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: $r^2$ is the proportion of variation in $y$ accounted for by the linear relationship with $x$ — the square of the correlation.

Common stuck point: The procedure for coefficient of determination is the easy part; the trap is reporting $r$ when the question asks for $r^2$ . Asking "Am I reporting the fraction of $y$ 's variation explained by the linear model (a 0-to-1 number), not the slope or the correlation's sign?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I reporting the fraction of $y$ 's variation explained by the linear model (a 0-to-1 number), not the slope or the correlation's sign?

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.

Answer

R^2 = 0.75

. The model explains 75% of variation in y.

First step

R^2 = 1 - \frac{SSE}{SST} = 1 - \frac{125}{500} = 1 - 0.25 = 0.75

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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

medium

A model has

SST=1200

and

r^2=0.7

. Find the explained sum of squares

SSR

and the residual

SSE

Example 4

medium

A model on

n=20

points has

SST=500

and

r^2=0.84

. Compute

SSE

and the residual standard deviation

s=\sqrt{SSE/(n-2)}

Example 5

hard

Five

y

-values are

4,5,7,8,11

with

\bar y=7

. The residuals from a fitted line are

1,-1,0,-1,1

. Compute

r^2

Example 6

challenge

A simple linear regression has

r^2=0.64

with

n=10

points. Compute the test statistic

t=r\sqrt{(n-2)/(1-r^2)}

for testing

H_0:\rho=0

assuming positive slope.

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.

Example 3

easy

The correlation is

r = 0.7

. Compute

r^2

Example 4

easy

r^2 = 0.64

and the slope is positive. Find

r

Example 5

easy

Interpret

r^2 = 0.85

in words.

Example 6

easy

What is the range of possible values for

r^2

Example 7

easy

r^2 = 1

, what does that say about the data and the line?

Example 8

easy

r^2 = 0

, how much of the variation in

y

does the line explain?

Example 9

easy

True or false: a high

r^2

guarantees the linear model is the correct model.

Example 10

easy

r^2 = 0.36

for a regression with a negative slope. Find

r

Example 11

medium

Total variation in

y

(SST) is

200

and residual variation (SSE) is

50

. Compute

r^2

Example 12

medium

Explained variation is

180

out of a total variation of

240

. Compute

r^2

Example 13

medium

A report states

r = 0.5

. A student claims '

50\%

of the variation is explained.' Correct them.

Example 14

medium

r^2 = 0.81

. A student says '

81\%

of the data points fall on the line.' What is wrong, and what is correct?

Example 15

medium

Model A has

r^2 = 0.9

but a strongly curved residual plot. Model B has

r^2 = 0.8

with random residuals. Which has the better-justified linear fit?

Example 16

medium

A regression has SST

= 500

and

r^2 = 0.6

. Find the residual sum of squares SSE.

Example 17

medium

r^2

rises from

0.49

0.64

, how does the magnitude of

r

change (slope stays positive)?

Example 18

medium

A linear model gives

r^2 = 0.95

, but extrapolating it predicts a negative weight for a real object. What does this reveal?

Example 19

medium

A regression has explained variation

SSR = 90

and residual variation

SSE = 60

. Compute

r^2

Example 20

challenge

Total variation SST

= 1000

. Adding a predictor reduces residual variation from SSE

= 400

to SSE

= 250

. By how much does

r^2

increase?

Example 21

challenge

Two variables have

r^2 = 0

yet are perfectly related by

y = x^2

over

[-2,2]

. Explain how both can be true.

Example 22

challenge

A model on

n=10

points has SST

= 360

and

r^2 = 0.75

. Find SSE and then the residual standard deviation

s = \sqrt{\frac{SSE}{n-2}}

Example 23

easy

Given

r=0.6

, compute

r^2

Example 24

easy

r^2=0.49

and the slope is positive, find

r

Example 25

easy

SST=400

SSE=80

. Compute

r^2

Example 26

easy

For a regression on a dataset,

r=-0.9

. Compute

r^2

Example 27

easy

A student says

r=0.4

means '40% of the variation is explained.' Correct them.

Example 28

medium

A regression gives

r^2=0.81

and a negative slope. Find

r

Example 29

medium

Two datasets have

r^2=0.64

and

r^2=0.49

. Which has the stronger linear association?

Example 30

medium

SSE=120

and

r^2=0.4

. Find

SST

Example 31

medium

Adding a predictor changes

SSE

from

300

180

with

SST=600

. How much does

r^2

rise?

Example 32

medium

The residuals from a line on

y

are exactly half the residuals from using

\bar y

. Find

r^2

Example 33

medium

r^2=0.04

. A student calls the relationship 'strong.' Correct them.

Example 34

medium

A least-squares line passes through every data point. State

r^2

and what that implies about

SSE

Example 35

medium

Data with strong curvature gives a regression line with

r^2=0.82

. Is the linear model appropriate? Justify.

Example 36

hard

SST=900

. The fitted line gives residual SS

SSE=144

. Compute

r^2

to two decimals.

Example 37

hard

For a regression with

n=12

points and

r^2=0.49

, find

SSE

when

SST=600

and the typical residual

s=\sqrt{SSE/(n-2)}

Example 38

hard

Explain why

r^2

is identical whether the regression line is

y

x

x

y

Example 39

hard

In a regression of

y

x

, doubling every

y

-value (no change to

x

) — what happens to

r^2

Example 40

challenge

In a multiple regression with

k=3

predictors and

n=20

R^2=0.7

. Compute the adjusted

R^2 = 1 - (1-R^2)\frac{n-1}{n-k-1}

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Related Concepts

Correlation Least Squares Regression Line Residuals Inference for Regression

Background Knowledge

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

correlationlinear regression lsrlresiduals