What a Residual Plot Adds
In “What to Do With an Unusual Point,” you saw why an observation that stands out deserves investigation rather than automatic removal. A residual plot gives another view of a regression: instead of plotting the observed response, it plots each residual against the explanatory variable or the fitted response. This can make a pattern in the model’s errors easier to see than it is in the original scatterplot.
A residual is the observed response minus the response predicted by the regression line. A positive residual means the observation is above the line; a negative residual means it is below. In a residual plot, the horizontal reference line at residual 0 marks observations whose responses equal their predictions.
A useful residual plot generally looks like a horizontal band of points scattered around zero. The points should not trace an obvious shape, and the vertical spread should be reasonably similar across the horizontal axis. This is not a requirement that all residuals be tiny or perfectly balanced. Random data produce variation. The aim is to look for a systematic feature, not to demand a perfect-looking plot.
Reading the Main Features
Scatter around zero: A roughly pattern-free band supports using a straight line to describe the association in the data. It suggests that the linear model has not left behind an obvious systematic pattern. It does not prove that the model is appropriate for every purpose, nor does it establish that the variables have a causal relationship.
Changing vertical spread: If the residuals form a funnel or fan—narrow in one part of the plot and wider in another—the variability of responses around the line may not be similar across the range of the explanatory variable. This is a warning about the model’s residual pattern. Describe what the plot shows rather than claiming that the plot alone identifies the reason for the changing spread.
A curved pattern: A visible arc or other systematic bend means residuals are not scattered randomly around zero. It signals that the straight-line model may be missing a feature of the relationship. The next tutorial examines what a curved residual pattern says about a nonlinear relationship.
Clusters or other structure: Separate bands or groups of residuals may indicate that cases from different groups behave differently. A pattern across time or measurement order can also be a clue that observations are not behaving alike in the plot. Note the feature and consider whether the data contain a relevant grouping or sequence; the plot alone does not tell you what caused it.
An isolated large residual: A point far above or below the rest has a residual that merits attention. As discussed in “Finding Outliers in the y-Direction,” its sign tells which side of the line it lies on. As discussed in “What to Do With an Unusual Point,” a large residual is a reason to investigate the observation, not proof that it is a recording error.
Worked Examples: Describing Residual Plots
Worked Example: A Pattern-Free Band Around Zero
Original AP-style question. In an invented greenhouse exercise, \(x\) is daily hours of supplemental light and \(y\) is seedling growth in centimeters. The observations are \((1,4)\), \((2,5)\), \((3,7)\), \((4,7)\), and \((5,9)\). Find the least-squares regression line and residuals, then describe the residual plot.
State. We want to judge whether the residual plot shows an obvious problem with using a line for these five observations.
Plan. Calculate the least-squares line, find each predicted growth value, and subtract that prediction from the observed growth. This is a descriptive check of a fitted model, so there are no inference conditions to check.
Do. The means are \(\bar{x}=3\) and \(\bar{y}=6.4\). The deviation sums are \(S_{xx}=10\) and \(S_{xy}=12\). Therefore, the slope is \(12/10=1.2\) centimeters per hour, and the intercept is \(6.4-1.2(3)=2.8\) centimeters. The fitted line is \(\hat{y}=2.8+1.2x\). For example, at \(x=1\), the prediction is \(2.8+1.2(1)=4.0\) cm and the residual is \(4-4.0=0\) cm. The predictions and residuals are:
| Hours of light, \(x\) | Observed growth, \(y\) (cm) | Predicted growth, \(\hat{y}\) (cm) | Residual, \(y-\hat{y}\) (cm) |
|---|---|---|---|
| 1 | 4 | 4.0 | 0.0 |
| 2 | 5 | 5.2 | -0.2 |
| 3 | 7 | 6.4 | 0.6 |
| 4 | 7 | 7.6 | -0.6 |
| 5 | 9 | 8.8 | 0.2 |
Conclude. The residuals alternate around zero, and their magnitudes are fairly similar. There is no obvious curve, fan, or isolated residual in this small example. The residual plot supports using a linear model as a description of these data, although five observations are too few to rule out other concerns.
Worked Example: A Possible Fan in Residual Spread
Original AP-style question. In an invented study of a device, \(x\) is hours of use and \(y\) is battery temperature in degrees Celsius. Six observed pairs are \((1,5.8)\), \((2,7.3)\), \((3,7.5)\), \((4,9.5)\), \((5,9.0)\), and \((6,11.9)\). Find the fitted line and residuals. What feature should the analyst notice?
State. We will check whether the residuals have a similar vertical spread across the observed hours of use.
Plan. Calculate the least-squares line and its residuals, then compare the residual magnitudes across \(x\). This is a visual, descriptive assessment; the six observations do not provide a formal test of changing spread.
Do. Here, \(\bar{x}=3.5\) and \(\bar{y}=8.5\). The deviation sums are \(S_{xx}=17.5\) and \(S_{xy}=18.8\). Thus the slope is \(18.8/17.5\approx1.0743\) degrees Celsius per hour, and the intercept is \(8.5-1.0743(3.5)\approx4.74\) degrees Celsius. The fitted line is \(\hat{y}=4.74+1.0743x\). At \(x=1\), for instance, the predicted temperature is \(4.74+1.0743(1)\approx5.8143\) degrees Celsius, giving a residual of \(5.8-5.8143\approx-0.0143\) degrees Celsius. The residuals, rounded to three decimal places, are approximately \(-0.014,\ 0.411,\ -0.463,\ 0.463,\ -1.111,\ 0.714\) degrees Celsius.
Conclude. The residuals are generally closer to zero at the low end and include greater departures at higher hours, although the pattern is not a perfectly smooth fan. This small plot suggests that vertical spread may change across the range. A careful report calls this a possible warning sign, not proof of a particular cause. More data and the context of how the measurements were collected matter when judging how serious it is.
Worked Example: One Residual Stands Apart
Original AP-style question. In an invented exercise, \(x\) is the number of practice sessions and \(y\) is a participant’s score on a short skills check. The observations are \((1,2)\), \((2,4)\), \((3,14)\), \((4,8)\), and \((5,10)\). Find the fitted line and residuals. What should be done about the point with \(x=3\)?
State. We will check whether the residual plot has an isolated point and describe what that feature does—and does not—tell us.
Plan. Calculate the least-squares line and residuals. Then identify any unusually large residual. As in “What to Do With an Unusual Point,” investigate a standout observation rather than removing it just because it makes the plot look less orderly.
Do. The means are \(\bar{x}=3\) and \(\bar{y}=7.6\), with \(S_{xx}=10\) and \(S_{xy}=20\). The slope is \(20/10=2\) score points per practice session, and the intercept is \(7.6-2(3)=1.6\) score points. The fitted line is \(\hat{y}=1.6+2x\). At \(x=3\), the prediction is \(1.6+2(3)=7.6\), so the residual is \(14-7.6=6.4\) score points. The predictions at \(x=1,2,3,4,5\) are \(3.6,5.6,7.6,9.6,11.6\), respectively. The residuals are \(-1.6,-1.6,6.4,-1.6,-1.6\) score points.
Conclude. The residual plot would show one point well above the zero line while the other four residuals are equal and below zero. This is a clear feature to investigate. The plot does not establish whether the score was recorded correctly, whether the case is valid, or why it differs. Check the source and measurement process before deciding how to handle the observation.
Common Mistakes and AP Exam Tip
- Calling every residual plot problem “a bad outlier.” A fan, curve, or cluster is a pattern across observations, not simply one point with a large residual. Name the feature you see.
- Thinking that points must sit exactly on zero. Residuals are usually not all zero. A good sign is a roughly pattern-free band around zero, not a perfect line.
- Claiming that a pattern proves its cause. A fan may suggest changing spread, and clusters may suggest groups, but a residual plot alone cannot establish why the feature appears. Use cautious wording such as “suggests” or “may indicate.”
- Concluding that no pattern proves the model is correct. A residual plot can reveal visible problems, but its appearance does not prove the model is appropriate for every use or population. Remember the concerns about sample reliability and model scope from “Sampling Method and Reliability of a Model” and “Scope of Inference for a Regression Model.”
- Forgetting the context and units. Residuals use the response variable’s units. Say, for example, that a point is about 6.4 score points above its predicted score, rather than calling it “6.4 units” without explanation.
For full-credit communication, identify the visible feature, describe where it occurs, and state what it suggests about the model. For example: “The residuals spread farther from zero at higher hours of use, suggesting that the vertical variability may not be constant across the observed range.” Do not say that the plot proves the model is invalid or identifies the cause.
Check Your Understanding
For each question, focus on what the residual plot supports and how to describe it carefully.
- A residual plot has points scattered above and below zero with no obvious shape and a similar vertical spread throughout. What does this suggest about the linear model?
- A residual plot is narrow on the left and much wider on the right. Describe the feature and what it may suggest.
- One point is far above zero, while the other residuals are near zero. What is the appropriate next step before deciding whether to remove it?
- Why is “the residual plot proves the model is correct” too strong a conclusion?
- Residuals are measured in which variable’s units, and what contextual details should you include when describing a large residual?