Statistics / Unit 4: Two-Variable Data
4.5 The Coefficient of Determination r-squared
Notes
The coefficient of determination $r^2$ is the fraction of the variation in $y$ that is explained by the linear relationship with $x$.
- Compute $r^2$ by squaring the correlation. It is always between 0 and 1.
- Interpretation: "$r^2\%$ of the variation in $y$ is explained by the linear relationship with $x$."
- $r^2$ does not tell the sign. Use the slope's sign to get the sign of $r$.
- Example: $r = -0.8$ gives $r^2 = 0.64$, so $64\%$ of the variation in $y$ is explained by the linear relationship.
Practice
If $r = 0.9$, then $r^2$ equals:
$0.9^2 = 0.81$.
For a regression, $r^2 = 0.49$. The best interpretation is:
$r^2$ is the proportion of variation in $y$ accounted for by the linear model.
A regression has a negative slope and $r^2 = 0.25$. The correlation is:
$\displaystyle r = \pm\sqrt{0.25} = \pm 0.5$, and the sign matches the slope.
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