Residuals and residual plots

About this lesson

Grade 8 › Associations in data

To save a PDF, choose “Save as PDF” in the print dialog.

Residuals and residual plots

Warm-up

Three quick questions from earlier lessons.

Discover

The model predicts 70 for a student who actually scored 76. Is the student above or below the line, and by how much?

  1. 6 above
  2. 6 below

Key idea

Residuals show what the line misses.

Residual = observed −- predicted =y−y^= y - \hat{y}. Positive: point above the line. A residual plot (residuals against xx) should look like random scatter around 0; a curve or a fan shape means a linear model is not appropriate.

Worked example

y^=12+3x\hat{y} = 12 + 3x; the data point is (5,30)(5, 30).

y^=27\hat{y} = 27
Predicted.
Residual =30−27=3= 30 - 27 = 3
Observed − predicted.
The point is 3 above the line
Positive.

Your turn, guided

Your turn: same line, point (4,20)(4, 20).

Residual −4-4
20 − 24.

Practice ★

  1. y^=30+3x\hat{y} = 30 + 3x. A data point is (7,46)(7, 46). Find its residual.
  2. y^=36+4x\hat{y} = 36 + 4x. A data point is (3,41)(3, 41). Find its residual.
  3. y^=32+6x\hat{y} = 32 + 6x. A data point is (7,70)(7, 70). Find its residual.

Watch out!

Practice ★★

  1. The residual plot of a linear model fitted to 56 points shows a clear U-shaped curve. What does that tell you?
    1. A straight line is not a good model: the relationship is curved
    2. The model is good, because residuals are both positive and negative
    3. There is one outlier
  1. A point has residual +8+8. Where is it relative to the regression line?
    1. 8 units above the line: the model underpredicts it
    2. 8 units below the line
    3. On the line

Exit ticket

Three questions to check you have got it.

  1. y^=30+3x\hat{y} = 30 + 3x. A data point is (7,46)(7, 46). Find its residual.
  2. y^=36+4x\hat{y} = 36 + 4x. A data point is (3,41)(3, 41). Find its residual.
  3. y^=32+6x\hat{y} = 32 + 6x. A data point is (7,70)(7, 70). Find its residual.

Summary card

Residuals

  • e=y−y^e = y - \hat{y}; positive is above the line.
  • Residual plot: no pattern means a linear model is appropriate.

Watch out: observed minus predicted, in that order.

End of the lesson

Next lesson: Interpreting a regression model: slope, r², extrapolation Back to the chapter

Answers: Residuals and residual plots

A

  1. −5-5
  2. −7-7
  3. −4-4

B

  1. A straight line is not a good model: the relationship is curved

C

  1. 8 units above the line: the model underpredicts it

D

  1. −5-5
  2. −7-7
  3. −4-4

CC BY-SA 4.0