Regression Slope from Correlation

Correlation and slope are the same relationship in different clothes. The correlation r is unit-free; multiply it by the ratio of the spreads and you get a slope carrying real units — change in y per unit of x. With r = 0.8, s_y = 5 and s_x = 2, the slope is 0.8 × 5/2 = 2.0. Francis Galton stumbled on this in the 1880s while plotting the heights of children against their parents: the best-fit line was flatter than the 45° line of pure heredity, and he called the effect "regression towards mediocrity" — the origin of the word regression itself.

Galton's discovery is also the trap. Because |r| ≤ 1, the slope is always shallower than s_y/s_x, so predicted values are pulled toward the mean of y. Tall parents have tall-but-shorter children; a company's best quarter is usually followed by a worse one — not because anything caused it, but because the extreme was partly luck. Note too that r is symmetric while the slope is not: regressing y on x gives 2.0 here, but regressing x on y gives 0.8 × 2/5 = 0.32, and 1/2.0 = 0.5 is not the answer.