R-Squared (Coefficient of Determination) Examples

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

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.

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

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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-Squared (Coefficient of Determination) asks whether the same cases connect two variables or groups in a pattern that can be described carefully.

Common stuck point: Students often know a procedure related to r-squared (coefficient of determination) but skip the recognition step: Am I studying a relationship between variables, and have I separated association from causation? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I studying a relationship between variables, and have I separated association from causation?

Worked Examples

Example 1

medium

R^2 = 0.81

for predicting weight from height. Interpret what the remaining

19\%

represents.

Answer

19\%

of the variation in weight is NOT explained by height — it is due to other factors and natural variability.

First step

R^2

tells what fraction of

y

-variation is explained by

x

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Example 2

medium

A regression of monthly sales ($) on advertising spend ($) gives

R^2 = 0.55

. Write a one-sentence interpretation.

Example 3

hard

Suppose

r = 0.30

for one dataset and

r = -0.30

for another. Compare their

R^2

values and the direction of the relationship.

Example 4

challenge

Two studies report

R^2 = 0.20

n = 100{,}000

and

R^2 = 0.85

n = 8

. Which model is 'better'?

Example 5

medium

Total variation in

y

200

; SSR (sum of squared residuals) is

50

. Find

R^2

Example 6

hard

Why does adding more predictors to a regression never decrease

R^2

Example 7

challenge

A small data set has

\bar{y} = 5

and observed

y

values

3, 5, 7, 5

. The model predicts

\hat{y}_i = 4, 5, 6, 5

. Find

R^2

Example 8

hard

A regression model has

R^{2} = 0.85

. Interpret this value.

Example 9

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

easy

A correlation is

r = 0.9

. Find

R^2

Example 2

easy

R^2 = 0.64

. What percent of the variation in y is explained by the model?

Example 3

easy

R^2

ranges between which two values?

Example 4

easy

R^2 = 0.80

. What proportion of variation is unexplained?

Example 5

easy

If a model explains all the variation in y, what is

R^2

Example 6

easy

If a model explains none of the variation in y, what is

R^2

When the model explains none of the variation, R²

Example 7

easy

r = -0.7

. Find

R^2

Example 8

easy

R^2 = 0.25

. Express the explained variation as a percent.

Example 9

medium

Model A has

R^2=0.81

; Model B has

R^2=0.49

on the same data. Which explains more variance, and by how many percentage points?

Example 10

medium

A regression has

R^2=0.36

. Find the magnitude of the correlation coefficient

|r|

Example 11

medium

A model has

R^2=0.95

. Explain why this alone does not guarantee good predictions on new data.

Example 12

medium

Why is comparing

R^2

between two models built on completely different datasets misleading?

Example 13

medium

R^2=0.49

and the regression slope is positive. Find

r

including its sign.

Example 14

medium

A model explains 70% of variance in y. The total variance of y is 200. How much variance is explained, in the same units?

Example 15

medium

Adding more predictors to a regression raised

R^2

from 0.82 to 0.83. Why is this not strong evidence the new predictors help?

Example 16

medium

R^2=0.9

is reported but the residual plot shows a strong curved pattern. Should you trust the model? Why?

Example 17

medium

A regression reports

R^2=0.49

. A student says 'the model is 49% accurate.' Why is that interpretation wrong?

Example 18

challenge

A simple regression has

R^2=0.64

and a negative slope. State

r

, and the percent of variation left unexplained.

Example 19

challenge

Total variance of y is 50. After regression, the residual (unexplained) variance is 20. Find

R^2

Example 20

challenge

Two simple regressions: Model P has

r=0.6

, Model Q has

r=0.8

. By what factor does Model Q explain more variance than Model P?

Example 21

easy

A correlation is

r = 0.6

. Find

R^2

Example 22

easy

A correlation is

r = -0.5

. Find

R^2

Example 23

medium

R^2 = 0.72

. Write a one-sentence interpretation in context of predicting test scores from study hours.

Example 24

medium

Total variation in

y

SS_{\text{tot}} = 200

. Residual variation is

SS_{\text{res}} = 50

. Compute

R^2

Example 25

medium

Total variation

SS_{\text{tot}} = 400

, residual

SS_{\text{res}} = 320

. Compute

R^2

Example 26

medium

R^2 = 0.64

. If correlation

r

is negative, what is

r

Example 27

medium

A model has

R^2 = 0.04

. Which of the following best describes the fit: strong, moderate, weak, or no linear fit?

Example 28

medium

Total variation in

y

1000

. The model leaves

250

unexplained. What is

R^2

as a percent?

Example 29

hard

A regression has

r = 0.7

but a clear curved residual plot. Why might using

R^2 = 0.49

be misleading?

Example 30

hard

R^2 = 0.90

for predicting house price from square footage. Is it valid to say square footage causes price differences?

Example 31

hard

A linear model has

R^2 = 0.36

. By how many percentage points does adding a predictor (giving

R^2 = 0.45

) increase the explained variation?

Example 32

hard

A regression line is

\hat{y} = 2 + 0.5x

and

R^2 = 0.64

. If we instead predicted

y

using only the mean

\bar{y}

, the prediction errors would be larger by what factor in sum-of-squares?

Example 33

medium

R^2

went from

0.25

0.81

after fitting a curve instead of a line. Did the new model improve the explained variation? By how much (in percentage points)?

Example 34

medium

R^2 = 0.16

, and the residual sum of squares is

84

, what is the total sum of squares?

Example 35

hard

A linear regression of

y

x

gives

R^2 = 0.49

. If we instead regress

x

y

, what is

R^2

Example 36

easy

A correlation is

r = 0.8

. Find

R^2

Example 37

easy

R^2 = 0.49

. What percent of variation in

y

is explained?

Example 38

easy

R^2 = 0.36

. What proportion of variation is unexplained?

Example 39

easy

r = 0

. Find

R^2

Example 40

easy

r = -0.5

. Find

R^2

Example 41

medium

R^2 = 0.81

. Find

|r|

Example 42

medium

Model A:

R^2 = 0.72

. Model B:

R^2 = 0.48

. By how many percentage points does A explain more variance?

Example 43

medium

r = 0.6

on a sample of

n=30

pairs. Find

R^2

Example 44

medium

R^2 = 0.04

. Find

|r|

Example 45

hard

Total sum of squares is

400

; SSR

= 100

. Find

R^2

Example 46

hard

A regression has

R^2 = 0.999

on the training data but predicts poorly on new data. What problem is most likely?

Example 47

hard

Two regression models are fit on different data sets. Model 1 has

R^2 = 0.9

; Model 2 has

R^2 = 0.7

. Can we conclude Model 1 fits its data more accurately?

Example 48

medium

|r| = 0.7

. Find

R^2

Example 49

hard

SST

= 500

; SSR

= 75

. Find the percent of variation explained.

Example 50

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 51

hard

A linear model has

R^{2} = 0.64

. What percentage of the variation is not explained by the model?

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Related Concepts

Linear Regression Standard Deviation

Background Knowledge

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

linear regressionstandard deviation intro