Key takeaway: The coefficient of determination, r squared, is the proportion of the variation in the response variable that is accounted for by its linear relationship with the explanatory variable.
Grade 12 – AP StatisticsAP Statistics Unit 5Statistics and Data
Definition of Coefficient of Determination
The coefficient of determination, written \(r^2\), tells how much better a regression line predicts than simply guessing the mean of \(y\) every time. It compares the scatter around the line with the scatter around the mean.
A value of 0.64 means that 64% of the differences in \(y\) from one individual to the next are accounted for by the linear link with \(x\); the other 36% is left to other factors and chance. It runs from 0 to 1, and higher values mean the line fits better.
Formula and notation
It is the square of the correlation coefficient, and it is also a ratio of variation:
\[ r^2=(r)^2=1-\frac{\text{variation left around the line}}{\text{total variation in }y}. \]
Because squaring removes the sign, \(r^2\) alone cannot tell whether the association is positive or negative.
Key facts
- A value near 1 means points hug the line; a value near 0 means the line explains little.
- To recover r, take the square root and borrow the sign of the slope.
- A high value does not prove causation, and it does not show that a linear model is appropriate.
- Always read it as a percent of variation in y, never as a percent of points on the line.
Coefficient of Determination example
For a sample of cars, the correlation between age and resale price is \(r=-0.80\). Find \(r^2\) and interpret it. What would \(r^2=0.49\) mean for the correlation?
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Square the correlation: \(r^2=(-0.80)^2=0.64\).
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Interpret: 64% of the variation in resale price is accounted for by the linear relationship with age. The remaining 36% is due to other factors such as mileage and condition.
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If \(r^2=0.49\), then \(r=\pm 0.70\), with the sign matching the direction of the slope.
About 64% of the variation in price is explained by age; an \(r^2\) of 0.49 corresponds to \(r=\pm 0.70\).
Common mistake
Watch out
A student interprets \(r^2=0.64\) as “64% of the cars lie on the regression line.”
Why it is wrong
Few points, if any, lie exactly on the line. The coefficient of determination describes variation: 64% of the differences in prices are accounted for by age through the linear model. It is a measure of how much of the spread in y is explained, not a count of points.
Related terms and lessons
- Least-Squares Regression Line
- Correlation Coefficient
- Residual
- Scatter Plot
- Correlation vs. Causation
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