Residuals Examples: 65 Problems with Answers

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

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

Concept Recap

A residual is the difference between an observed data value and the value predicted by a statistical model, calculated as $\text{residual} = y_{\text{observed}} - y_{\text{predicted}}$ . Positive residuals mean the model underestimated; negative residuals mean it overestimated.

If your model predicts 80 but the actual value is 85, the residual is +5. Residuals are 'leftovers' - what the model couldn't explain. Patterns in residuals reveal model problems.

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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: Residuals 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 residuals 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

Regression line

\hat{y} = 0.5x + 2

. For the point

(10, 9)

, compute the residual and explain its meaning.

Find the residual at (10, 9) on the line ŷ = 0.5x + 2

Answer

residual

= +2

; the actual

y

2

units above the line at

x = 10

First step

Predict

\hat{y} = 0.5(10) + 2 = 7

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Example 2

medium

Predict using

\hat{y} = 3x + 2

for

(5, 20)

. State the residual and whether the model over- or under-predicted.

Find the residual at (5, 20) on ŷ = 3x + 2; did the model over- or under-predict?

Example 3

hard

Four points have residuals

+5, -4, +2, r_4

. Their sum is

0

. Find

r_4

and explain.

Example 4

hard

Two regressions: residual SE

s = 1.0

s = 5.0

for the same response variable. Which has tighter predictions, and why?

Example 5

challenge

An influential observation has very high leverage (extreme

x

) and a small residual on the fitted line. Why is it still influential, despite the small residual?

Example 6

medium

Using

\hat{y} = 0.5x + 2

, complete the residual table for

(2,3),(4,5),(6,4),(8,7)

Complete the residual table for points A–D on the line ŷ = 0.5x + 2

Example 7

hard

Using

\hat{y} = x + 2

, compute SSR for

(1,4),(2,3),(3,6),(4,5)

Compute SSR for the four points A–D on the line ŷ = x + 2

Example 8

hard

A point has residual

+10

and lies far from the rest of the data. Why is it called influential?

Example 9

hard

A regression model predicts

\hat{y} = 30

for a data point, but the actual value is

y = 35

. Calculate the residual and interpret it.

Example 10

hard

A residual plot shows a clear U-shaped pattern. What does this indicate about the regression model?

Practice Problems

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

Example 1

easy

Observed value 50, predicted value 47. Find the residual.

Example 2

easy

Observed 30, predicted 35. Find the residual.

Example 3

easy

A residual of

+8

means the model did what?

Example 4

easy

A residual of

-4

means the model did what?

Example 5

easy

Predicted

\hat{y}=12

and observed

y=12

. Find the residual.

Example 6

easy

Using

\hat{y}=2x+1

, find the residual at the point

(3, 10)

Find the residual at the point (3, 10) on the line ŷ

Example 7

easy

A residual is the difference between which two values?

Example 8

easy

Using

\hat{y}=3x-2

, find the residual at

(4, 8)

Find the residual at (4, 8) on the line ŷ

Example 9

medium

For

\hat{y}=x+2

, find the residuals at

(1,4)

and

(5,5)

, then state which point the line fits worse.

Compute residuals at P₁(1,4) and P₂(5,5); which point fits worse?

Example 10

medium

Why does the sum of all residuals from a least-squares line equal zero?

Example 11

medium

Residual plot of a linear fit shows a clear funnel shape (spread grows with x). What does this reveal?

Example 12

medium

Using

\hat{y}=2x+3

, the points are

(1,6),(2,8),(3,9)

. Find the largest residual.

Find the residuals at each point; identify the largest residual

Example 13

medium

How is a residual different from the model's 'error' term?

Example 14

medium

A residual plot scatters randomly around zero with no pattern. What does this suggest about the linear model?

Example 15

medium

For

\hat{y}=-x+10

, find the residual at

(2, 5)

and state over- or under-estimate.

Find the residual at (2, 5) on ŷ

Example 16

medium

Two models fit the same point

(4, 20)

. Model A predicts 18, Model B predicts 23. Which model fits this point better?

Example 17

medium

For

\hat{y}=4x-1

, find the residual at

(2, 6)

and state whether the point lies above or below the line.

Find the residual at (2, 6); is the point above or below the line?

Example 18

challenge

Dataset

(1,3),(2,5),(3,4)

fit by

\hat{y}=x+2

. Compute the sum of squared residuals (SSR).

Compute residuals at A, B, C and find the sum of squared residuals (SSR)

Example 19

challenge

A line gives residuals

+4, -1, -3

at three points. Verify these come from a least-squares fit, and find the missing fourth residual if the four must sum to zero.

Example 20

challenge

For

\hat{y}=2x+1

(1,3),(2,6),(3,7)

, find the point with the largest squared residual and its contribution to SSR.

Find the point with the largest squared residual and its SSR contribution

Example 21

easy

Observed

y = 22

, predicted

\hat{y} = 18

. Find the residual.

Example 22

easy

Observed

y = 6

, predicted

\hat{y} = 11

. Find the residual.

Example 23

easy

Regression line:

\hat{y} = 2x + 3

. Observed point

(4, 13)

. Find the residual.

Find the residual at (4, 13) on the line ŷ

Example 24

easy

Regression line

\hat{y} = -x + 10

. Observed

(3, 5)

. Find the residual.

Find the residual at (3, 5) on the line ŷ

Example 25

medium

The sum of residuals for the least-squares regression line is always what number?

Example 26

medium

For five data points the residuals are

+2, -3, +1, -1, +x

. Find

x

Example 27

medium

A residual plot shows a clear curved (U-shaped) pattern. What does this tell you about the regression?

Example 28

medium

A residual plot shows residuals fanning out (increasing spread as

x

increases). What is this pattern called?

Example 29

medium

Regression line:

\hat{y} = 4x - 1

. Three observed points are

(1, 5), (2, 6), (3, 12)

. Find each residual.

Compute the residual at each of the three points A, B, and C

Example 30

medium

If the largest residual is

+12

, what does this point look like on a scatterplot?

Example 31

medium

Why do we square residuals when fitting the least-squares line?

Example 32

hard

A residual plot shows residuals scattered randomly with no pattern. What does this suggest?

Example 33

hard

The residual at

(x, y) = (10, 50)

-8

. What is the predicted value

\hat{y}

x = 10

Example 34

hard

Sum of squared residuals is

80

for

n = 10

points (regression with

1

predictor). Estimate the residual standard error

s

Example 35

medium

\hat{y} = 0.8x + 4

. Observed

(20, 22)

. Find the residual.

Example 36

medium

For a dataset, the sum of squared residuals is

144

with

n = 10

. The total sum of squares is

400

. Compute

R^2

Example 37

easy

If a point lies exactly on the regression line, what is its residual?

Example 38

easy

Observed value

25

, predicted value

19

. Find the residual.

Example 39

easy

Observed

7

, predicted

11

. Find the residual.

Example 40

easy

Using

\hat{y} = 2x + 1

, find the residual at

(5, 12)

Find the residual at (5, 12) on the line ŷ

Example 41

easy

Using

\hat{y} = 3x - 5

, find the residual at

(2, 0)

Find the residual at (2, 0) on the line ŷ

Example 42

easy

Using

\hat{y} = x + 4

, find the residual at

(6, 12)

Find the residual at (6, 12) on the line ŷ

Example 43

medium

For

\hat{y} = 2x + 3

, find residuals at

(1, 6), (2, 8), (3, 10)

and identify the worst-fit point.

Find the residuals at A, B, C; all three are equal — identify the worst-fit point

Example 44

medium

A residual plot shows residuals randomly scattered around

0

. What does this suggest?

Example 45

medium

A residual plot shows a clear curved arc. What does this suggest about the linear model?

Example 46

medium

Using

\hat{y} = 4x - 1

, find the residual at

(3, 9)

Find the residual at (3, 9) on the line ŷ

Example 47

medium

For

\hat{y} = -x + 8

, find the residual at

(2, 7)

Find the residual at (2, 7) on the line ŷ

Example 48

hard

A residual plot shows a funnel widening to the right. What problem does this indicate?

Example 49

hard

For

\hat{y} = 3x

, residuals at

(1, 5)

and

(2, 4)

are computed. Which point has the larger absolute residual?

Example 50

hard

For

\hat{y} = 2x + 1

, three points are

(1,3),(2,4),(3,8)

. Which point has the largest residual?

Find residuals at A, B, C; which has the largest residual?

Example 51

medium

A line predicts

\hat{y}=50

x=10

. The observed

y

46

. Find the residual.

Example 52

hard

Two regression models give SSR values of

200

and

90

on the same data. Which fits better, and why?

Example 53

challenge

For the line

\hat{y} = 2x + 1

applied to

(1,4),(2,5),(3,8)

, compute the sum of residuals and explain why it is not zero.

Example 54

hard

Given

\hat{y} = 20 + 4 x

and actual data points (2, 30), (3, 33), (4, 35), find the residual for each point.

Example 55

hard

A model predicts values 18, 22, and 26 for three points, while the actual values are 20, 21, and 30. Find the residuals and identify which point has the largest underprediction.

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Related Concepts

Linear Regression

Background Knowledge

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

linear regression