Two Plots, Two Viewpoints
A scatterplot and a residual plot can display the same data, but they help answer different questions. The scatterplot shows the original values of the explanatory variable \(x\) and response variable \(y\). A residual plot shows how far the observed responses are from the predictions of a fitted line. The first gives the broad view of the relationship; the second focuses attention on the model’s errors.
In “Reading a Residual Plot for Model Fit,” you learned to look for a roughly pattern-free band of residuals around zero. The tutorials “Curved Residual Pattern Means Nonlinear Relationship” and “Non-Constant Spread in Residuals” examined two patterns that warn about fit. Here, the key skill is comparing those clues with what you can see in the original scatterplot. Sometimes the scatterplot already makes a problem obvious. Other times, the overall trend dominates the view, and the residual plot makes a smaller but systematic departure easier to notice.
The residual plot is not a different analysis of different cases: it is a transformed view of how those cases compare with the fitted line. A positive residual means the observed response is above the line, and a negative residual means it is below. The residual plot’s horizontal axis locates each case in the data, while its vertical axis measures the signed prediction error in response units.
A useful comparison is to ask two questions in order. First, what overall association does the scatterplot show—its direction, form, and strength? Second, after accounting for the fitted line, do the residuals show a pattern that the broad association might conceal? The second question does not replace the first. A residual plot does not display the original response values or tell the whole story about the relationship.
A Paired-Plot Reading Strategy
Use the same cases and the same fitted line when comparing the two plots. A point’s vertical position in the scatterplot is its observed \(y\)-value. Its vertical position in the residual plot is the difference between that observed value and its fitted value. That difference can make departures easier to compare: the fitted trend has been subtracted away, leaving the direction and size of the errors visible around zero.
Describe the overall direction, form, and strength of the association, and note any points that stand out in the original scale.
For a point with observed response \(y\), compare it with its predicted response \(\hat{y}\). Its residual is \(y-\hat{y}\).
In the residual plot, check whether residuals form a pattern-free band, bend, fan, or contain one or more unusually large vertical distances.
Explain what the scatterplot shows about the overall relationship and what the residual plot clarifies about the line’s errors.
For example, a scatterplot can show a strong positive association while the residual plot reveals a gentle curve. There is no contradiction: the data can follow an overall upward trend and still depart systematically from a straight line. Similarly, a cloud that looks tightly grouped relative to a large response scale may show a fan when the residuals are plotted with zero as a clear reference.
Worked Examples: Comparing the Views
Worked Example: Overall Trend with a Hidden Bend
Original AP-style question. In an invented plant-growth exercise, \(x\) is the number of days after planting and \(y\) is plant height in centimeters. A fitted line is \(\hat{y}=10+2x\). The observations for \(x=1,2,3,4,5,6,7\) are \(y=16,14,14,14,18,22,28\). Compare what the scatterplot and residual plot show.
State. We want to describe both the overall association between days and height and whether the fitted straight line captures the pattern adequately.
Plan. First identify the broad trend in the original \((x,y)\) pairs. Then calculate \(y-\hat{y}\) for each case and inspect the residuals in order of \(x\). A sequence of positive residuals at both ends and negative residuals in the middle would indicate a bend around the line.
Do. The fitted values and residuals are:
| \(x\) (days) | Observed \(y\) (cm) | Predicted \(\hat{y}=10+2x\) (cm) | Residual \(y-\hat{y}\) (cm) |
|---|---|---|---|
| 1 | 16 | 12 | 4 |
| 2 | 14 | 14 | 0 |
| 3 | 14 | 16 | -2 |
| 4 | 14 | 18 | -4 |
| 5 | 18 | 20 | -2 |
| 6 | 22 | 22 | 0 |
| 7 | 28 | 24 | 4 |
For the first case, the residual calculation is \(16-12=4\) centimeters. For the fourth case, it is \(14-18=-4\) centimeters. Across the data, height generally increases with days, so the scatterplot shows a positive association. But the residuals move from positive at the low end to negative in the middle and positive again at the high end.
Conclude. The scatterplot shows an overall increasing relationship, but the residual plot makes a curved pattern clearer: the line underpredicts at the low and high ends and overpredicts in the middle. The positive association does not, by itself, establish that a straight line adequately describes the relationship.
Worked Example: One Point That Stands Out More Clearly in Residuals
Original AP-style question. In an invented delivery-route exercise, \(x\) is the number of stops on a route and \(y\) is the route’s fuel use in liters. A fitted line is \(\hat{y}=12+2x\). For routes with \(x=1,2,3,4,5,6,7\), fuel uses are \(14,16,20,16,24,24,26\) liters. Compare the scatterplot and residual plot, paying particular attention to the route with four stops.
State. We will check whether the route with four stops has a response far from the line relative to the other routes.
Plan. Calculate each fitted value and residual. In the scatterplot, compare the point with the general upward pattern and fitted line. In the residual plot, compare its vertical distance from zero with the other residuals.
Do. For the four-stop route, the fitted fuel use is \(12+2(4)=20\) liters, so its residual is \(16-20=-4\) liters. The full set of residuals is:
The scatterplot shows an overall increasing pattern, but the point at four stops lies below the fitted line. The residual plot puts that comparison on a shared zero-centered scale: its \(-4\)-liter residual is farther from zero than the other displayed residuals, whose magnitudes are \(0\) or \(2\) liters. It is therefore the clearest departure from the line among these cases.
Conclude. Both plots locate the four-stop route below the line, but the residual plot makes its relative vertical distance especially easy to compare. The residual identifies a response unusually low relative to the fitted prediction in this data set; the plot alone does not establish why that route differs or whether its measurement is an error.
Worked Example: A Fan Hidden by a Strong Upward Trend
Original AP-style question. In an invented greenhouse study, \(x\) is the number of days since a lighting adjustment and \(y\) is daily water use in liters. A fitted line is \(\hat{y}=50+10x\). At each \(x=1,2,3,4\), two observations have residuals \(-1\) and \(1\), \(-2\) and \(2\), \(-4\) and \(4\), and \(-6\) and \(6\) liters, respectively. Explain what each plot emphasizes.
State. We will compare the overall association with the pattern in the prediction errors as \(x\) increases.
Plan. The scatterplot displays observed water use against days; the residual plot displays the stated residuals against days. Compare the residual magnitudes at each \(x\) to see whether the vertical spread stays similar or changes.
Do. The fitted values at \(x=1,2,3,4\) are \(60,70,80,90\) liters. Adding the paired residuals gives observed water-use values of \(59\) and \(61\), \(68\) and \(72\), \(76\) and \(84\), and \(84\) and \(96\) liters. The scatterplot therefore shows a clear upward trend, with the points increasingly separated around the fitted line.
In the residual plot, the spread at \(x=1\) runs from \(-1\) to \(1\) liter, while at \(x=4\) it runs from \(-6\) to \(6\) liters. The residuals occur on both sides of zero at every \(x\), but their vertical spread widens as days increase.
Conclude. The scatterplot emphasizes the strong increasing association; the residual plot isolates a widening fan that may be less obvious against the rising response scale. Individual water-use predictions are less dependable at higher \(x\)-values because the residuals are more spread out there. This pattern concerns changing error size, not a consistent direction of overprediction or underprediction.
What Each Plot Can—and Cannot—Tell You
The original scatterplot is the place to describe the observed association. It shows the variables in their original units, the direction and broad form of the relationship, and how tightly the points cluster around a possible trend. It also helps you judge whether a point’s original response value is unusual. Keep the context and units attached to that description, as emphasized in “Keeping Context in Every Regression Sentence.”
The residual plot is more specifically a diagnostic view of the fitted model. It can clarify whether errors show a systematic bend, changing spread, or a point with an unusually large residual. Because the vertical axis is measured in residual units, it makes positive and negative departures from the line directly comparable around zero. A reasonably pattern-free band supports the line’s fit; it does not prove the model is perfect or guarantee accurate predictions for every case.
Neither graph alone gives the whole assessment. A residual plot can make the overall relationship hard to reconstruct because the original response values have been replaced by differences from predictions. A scatterplot can make a model-fit pattern hard to see because the overall movement of \(y\) may dominate the visual. Read them together, and be precise about which graph supports each claim.
Common Mistakes and AP Exam Tip
- Calling a strong association proof of a good linear fit. A scatterplot can show a clear trend and still have curvature or changing residual spread. Check the residual plot before concluding that a line describes the pattern well.
- Describing residuals as observed responses. A residual is \(y-\hat{y}\), not \(y\). State whether a residual is positive or negative and use the response units when interpreting its size.
- Claiming a residual plot shows the whole relationship. It shows deviations from the fitted model, not the original values or the full association. Use the scatterplot for the overall relationship and the residual plot for model-fit clues.
- Confusing changing spread with a directional error. A fan indicates that residual magnitudes vary across the plot. To claim systematic overprediction or underprediction in a region, there must also be a tendency for residuals to have one sign there.
- Treating one large residual as a broad pattern. A single point far from zero may be an unusual residual; a curve or fan involves a pattern across a range. Describe the evidence visible in the plot rather than generalizing from one case.
For a full-credit comparison, name what the scatterplot shows, identify the specific residual pattern, and explain what that pattern means for the fitted line. For example: “The scatterplot shows a strong positive association. However, the residual plot has a curved pattern, with residuals positive at both ends and negative in the middle, suggesting that a straight line misses systematic structure in the relationship.”
Check Your Understanding
For each question, distinguish the overall relationship from the evidence about the fitted line.
- A scatterplot shows a strong upward trend. Its residual plot has positive residuals at both ends and negative residuals in the middle. What does each plot contribute to the diagnosis?
- For one case, \(y=31\) and \(\hat{y}=35\) minutes. Find its residual and interpret its sign in context.
- Why might a residual plot make a changing-spread pattern easier to see than the original scatterplot?
- A point appears far from the overall cloud in the scatterplot and has a large negative residual. What does the residual say about the point relative to the fitted line?
- What important information about the data does a residual plot not display directly?