Getting the slope from correlation and spread

Question

An analyst studies how weekly study hours ( $x$ ) relate to a test score ( $y$ ). She has computed the correlation and the two standard deviations, but not the regression line itself:

$r = 0.6, \qquad s_x = 4, \qquad s_y = 20.$

What is the slope of the least-squares line predicting score from study hours?

Accepted Answer

You do not need the raw data to get the slope. There is a clean shortcut linking the slope to the correlation and the two spreads: b = r \cdot \frac{sy}{sx}. In words: the slope is the correlation scaled by how spread out y is compared with x. Correlation is a pure "tightness" number with no units; multiplying by the ratio of standard deviations puts it back into real units (score per hour). Plug in the values: b = 0.6 \cdot \frac{20}{4} = 0.6 	imes 5 = 3. So each extra hour of study is associated with about 3 more points. Notice how the pieces play their roles: the correlation 0.6 sets the …

Getting the slope from correlation and spread

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