Regression Coefficient

Linear Regression — Chapter 3, GSEB Class 12 Statistics

What is the Regression Coefficient?

The regression coefficient (b) is the slope of the regression line. It tells us how much Y changes for every one-unit increase in X.

byx (Y on X)

Change in Y for unit change in X. Used when predicting Y from X.

b > 0 → positive slope (Y increases with X)

bxy (X on Y)

Change in X for unit change in Y. Used when predicting X from Y.

b < 0 → negative slope (X decreases as Y increases)

byx (Y on X)

b_{yx} = r \times \frac{\sigma_y}{\sigma_x}

bxy (X on Y)

b_{xy} = r \times \frac{\sigma_x}{\sigma_y}

byx — Deviation Form

b_{yx} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}

bxy — Deviation Form

b_{xy} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (y - \bar{y})^2}

Key relationship: byx × bxy = r². The product of the two regression coefficients equals the square of the correlation coefficient.

The sign of b always matches the sign of r

r > 0 → b > 0

Positive correlation → positive slope → line goes up

r < 0 → b < 0

Negative correlation → negative slope → line goes down

Important: b is NOT bounded between −1 and +1 (unlike r). The slope can be any real number — it depends on the units of measurement.

Regression coefficient (b) = slope of the regression line — change in Y per unit change in X.
byx = r × (σy/σx) — used to predict Y from X (Y on X line).
bxy = r × (σx/σy) — used to predict X from Y (X on Y line).
byx × bxy = r² — product of coefficients equals squared correlation.
The sign of b always matches the sign of r — both positive or both negative.
b is NOT bounded between −1 and +1 — it can exceed 1 in absolute value.