AP Statistics · Topic 5.4 · Unit 5

AP Stats 5.4: Residuals and Residual Plots

By Jude Wallis · Published July 28, 2026

Topic 5.4 in the Fall 2026 AP Statistics course is residuals. A residual is observed minus predicted, y minus y-hat. A positive residual means the model underpredicted; a negative one means it overpredicted. A residual plot with random scatter supports a linear model; curvature warns against one.

AP Statistics: Unit 5 (topics 5.4). Topic 5.4 (Residuals) is part of Unit 5 in the Fall 2026 CED. It covers computing residuals, reading their sign, and using residual plots to judge whether a linear model fits.

What a residual is

A residual measures how far off a single prediction was. The AP course defines it as the observed response value minus the predicted response value for a given $x$ :

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

In words, residual equals observed minus predicted. Because it is a signed distance, a residual is measured in the same units as the response variable. Every point in the data has its own residual, and the least-squares line is the line that makes the sum of the squared residuals as small as possible.

Reading the sign

The sign of a residual tells you which way the model missed. A positive residual means the observed value sits above the line, so the model underpredicted (underestimated) the response. A negative residual means the observed value sits below the line, so the model overpredicted (overestimated) the response.

The size tells you by how much. A residual of $+2$ thousand dollars means the actual value came in two thousand dollars above what the line predicted. Reading a residual well means naming both its direction and its amount in context, not just its number.

Residual plots and checking the model

A residual plot is a scatterplot of the residuals against the predicted values (or against the explanatory variable). Its job is to check whether a straight line was the right choice in the first place.

The rule is about pattern. Apparent randomness in the residual plot, with points scattered above and below zero and no clear shape, confirms that the linear form fits and that a simple linear model is appropriate. Curvature in the residual plot suggests the opposite: a line is not the most appropriate model, and the underlying relationship bends. You never judge a model from the correlation alone, because the residual plot is the direct check; the residual plot guide shows worked examples of both cases.

How residuals connect to the rest of Unit 5

Residuals are the hinge of Unit 5. A single residual measures one point's miss, while the residual plot gathers every miss into one picture that tells you whether a line belongs on the data at all. The AP course says to fit a linear model only when the data show a linear trend, and the residual plot is how you make that call.

Residuals also define the line itself. The least-squares regression line in topic 5.5 is the one line that makes the sum of the squared residuals as small as possible, so understanding residuals here is what makes least-squares regression more than a calculator button.

Compute a residual and interpret its sign

A least-squares model predicts a used car's price (in thousands of dollars) from its age in years: $\hat{y} = 30 - 2.5x$ . The model was built from cars aged 1 to 8 years. A particular 4-year-old car actually sold for 22 thousand dollars. Find the residual and interpret it.

Find the predicted price at $x = 4$ . $\hat{y} = 30 - 2.5 \times 4 = 30 - 10 = 20$ thousand dollars.
Find the residual, observed minus predicted. $\text{residual} = y - \hat{y} = 22 - 20 = 2$ thousand dollars.
Interpret the sign. The residual is positive, so the observed price is above the line and the model underpredicted this car's price.

The residual is $+2$ thousand dollars. Because it is positive, the model underpredicted: this 4-year-old car sold for two thousand dollars more than the line predicted.

Frequently asked questions

Does a positive residual mean the point is above or below the line?

Above. A positive residual means the observed $y$ is larger than the predicted $\hat{y}$ , so the point sits above the regression line and the model underpredicted. A negative residual sits below the line, where the model overpredicted.

If I know the residual, can I find the observed value?

Yes. Rearrange residual equals observed minus predicted into observed equals predicted plus residual. For example, if the line predicts 15 and the residual is $-3$ , the observed value is $15 + (-3) = 12$ , and the negative residual tells you the model overpredicted.