Practice Coefficient of Determination in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

The proportion of the total variation in the response variable y that is explained by the linear relationship with the explanatory variable x. It equals the square of the correlation coefficient: r^2.

Total variation in y has two parts: what the regression line explains and what's left over (residual variation). If r^2 = 0.85, the regression line accounts for 85\% of why y values differ from each other, and 15\% is unexplained. Think of r^2 as a report card for how well x predicts y.

Example 1

medium
A regression model has SST = 500 (total variation) and SSE = 125 (unexplained variation). Calculate R^2 and interpret its meaning.

Example 2

hard
Two models predict house prices: Model 1 (size only): R^2=0.60. Model 2 (size + neighborhood + age): R^2=0.85. Explain what the increase in R^2 means and what caution should be applied with multi-variable R^2.

Example 3

easy
The correlation between study hours and test score is r=0.8. Calculate R^2 and interpret it.

Example 4

hard
A model has R^2=0.95. A researcher concludes 'the model is perfect and ready for deployment.' Identify two potential problems with this conclusion.
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