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The Pearson correlation coefficient is the geometrical mean of the slopes of the two regression lines. For me it is difficult to understand how a geometrical mean of the slope of the two regression lines can be a measure of the linear correlation between two variables X and Y.

I found this paper fantastic, Bravais-Pearson and Spearman correlation coefficients: meaning, test of hypothesis and confidence interval by Artusi and coll., in explaining the Pearson coefficient, but I have difficulties in doing the last deduction.

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Let's regress $Y$ on $X$ and write $b_{0x} + b_{1x} x$ for the regression line, where $b_{1x}$ is the slope. Then Pearson correlation coefficient is $r = b_{1x} \frac{s_x}{s_y}$, $s_x$ and $s_y$ being the standard deviations of $X$ and $Y$, respectively. If, on the contrary, we regress $X$ on $Y$, the regression line is defined by $b_{0y} + b_{1y} y$, thus $r = b_{1y} \frac{s_y}{s_x}$.

Then it follows that $r^2 = b_{1x} b_{1y}$, i.e. $r = \sqrt{b_{1x} b_{1y}}$, which is the geometric mean of the two slopes.

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This is an interesting article with different interpretation of the corr. coeff. https://www.stat.berkeley.edu/~rabbee/correlation.pdf This does not give an answer to your question, but maybe you can find other interpretation

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    $\begingroup$ Thank you for your answer. But also there I cannot find the last deduction that I have asked $\endgroup$
    – Ashish Bhigah
    Commented Dec 6, 2020 at 19:12

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