How to interpret a residual plot (AP Stats)

By Jude Wallis · Published July 28, 2026

A residual is the actual y value minus the value the line predicts. A residual plot graphs those residuals against x or the predicted values. If the dots scatter randomly around zero, a line fits the data. A clear curve means a line is the wrong model.

AP Statistics: Unit 5 (topics 5.4 Residuals, 5.5 Least-Squares Regression). Residual plots are Topic 5.4 in Unit 5 of the Fall 2026 AP Statistics course, where you calculate residuals as observed minus predicted and use their pattern to judge whether a linear model is appropriate.

How to interpret a residual plot

A residual plot helps you check one thing: is a straight line a good model for your data? You read it by studying the pattern of the dots around a horizontal line at zero.

Here is the quick version. If the dots are scattered randomly above and below zero with no clear shape, a line fits the data well. If the dots bend into a curve, a line is the wrong model. If the dots widen into a fan, the amount of scatter is not constant across the data.

Two symbols show up throughout this page. The value the regression line predicts is written $\hat{y}$ (y-hat), and the actual value you observed is written $y$ . A residual measures the gap between them.

Residuals explained: observed value minus predicted value

A residual is the observed value of the response variable minus the value the regression line predicts for it.

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

Suppose a line predicts a plant will be 12 cm tall, so $\hat{y} = 12$ , but the plant you measured is actually 14 cm tall, so $y = 14$ . The residual is $14 - 12 = 2$ cm. The line missed low by 2 cm for that plant.

Every point in your dataset has its own residual, one gap for each observation. A residual of zero means the line passed exactly through that point. Large residuals mark the points the line predicts poorly.

For the least-squares line, the residuals always add up to zero, so their mean is zero too. That is why the reference line on a residual plot sits at zero. To see where the line itself comes from, read how to find the least-squares regression line.

How to build a residual plot

A residual plot is a scatterplot of the residuals on the vertical axis against either the explanatory variable $x$ or the predicted values $\hat{y}$ on the horizontal axis. Both choices are standard, and for a simple linear regression they produce the same overall shape.

To build one by hand:

Fit the least-squares line and write down its equation.
For each data point, compute the predicted value $\hat{y}$ .
Subtract to get each residual, $y - \hat{y}$ .
Plot each residual above its $x$ value (or its $\hat{y}$ value).
Draw a horizontal reference line at zero.

Then you study the shape of the cloud of points around that zero line. The next sections cover what each shape means. You can generate a residual plot automatically with the regression calculator.

Random scatter means a line is a good model

The pattern you want to see is no pattern. When the residuals are scattered randomly above and below zero, with roughly even spread and no bending, the linear model is doing its job.

Apparent randomness in the residual plot confirms that the association between the two variables really is linear, so a straight line is an appropriate model. The line misses high about as often as it misses low.

Random scatter does not mean every residual is tiny. It means the misses have no leftover structure, so there is nothing systematic left for a different curve to capture.

One caution with small datasets: with only a handful of points it is hard to judge randomness, so read the plot together with the original scatterplot rather than on its own. The more points you have, the more trustworthy the pattern you see.

Patterns that signal a problem

Two shapes are worth watching for, and they point to different problems. Curvature tells you a straight line is the wrong model for the trend. A fan shape tells you the spread of the data is not constant, even when the line still describes the average trend.

Curvature. If the residuals bend into a U shape or an upside-down U, the linear model is not the most appropriate model for the data. The line runs too high in some regions and too low in others in a systematic way, which means the true relationship curves. Your eye can miss a gentle curve in the original scatterplot, but a residual plot stretches it out so it is easy to spot.

A fan or megaphone shape. If the residuals cluster tightly near zero at one end and spread far from zero at the other, the amount of scatter is not constant across the data. The line may still capture the average trend, but predictions are much less reliable where the fan is wide than where it is narrow.

In both cases the residual plot has revealed structure that the fitted line alone would hide. A high correlation does not rule either problem out, so the residual plot is worth checking even when the line looks like a strong fit.

Positive and negative residuals: underprediction and overprediction

The sign of a residual tells you which way the line missed.

A positive residual means the observed value is above the line, so the model underpredicts (underestimates) the response for that point. The actual $y$ came in higher than $\hat{y}$ .

A negative residual means the observed value is below the line, so the model overpredicts (overestimates) the response. The actual $y$ came in lower than $\hat{y}$ .

On the residual plot, positive residuals sit above the zero line and negative residuals sit below it. A point that lands right on the zero line was predicted exactly.

Residual plots on the AP Statistics exam

Residuals are Topic 5.4 in Unit 5, Regression Analysis, of the Fall 2026 AP Statistics course. Unit 5 makes up 10 to 20 percent of the multiple-choice section.

For this topic you are expected to calculate residuals as observed minus predicted, interpret the sign as underprediction or overprediction, and describe the form of an association using a residual plot. Random scatter supports a linear model, and curvature suggests a line is not the most appropriate model.

A common exam twist reverses the calculation. Instead of computing a residual, you are given the residual and the regression equation and asked to recover the observed value or make a prediction. The second worked example below walks through that version. For the full unit, see Unit 5: Regression Analysis or the official AP Statistics home page.

Computing residuals and describing the plot

A small dataset has five points: (1, 2), (2, 5), (3, 5), (4, 8), (5, 10). The least-squares regression line for this data is $\hat{y} = 0.3 + 1.9x$ . Find the residual for each point and describe what the residual plot shows.

Recall the definition. A residual is the observed $y$ minus the predicted $\hat{y}$ (y-hat), where $\hat{y}$ is the value the line gives at that $x$ .
At $x = 1$ : predicted $\hat{y} = 0.3 + 1.9(1) = 2.2$ , so the residual is $2 - 2.2 = -0.2$ .
At $x = 2$ : predicted $\hat{y} = 0.3 + 1.9(2) = 4.1$ , so the residual is $5 - 4.1 = 0.9$ .
At $x = 3$ : predicted $\hat{y} = 0.3 + 1.9(3) = 6.0$ , so the residual is $5 - 6.0 = -1.0$ .
At $x = 4$ : predicted $\hat{y} = 0.3 + 1.9(4) = 7.9$ , so the residual is $8 - 7.9 = 0.1$ .
At $x = 5$ : predicted $\hat{y} = 0.3 + 1.9(5) = 9.8$ , so the residual is $10 - 9.8 = 0.2$ .
Check the total: $-0.2 + 0.9 - 1.0 + 0.1 + 0.2 = 0$ , which matches the rule that least-squares residuals sum to zero.
Plot the residuals ( $-0.2$ , $0.9$ , $-1.0$ , $0.1$ , $0.2$ ) against $x$ . They fall on both sides of the zero line and do not bend into a curve, so a straight line is a reasonable model for this data.

The residuals are $-0.2$ , $0.9$ , $-1.0$ , $0.1$ , and $0.2$ . They scatter above and below zero with no curved pattern, so the linear model fits. The point at $x = 3$ has the largest negative residual ( $-1.0$ ), meaning the line overpredicts there, while the point at $x = 2$ has the largest positive residual ( $0.9$ ), meaning the line underpredicts there.

Working backward from a residual to the observed value

Using the same line $\hat{y} = 0.3 + 1.9x$ , a new observation at $x = 6$ has a residual of $-0.5$ . What was the observed $y$ value, and does the model over- or underpredict there?

Find the predicted value at $x = 6$ : $\hat{y} = 0.3 + 1.9(6) = 0.3 + 11.4 = 11.7$ .
Rearrange the definition. Since residual $= y - \hat{y}$ , the observed value is $y = \hat{y} + \text{residual}$ .
Substitute the numbers: $y = 11.7 + (-0.5) = 11.2$ .
Read the sign. The residual is negative, so the observed value sits below the line and the model overpredicts.

The observed value was $y = 11.2$ . Because the residual is negative, the model overpredicts (overestimates) the response at $x = 6$ by 0.5.

Frequently asked questions

What does a residual plot tell you?

It tells you whether a straight line is an appropriate model for the data. Random scatter around zero supports a linear model, while a curved pattern suggests the relationship is not linear and a line is not the best fit.

What does a good residual plot look like?

The residuals are scattered randomly above and below the horizontal line at zero, with no curve and roughly even spread. There is no leftover pattern for another model to explain.

Do residuals always add up to zero?

For the least-squares regression line, yes. The residuals sum to zero and their mean is zero, which is why the reference line on a residual plot is drawn at zero.

Should I plot residuals against x or against the predicted values?

Either works. A residual plot is a scatterplot of residuals versus the explanatory variable $x$ or versus the predicted values $\hat{y}$ , and for a simple linear regression the two versions have the same shape.