Practice R-Squared (Coefficient of Determination) in Statistics

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

R-squared (the coefficient of determination) is the proportion of variance in the dependent variable that is explained by the independent variable(s) in a regression model. It ranges from 0 to 1, where 0 means the model explains none of the variability and 1 means it explains all of it.

R2=0.80R^2 = 0.80 means the model explains 80% of why YY values differ. The other 20% is unexplained variation. Higher R2R^2 = better predictions.

Showing a random 20 of 76 problems.

Example 1

easy
R2=0.25R^2 = 0.25. Express the explained variation as a percent.

Example 2

medium
Total variation in yy is 200200; SSR (sum of squared residuals) is 5050. Find R2R^2.

Example 3

medium
As R2R^2 increases from 00 to 11, residuals tend to do what?

Example 4

easy
If R2=0R^2 = 0, what does that mean about the regression?

Example 5

medium
R2=0.49R^2=0.49 and the regression slope is positive. Find rr including its sign.

Example 6

hard
A regression has R2=0.999R^2 = 0.999 on the training data but predicts poorly on new data. What problem is most likely?

Example 7

easy
A correlation is r=0.6r = 0.6. Find R2R^2.

Example 8

medium
R2=0.04R^2 = 0.04. Find ∣r∣|r|.

Example 9

easy
r=0r = 0. Find R2R^2.

Example 10

easy
R2R^2 cannot be negative. True or false?

Example 11

hard
Total sum of squares is 400400; SSR =100= 100. Find R2R^2.

Example 12

medium
Total variation in yy is SStot=200SS_{\text{tot}} = 200. Residual variation is SSres=50SS_{\text{res}} = 50. Compute R2R^2.

Example 13

easy
A correlation is r=0.8r = 0.8. Find R2R^2.

Example 14

easy
r=1r = 1. What is R2R^2?

Example 15

hard
A regression model has R2=0.85R^2 = 0.85. Interpret this value.

Example 16

medium
Model A: R2=0.72R^2 = 0.72. Model B: R2=0.48R^2 = 0.48. By how many percentage points does A explain more variance?

Example 17

medium
r=0.6r = 0.6 on a sample of n=30n=30 pairs. Find R2R^2.

Example 18

challenge
Two simple regressions: Model P has r=0.6r=0.6, Model Q has r=0.8r=0.8. By what factor does Model Q explain more variance than Model P?

Example 19

medium
A model has R2=0.95R^2=0.95. Explain why this alone does not guarantee good predictions on new data.

Example 20

hard
A linear model has R2=0.64R^2 = 0.64. What percentage of the variation is not explained by the model?
← Back to R-Squared (Coefficient of Determination) Worked Examples