac{σ_{y}}{σ_{x}} (x - \bar{x}) \dots (ii)

Let m_{1} and m_{2} be the slopes of (i) and (ii) res .

Then, m_{1} = \frac{σ_{y}}{r \cdot σ_{x}} and m_{2} = \frac{r \cdot σ_{y}}{σ_{x}}

∴ \tan θ = \frac{(m_{1} - m_{2})}{(1 + m_{1} m_{2})}

= \frac{\frac{σ_{y}}{r \cdot σ_{x}} - \frac{r \cdot σ_{y}}{σ_{x}}}{1 + \frac{{(σ_{y})}^{2}}{{(σ_{x})}^{2}}} = {\frac{(1 - r^{2})}{r} \times \frac{σ_{x} \cdot σ_{y}}{(σ_{x}^{2} + σ_{y}^{2})}}

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