Residual Plots (College Board AP® Statistics)

Revision Note

Author

Mark Curtis

Expertise

Maths

Residual plots

What is a residual plot?

Recall that a residual is the vertical distance between a data point and the regression line
A residual plot is a graph that shows all the residuals from a scatterplot
- The vertical axis shows the value of the residual
- The horizontal axis shows the $x$ -values

What is a residual plot used for?

When using the least-squares regression line, a residual plot gives a good indication as to whether the data points follow a linear model or not
- If the residuals vary randomly from positive to negative then this suggests a linear model is a good fit for the data
- If the residuals do not vary randomly (i.e. they follow a curve or form a pattern) then this suggests a linear model is not a good fit for the data
  - In this case, a non-linear (curved) model may be more appropriate

Worked Example

A scatterplot is shown below. The equation of the least-squares regression line is $\hat{y} = 2.4 + 2.8 x$ , where $\hat{y}$ is the predicted $y$ -value. By constructing a residual plot, determine whether or not a linear model is appropriate for the data.

A scatterplot with points shown and a dashed regression line on a grid.

Answer:

First calculate the residuals (by finding the vertical distance of each data point above or below the regression line, $y - \hat{y}$ )

To find the $\hat{y}$ -values from the regression line, it is more accurate to substitute the $x$ -values into the equation of the line $\hat{y} = 2.4 + 2.8 x$ , rather than read off their values from the graph

The residual at $x = 0$ is $4 - (2.4 + 2.8 \times 0) = 1.6$

The residual at $x = 1$ is $2 - (2.4 + 2.8 \times 1) = - 3.2$ etc.

Scatterplot with a regression line with equation ŷ = 2.4 + 2.8x. Data points include he residuals: (+3.2), (+1.6), (–3.2), and (–1.6).

Then construct a residual plot by plotting the values of the residuals on the vertical axis and the $x$ -values on the horizontal axis

A residual plot of residuals on the vertical axis against x-values on the horizontal axis.

If the residual plot shows that residuals vary randomly from positive to negative (without following a curve or a pattern), then a linear model is a good fit

The residual plot shows that residuals appear to vary randomly which suggests a linear model is appropriate for the data

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