R-Squared (Coefficient of Determination) Examples in Statistics

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

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

Concept Recap

The proportion of variance in the dependent variable that is explained by the independent variable(s) in a regression model, ranging from 0 to 1.

R^2 = 0.80 means the model explains 80% of why Y values differ. The other 20% is unexplained variation. Higher R^2 = better predictions.

Read the full concept explanation →

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-squared is the proportion of variability in Y that is explained by the regression model. An R-squared of 0.80 means 80% of the variation is accounted for.

Common stuck point: Students think R-squared tells you if the model is correct. A high R-squared can result from overfitting or a spurious relationship — always check residuals too.

Worked Examples

Example 1

hard
A regression model has R^2 = 0.85. Interpret this value.

Solution

  1. 1
    Step 1: R^2 = 0.85 means 85% of the variability in the response variable is explained by the linear relationship with the explanatory variable.
  2. 2
    Step 2: The remaining 15% is due to other factors or random variation.
  3. 3
    Step 3: An R^2 of 0.85 indicates a strong linear fit.

Answer

85% of the variation in y is explained by the linear model.
R^2 (coefficient of determination) ranges from 0 to 1. Higher values mean the model explains more variability. It is the square of the correlation coefficient r.

Example 2

hard
If the correlation coefficient is r = -0.9, find R^2 and interpret both values.

Practice Problems

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

Example 1

hard
Two models are compared: Model A has R^2 = 0.72 and Model B has R^2 = 0.58. Which model provides a better fit and why?

Example 2

hard
A linear model has R^2 = 0.64. What percentage of the variation is not explained by the model?
← Back to R-Squared (Coefficient of Determination) Practice Mode

Related Concepts

Linear Regression

Background Knowledge

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

linear regressionvariance