Properties of Regression Coefficient
Linear Regression — Chapter 5, GSEB Class 12 Statistics
Properties
Sign of b Matches Sign of r
If r > 0, both byx and bxy > 0. If r < 0, both byx and bxy < 0.
The regression slope always moves in the same direction as the correlation. Positive r → upward-sloping line; negative r → downward-sloping line.
byx × bxy = r²
The product of the two regression coefficients equals the square of the correlation coefficient.
This is always ≤ 1 since r² ≤ 1. It links regression back to correlation.
Not Independent of Change of Origin
Adding or subtracting a constant from X or Y changes the value of b.
Unlike r, the regression coefficient IS affected by shifting the origin. If you add 10 to all X values, b changes.
Not Independent of Change of Scale
Multiplying or dividing X or Y by a constant changes the value of b.
If you convert X from metres to centimetres (×100), b changes by a factor of 100. The slope depends on the units used.
r = √(byx × bxy)
The correlation coefficient is the geometric mean of the two regression coefficients.
This follows from byx × bxy = r². Taking square root gives r = √(byx × bxy).
b Can Exceed 1 in Absolute Value
Unlike r (bounded −1 to +1), b has no fixed range. |b| can be greater than 1.
b depends on the units of measurement. If Y changes by 100 when X changes by 1, b = 100. This is perfectly valid.
Comparison with r
| Property | r (Correlation) | b (Regression) |
|---|---|---|
| Range | −1 to +1 (bounded) | No fixed range |
| Independent of origin | ✓ Yes | ✗ No |
| Independent of scale | ✓ Yes | ✗ No |
| Sign | ± (direction) | Same as r |
| Units | Unitless | Has units (Y units / X units) |
Key Takeaways
Key Takeaways
- Sign of b always matches sign of r — both positive or both negative.
- byx × bxy = r² — product of regression coefficients equals squared correlation.
- b is NOT independent of change of origin — shifting values changes b.
- b is NOT independent of change of scale — changing units changes b.
- r = √(byx × bxy) — correlation is the geometric mean of the two coefficients.
- Unlike r (bounded −1 to +1), b can exceed 1 in absolute value.