Coefficient of Determination Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Coefficient of Determination.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: r^2 close to 1 means the model explains most of the variation; close to 0 means it explains very little. But a high r^2 does NOT prove the model is correct—always check the residual plot.

Common stuck point: Students confuse r and r^2. If r = 0.7, the model explains r^2 = 0.49 or only 49\% of variation—much less impressive than r sounds.

Worked Examples

Example 1

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

Solution

  1. 1
    R^2 = 1 - \frac{SSE}{SST} = 1 - \frac{125}{500} = 1 - 0.25 = 0.75
  2. 2
    Alternatively: R^2 = \frac{SSR}{SST} = \frac{SST - SSE}{SST} = \frac{375}{500} = 0.75
  3. 3
    Interpretation: the regression model explains 75% of the variation in y; 25% remains unexplained
  4. 4
    For the square root: r = \sqrt{0.75} \approx 0.866 (if positive association)

Answer

R^2 = 0.75. The model explains 75% of variation in y.
R^2 = 1 - SSE/SST measures the proportion of total variation in y explained by the model. SST = total variation; SSE = residual (unexplained) variation; SSR = explained variation. R^2 = r^2 for simple linear regression.

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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

Example 2

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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Background Knowledge

These ideas may be useful before you work through the harder examples.

correlationlinear regression lsrlresiduals