AP Statistics 5.4 Residuals Study Notes - New Syllabus

5.4.C.1 Residual Plots

After fitting a linear regression model, statisticians use a residual plot to evaluate how well the regression line fits the observed data.

A residual plot is a scatterplot that displays the residuals for each observation against either:

• The predicted response values (\( \hat{y} \)), or
• The explanatory variable values (\( x \)).

Each point on a residual plot represents one observation from the original data set.

The vertical coordinate is the residual, which is calculated as

\( \mathrm{Residual}=y-\hat{y} \)

The horizontal coordinate is either the predicted response value \( \hat{y} \) or the explanatory variable value \( x \).

Unlike the original scatterplot, a residual plot focuses on the prediction errors made by the regression model rather than the original data values.

5.4.C.2 Using Residual Plots to Evaluate a Linear Regression Model

Residual plots help statisticians determine whether a linear regression model is appropriate for the observed data.

Because residuals measure the prediction errors of the regression model, the pattern of points in a residual plot reveals whether the linear model adequately describes the relationship.

A residual plot should always be examined after fitting a regression line because the value of the correlation coefficient alone cannot determine whether a linear model is appropriate.

Residual plots are useful for:

• Evaluating the appropriateness of a linear regression model.
• Detecting non-linear relationships.
• Identifying unusual observations or outliers.
• Checking whether the residuals display any systematic pattern.

Important AP Exam Notes

• A residual plot evaluates the fit of a regression model, not the strength of the association.
• The points in a residual plot should be examined for patterns.
• A clear pattern in the residuals often suggests that the linear model does not adequately describe the data.
• Residual plots provide evidence about whether a linear regression model is appropriate.

5.4.C.3 Interpreting Residual Plots for a Linear Regression Model

A residual plot helps determine whether a linear regression model is appropriate for a set of bivariate quantitative data.

The linear regression model should be used only if the relationship between the explanatory and response variables is approximately linear.

If the residual plot shows an apparently random scatter of points around the horizontal line at \( \mathrm{Residual}=0 \), then the linear model is appropriate.

A random pattern indicates that the regression model has accounted for the linear trend in the data and that no systematic pattern remains in the residuals.

Random Residual Plot

• The residuals are randomly scattered above and below \( \mathrm{Residual}=0 \).
• No obvious pattern or trend is present.
• The spread of the residuals is approximately constant.
• This provides evidence that a simple linear regression model is appropriate.

5.4.C.4 Curvature in a Residual Plot

If the residual plot displays a noticeable curved pattern, the residuals are not randomly distributed.

This indicates that the linear regression model has failed to capture an important feature of the relationship between the variables.

A curved residual plot suggests that the relationship between the variables is non-linear, and a different model may better describe the data.

Curved Residual Plot

• The residuals form a curved or systematic pattern.
• The points are not randomly scattered around zero.
• The relationship between the variables is likely non-linear.
• A simple linear regression model is not the most appropriate model.

Important AP Exam Notes

• A random residual plot confirms that a linear model is appropriate.
• A curved residual plot suggests that the relationship is non-linear.
• Residual plots should always be interpreted together with the original scatterplot.
• When describing a residual plot, focus on whether the residuals are random or show a systematic pattern.