When the spreads match, slope equals correlation

Question

Two standardized aptitude sub-scores, verbal ( $x$ ) and quantitative ( $y$ ), are related with these summaries:

$r = 0.75, \qquad s_x = 8, \qquad s_y = 8.$

What is the slope of the line predicting the quantitative score from the verbal score?

Accepted Answer

Use the slope formula: b = r \cdot \frac{sy}{sx} = 0.75 \cdot \frac{8}{8} = 0.75 	imes 1 = 0.75. Here is the point worth remembering: when the two variables are equally spread (sx = sy), the ratio of spreads is 1, and the slope equals the correlation itself. So a correlation of 0.75 becomes a slope of exactly 0.75. This is why people sometimes blur "slope" and "correlation," but they are only equal in this special matched-spread case (which is also what happens if you standardize both variables first). In general the two differ by the ratio sy / sx, and only the correlation is unitless. Take…

When the spreads match, slope equals correlation

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