1. Special Relationship: The Product of Regression Coefficients

There is an important relationship between the correlation coefficient and regression coefficients:

byx × bxy =r^2

and since −1 ≤ r ≤ 1
0 ≤ r^2 ≤ 1
byx × bxy ≤ 1

If one of the regression coefficients is greater than unity (1), the other must be less than unity.

2. Relationship Between Regression Coefficienct, Correlation Coefficient & Standard Deviation

The correlation coefficient is directly related to standard deviation because it is calculated using standardized values of X and Y.

We can rewrite r in terms of the regression coefficients and standard deviations:

byx = r × [σY / σX]​​

bxy = r× [σX / σY]​​

Where:

• byx​ → Regression coefficient of Y on X
• bxy​ → Regression coefficient of X on Y

3. Key Takeaways:

✅ Correlation is unitless because it is derived from standard deviations.
✅ Regression coefficients depend on standard deviations, but correlation does not change with scale transformation (e.g., changing units).
✅ If σX​ or σY​ changes, the regression coefficients change, but r remains the same.