When Residuals Spread Out
In “Curved Patterns in a Residual Plot,” you learned to look for an organized change in residual signs as you move across the horizontal axis. A different warning sign is a change in the vertical spread of the residuals. The residuals may stay roughly centered around zero, yet become more spread out toward one end of the plot. This pattern often looks like a fan opening or closing.
A fan-shaped residual plot is useful because it reveals that prediction errors do not have about the same variability across the predictor range. If the fan opens as \(x\) increases, residuals tend to be smaller near the lower values of \(x\) and larger near the higher values. Predictions may therefore be more consistent at the lower end and less consistent at the higher end. This describes typical variability, not a guarantee about any individual prediction.
The central location and the spread tell different things. Residuals mostly above or below zero in a region suggest that the line tends to underpredict or overpredict there. A fan, by itself, is about how far residuals tend to lie from zero—not about a consistent shift to one side. It can be possible for residuals to remain centered around zero while their distances from zero grow.
This distinction matters when assessing a linear model. A fan does not, by itself, show that the mean relationship is curved or that the least-squares line was calculated incorrectly. It does show that the errors have changing variability, rather than a roughly similar spread across the range. Describe that feature and explain what it means for predictions; do not claim more than the plot supports.
Read the Fan from Narrow End to Wide End
A residual plot places residuals vertically and the predictor \(x\) or fitted value \(\hat{y}\) horizontally, as discussed in “Constructing a Residual Plot.” The zero line is the reference for interpreting signs. The key question for a fan is: Does the typical vertical distance from zero change as you move across the plot?
Use it to judge whether residuals are positive or negative and how far they lie from zero.
Scan from left to right and compare the typical distances from zero in different regions.
If the spread grows to the right, errors tend to vary more at larger horizontal values. If it narrows, they tend to vary less.
State where predictions tend to be more or less consistent, and keep the conclusion within the observed range.
Do not compare only the highest and lowest individual residuals. Individual points can be unusually close to or far from zero by chance. Look at the overall width of the cloud in regions of the horizontal axis. A fan is a pattern in that typical width, not simply one large residual.
Worked Example: A Widening Fan in Battery Testing
A fictional lab tests battery packs at different output levels, \(x\), and uses a fitted line to predict operating time, \(y\), in hours. The fitted line is \(\hat{y}=10+2x\). The table shows the residuals from two tests at each output level.
| Output level \(x\) | Fitted time \(\hat{y}\) (hours) | Residuals (hours) | Mean absolute residual (hours) |
|---|---|---|---|
| 1 | 12 | −0.5, 0.5 | 0.5 |
| 2 | 14 | −1, 1 | 1 |
| 3 | 16 | −2, 2 | 2 |
| 4 | 18 | −4, 4 | 4 |
State. Describe the residual pattern and explain what it says about prediction accuracy at different output levels.
Plan. Compare the residuals’ vertical spread across output levels. Check whether the residuals are centered near zero or instead tend to one side, since a consistent shift would suggest overprediction or underprediction rather than just changing spread.
Do. At each output level, the residuals have opposite signs and sum to zero. For example, at \(x=4\), the residuals are \(-4\) and \(4\) hours, so their mean is \((-4+4)/2=0\) hours. The mean absolute residual there is \((|-4|+|4|)/2=4\) hours. At \(x=1\), the mean absolute residual is \((|-0.5|+|0.5|)/2=0.5\) hours. Across the four output levels, the mean absolute residuals are \(0.5,1,2,\) and \(4\) hours: the typical distance from zero increases. The residuals are balanced around zero at each level, so this example does not show a consistent tendency to overpredict or underpredict at a particular output level.
Conclude. The residual plot would show a widening fan as output level increases. Prediction errors tend to be less variable, and predictions more consistent, at the lower output levels; errors tend to vary more at the higher levels. This describes the observed tests and does not mean every high-level prediction will be farther from its observed value than every low-level prediction.
A Fan Is Not the Same as a Curve
A curved residual pattern and a fan-shaped pattern are different visual clues. In a curve, the residuals’ central position tends to change across the horizontal range—for example, they may be mostly positive at both ends and mostly negative in the middle. In a fan, the central position can remain near zero while the cloud becomes wider or narrower. Both features can occur in the same plot, so check the direction of the residuals and their spread separately.
One useful check is to imagine a middle path through the residual cloud. Does that path bend above and below zero, or does it remain roughly near zero while the cloud’s width changes? This is a visual guide, not a separate calculation. A fan can be somewhat uneven, and real plots rarely form perfect geometric shapes. Look for an overall increase or decrease in typical vertical distance, not exact symmetry.
The residual plot can also reveal that prediction accuracy varies by region even when the fitted line is useful for describing the overall trend. Say specifically where predictions tend to be more or less variable. Avoid saying that all predictions in one region are accurate or inaccurate; residuals vary from observation to observation.
Worked Example: Narrowing Spread in a Water-Flow Sensor
A fictional technician studies a sensor’s measured flow rate, \(y\), in liters per minute, against its operating setting, \(x\). The fitted line is \(\hat{y}=30+5x\). At each setting, two observed measurements have the following residuals.
| Setting \(x\) | Residuals (liters per minute) | Mean absolute residual (liters per minute) |
|---|---|---|
| 0 | −4, 4 | 4 |
| 1 | −2, 2 | 2 |
| 2 | −1, 1 | 1 |
| 3 | −0.5, 0.5 | 0.5 |
State. Explain how the residual spread changes and what that pattern suggests about prediction consistency.
Plan. Compare the residuals’ distances from zero across the settings. Check whether this is a narrowing fan, and describe prediction variability in the correct direction across the observed settings.
Do. At \(x=0\), the mean absolute residual is \((|-4|+|4|)/2=4\) liters per minute. At \(x=3\), it is \((|-0.5|+|0.5|)/2=0.5\) liters per minute. The values decrease across the settings from \(4\) to \(2\) to \(1\) to \(0.5\) liters per minute. The residuals at each setting are balanced around zero, so the main pattern is shrinking spread rather than a shift toward positive or negative residuals.
Conclude. The residual plot would show a fan narrowing as the operating setting increases. Prediction errors tend to be more variable at lower settings and less variable at higher settings, so predictions are more consistent toward the higher end of the observed setting range. This conclusion does not establish how the sensor behaves outside the settings studied.
Describe What the Pattern Means in Context
A strong interpretation names the changing spread and connects it to the response variable’s prediction errors. For a fan opening to the right, an AP-style explanation might say: “The residuals are more widely spread at larger values of the predictor. Predictions of the response therefore tend to have greater variability at the higher observed predictor values and are more consistent at the lower observed values.” Substitute the variables and units from the situation whenever possible.
Keep the conclusion about prediction variability separate from the sign interpretation. Positive residuals mean the line underpredicted; negative residuals mean it overpredicted, as explained in “Sign of a Residual: Over- and Underprediction.” A fan opening upward and downward around zero does not mean that the line increasingly underpredicts. The opening describes larger typical errors in either direction.
Worked Example: Changing Spread in Delivery Times
A fictional delivery team models delivery time, \(y\), in minutes, using route distance, \(x\), in kilometers. A residual plot shows the following residuals for three ranges of route distance. The values are residuals from the fitted line, not raw delivery times.
| Distance range (kilometers) | Residuals (minutes) | Mean residual (minutes) | Mean absolute residual (minutes) |
|---|---|---|---|
| Short routes | −1, 0, 1 | \((-1+0+1)/3=0\) | \((1+0+1)/3=0.67\), rounded |
| Medium routes | −2, 0, 2 | \((-2+0+2)/3=0\) | \((2+0+2)/3=1.33\), rounded |
| Long routes | −5, 0, 5 | \((-5+0+5)/3=0\) | \((5+0+5)/3=3.33\), rounded |
State. Determine whether this residual plot suggests that prediction consistency is similar across route lengths.
Plan. Compare the spread around zero in each distance range, using the mean absolute residual as a simple summary of the listed values. Also check whether the group means suggest a consistent direction of prediction error.
Do. The mean absolute residuals increase from about \(0.67\) minutes for short routes to \(1.33\) minutes for medium routes and \(3.33\) minutes for long routes. In contrast, the mean residual is \(0\) minutes in each range. Thus, the listed residuals become more spread out without showing an average shift above or below zero in any range. These calculations summarize the example’s listed residuals; the plot’s overall widening pattern is the main evidence about changing spread.
Conclude. The residual spread increases with route distance, so delivery-time predictions tend to be less consistent for longer routes than for shorter routes in the observed data. The residuals do not show a directional pattern of increasingly underpredicting or overpredicting long-route times. The conclusion concerns variation in prediction errors, not a claim that every long route has a larger error.
Common Mistakes and AP Exam Tips
- Calling every fan a curve. A curve describes a systematic change in the residuals’ central position; a fan describes changing vertical spread. Check both features before deciding what the plot shows.
- Saying a widening fan means the line underpredicts at high \(x\). Greater spread means residuals tend to be farther from zero, not that they tend to be positive. Use the residual signs to assess overprediction or underprediction.
- Claiming every prediction at one end is more accurate. A fan describes typical variability across a region. Individual residuals can still be large in the narrow region or small in the wide region.
- Ignoring which way the fan opens. Name the region with greater spread. If it opens to the right, errors tend to be more variable at larger horizontal values; if it narrows to the right, the greater variability is toward the left.
- Declaring the line useless solely because the spread changes. The plot shows that prediction errors have changing variability. It does not, by itself, prove that the line fails to describe the overall trend. State the limitation that the plot actually reveals.
- Claiming the pattern applies beyond the data. Describe prediction consistency across the observed range. Do not extend the conclusion to predictor values the data do not cover.
For full-credit communication, identify whether the residual cloud widens or narrows, say where the spread is greater, and explain how that affects the typical variability of predictions in context. If the residuals remain centered near zero, avoid describing a directional bias unless the signs support one. Keep the conclusion about the observed range.
Check Your Understanding
Use the direction of the spread and the signs of residuals to answer each question.
- A residual plot is narrow at low \(x\) and wide at high \(x\), with residuals on both sides of zero. What does the pattern suggest about prediction consistency across the observed range?
- Why does a widening fan not necessarily mean that the regression line increasingly underpredicts?
- A residual plot has a curved middle path near zero, but its vertical spread is similar throughout. Is this primarily evidence of curvature, changing spread, or both?
- A fan narrows as \(x\) increases. Where do prediction errors tend to have greater variability?
- Write one sentence interpreting a widening fan for a model that predicts repair time from machine age. Include which region has greater variability and limit the statement to the observed range.