17.2: Relationship between the slope and the correlation
- Page ID
- 45250
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Product moment correlation is used to indicate the strength of the linear association between two ratio-scale variables; the slope tells you the rate of change between the two variables. When the correlation is negative, the slope will be negative; when correlation is positive, so too will the slope.
As you might suspect, there is a mathematical relationship between the product moment correlation, \(r\), and the regression slope, \(b_{1}\). We haven’t spent much time explaining the equations presented in this text, but correlation and linear regression are such important tools it’s worth a closer look.
Recall the equation of the correlation is \[r_{XY} = \frac{\left(X - \bar{X}\right) \left(Y - \bar{Y}\right)}{(n-1) s_{X} s_{Y}} \nonumber\]
where the numerator is termed the covariance between X and Y and the denominator contains the standard deviations of X and Y variables. We can say the at the covariance is standardized by the variability in X and Y. In contrast, the regression slope is equal to the covariance divided by the variance in X. \[b_{1} = \frac{\sum_{i=1}^{n} \left(X - \bar{X}\right) \left(Y - \bar{Y}\right)}{\sum_{i=1}^{n} (n-1) s_{X} s_{Y}} \nonumber\]
Thus, with a little algebra, we can see that the slope and correlation are equal to each other as \[b_{1} = r \cdot \frac{s_{X}}{s_{Y}} \nonumber\]
This should drive home the following statistical reasoning point. You can always calculate a slope from a correlation, but recall that correlation analysis is intended as a test of the hypothesis of a linear association between variables for which cause and effect model — though perhaps reasonable — should not always be implied. Just because it is mathematically possible does not mean the analysis is correct for the problem.
Questions
- If the correlation is 0.6, \(s_{\bar{X}} = 2.3\), and \(s_{\bar{Y}} = 1.67\), what is the slope?