AP Statistics Unit 5: Residuals and residual plots

Residuals measure how far each actual data point falls from the regression line, and residual plots reveal whether a linear model fits the data appropriately. On the AP Statistics exam, these concepts can appear in the Section I three-question set on regression analysis and in Section II free-response questions (for example Question 2 or 4).

What the exam asks

Expect to be tested on four specific skills using tables, scatterplots, or computer regression output:

Calculate a residual ( $y - \hat{y}$ ) or work backward to find an actual response value $y$ given a residual and a regression equation.
Interpret a residual in context, stating whether the linear model overpredicted or underpredicted the response variable.
Identify a residual plot and its axes.
Justify whether a linear model is appropriate by examining the pattern of points in a residual plot.

Core ideas

Calculating residuals

A residual is the vertical distance between an observed data point $(x, y)$ and the value predicted by the regression model $(x, \hat{y})$ .

\text{residual} = y - \hat{y}

$y$ is the actual (observed) value of the response variable from the dataset.
$\hat{y}$ is the predicted value of the response variable, calculated by substituting $x$ into the regression equation $\hat{y} = a + bx$ .

By rearranging the formula, you can also find the actual observed value if you know the predicted value and the residual:

y = \hat{y} + \text{residual}

Interpreting residuals

The sign of a residual tells you whether the actual data point lies above or below the regression line:

Sign of residual	Position of point relative to line	Comparison of $y$ and $\hat{y}$	What the model did
Positive ( $y - \hat{y} > 0$ )

Residuals and residual plots

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What the exam asks

Core ideas

Calculating residuals

Interpreting residuals