Look at the Vertical Distance from the Fitted Line
In “Outlier, High-Leverage, and Influential Points Defined,” you learned that these labels describe different features of a point. This tutorial focuses on finding an outlier in the y-direction: an observation whose response is unusually far from the pattern represented by the regression line. The key evidence is its residual, not whether its explanatory-variable value looks unusual.
For a case with explanatory-variable value \(x\), the fitted line gives a predicted response \(\hat{y}\). The observed response is \(y\). Their difference, \(y-\hat{y}\), is the residual. It measures the vertical distance from the point to the line in response units.
The word “large” refers to the residual’s magnitude, or distance from zero, not just its numerical sign. For example, \(-5\) has a larger magnitude than \(2\), so it represents a greater vertical distance. Whether that distance is unusual depends on how much residual scatter is typical in the data. There is no universal cutoff that identifies every outlier.
Calculate and Interpret a Residual
To examine a particular case, find the fitted response at its \(x\)-value and subtract that prediction from the observed response. Preserve the units of \(y\): a residual for a response measured in minutes is in minutes, while one for a response measured in centimeters is in centimeters.
A positive result means the observed response is greater than predicted, so the point is above the line. A negative result means the observed response is less than predicted, so the point is below the line. The absolute value \(|y-\hat{y}|\) is the vertical distance without direction.
A residual is not automatically an outlier just because it is nonzero. Most real data have residuals on both sides of the fitted line. Instead, compare the candidate residual with other residuals or with the overall vertical scatter. A residual of 4 units may be striking if almost all other residuals are within 1 unit of zero; it may be ordinary if residuals commonly range from \(-8\) to \(8\).
Worked Examples: Finding Large Residuals
Worked Example: A Tomato Plant Above the Yield Pattern
Original AP-style question. A class records watering amount, \(x\), in liters per week, and tomato yield, \(y\), in kilograms per plant. A regression output for the class data gives \(\hat{y}=2.4+1.8x\). For a plant watered 4 liters per week, the observed yield is 15.1 kg. Most of the other residuals are between \(-1.2\) kg and \(1.4\) kg. Is this plant an outlier in the y-direction?
Find the predicted yield. At \(x=4\), the fitted line predicts
Calculate the residual.
Check the subtraction by adding the residual back to the prediction: \(9.6+5.5=15.1\) kg, the observed yield. The positive residual means the plant produced 5.5 kg more than the line predicted. Since the other residuals are within 1.4 kg of zero, 5.5 kg is unusually large in this comparison. The plant is an outlier in the y-direction relative to the fitted pattern. This conclusion concerns its vertical distance; it does not establish that its watering amount is unusual or that it substantially changes the line.
Worked Example: A Battery Below the Runtime Pattern
Original AP-style question. A technician studies the relationship between the number of charge cycles, \(x\), and battery runtime, \(y\), in hours. For a set of batteries, the fitted line is \(\hat{y}=18.6-0.021x\). A battery at 300 charge cycles runs for 8.9 hours. The other residuals are generally between \(-1.0\) and \(0.8\) hour. Describe the battery’s residual and assess whether it is a y-direction outlier.
Calculate the predicted runtime.
Calculate and check the residual.
The check \(12.3+(-3.4)=8.9\) confirms the residual matches the observed runtime. Its negative sign places the battery below the fitted line: its runtime is 3.4 hours less than predicted. The magnitude, 3.4 hours, is much greater than the usual residual range described for the other batteries. This observation is therefore an outlier in the y-direction relative to the pattern. The conclusion is based on vertical distance, not simply on the fact that 8.9 hours is a low runtime.
Worked Example: Find the Outlier in a Residual Plot
Original AP-style question. A class records a device setting \(x\) and an output measurement \(y\). The values below are the residuals from the fitted line \(\hat{y}=20+2x\). Identify any observation that appears to be an outlier in the y-direction.
| \(x\) | Observed \(y\) | Predicted \(\hat{y}=20+2x\) | Residual \(y-\hat{y}\) |
|---|---|---|---|
| 1 | 21 | 22 | \(-1\) |
| 2 | 23 | 24 | \(-1\) |
| 3 | 25 | 26 | \(-1\) |
| 4 | 34 | 28 | 6 |
| 5 | 29 | 30 | \(-1\) |
| 6 | 31 | 32 | \(-1\) |
| 7 | 33 | 34 | \(-1\) |
Check the residual calculations. At \(x=4\), for example, the fitted response is \(20+2(4)=28\), so the residual is \(34-28=6\). Adding the residual back gives \(28+6=34\), the observed response. At \(x=1\), the residual is \(21-22=-1\), and \(22+(-1)=21\). The other rows follow the same observed-minus-predicted calculation.
Confirm that the stated line fits these data by least squares. The mean of the \(x\)-values is 4, and the mean of the \(y\)-values is 28. The residuals sum to \(6+6(-1)=0\). Their products with the centered \(x\)-values also sum to zero:
Thus the residuals sum to zero and have zero cross-product with \(x-\bar{x}\). The fitted line passes through \((\bar{x},\bar{y})=(4,28)\), has slope 2, and has intercept \(28-2(4)=20\), consistent with \(\hat{y}=20+2x\).
Identify and interpret. The observation at \(x=4\) has residual 6 response units, while every other observation has residual \(-1\) response unit. Its vertical distance is much greater than the others, so the observation at \(x=4\) is a y-direction outlier. It is above the line because its residual is positive. This comparison identifies the vertical outlier; it does not answer whether \(x=4\) is unusual among the explanatory-variable values.
Use a Residual Plot to See the Vertical Departures
A residual plot displays residuals on the vertical axis and the explanatory-variable values on the horizontal axis. The horizontal reference line is at residual 0. A point far above that reference line has a large positive residual; a point far below it has a large negative residual. In either case, its vertical distance from zero is the residual’s magnitude.
For identifying a y-direction outlier, look for a residual point that is isolated vertically from the rest. Compare its distance from zero with the general size of the other residuals. If the residual plot has several points with similar vertical distances, one may not be especially unusual even if its residual looks large when considered alone.
A residual plot can also display patterns that call the linear model into question, such as a curved arrangement or changing spread. Those patterns are broader assessments of model fit, not the same thing as identifying one unusually large residual. Keep the question in view: when asked for a y-direction outlier, point to the observation with a residual unusually far from zero and describe the response-unit distance.
Keep Vertical Outliers Separate from Other Labels
As in “Outlier, High-Leverage, and Influential Points Defined,” an outlier concerns vertical distance, high leverage concerns an unusual \(x\)-value, and influence concerns how much the fitted line changes when the observation is removed. A large residual supports a y-direction outlier description. By itself, it does not show that the point has high leverage or is influential.
The distinction matters because a point can be vertically far from the line while its \(x\)-value is near the center of the observed \(x\)-values. It can also have an unusual \(x\)-value but a small residual because it lies close to the fitted line. When the question asks only about a y-direction outlier, do not infer the other properties without their own evidence.
Common Mistakes and AP Exam Tip
- Reversing the subtraction. The residual is \(y-\hat{y}\), not \(\hat{y}-y\). Reversing it changes the sign and therefore changes whether you say the point is above or below the line.
- Calling every nonzero residual an outlier. Residuals are commonly nonzero. Say a residual is unusually large only after comparing its magnitude with the other residuals or the typical vertical scatter.
- Ignoring the sign. A positive residual means the observed response is above the predicted response; a negative residual means it is below. The magnitude tells how far, while the sign tells which direction.
- Using only the observed response value. A response that seems large in isolation is not necessarily far from the regression pattern. Compare it with the predicted response at that case’s \(x\)-value.
- Confusing a large residual with high leverage or influence. A residual describes vertical departure. High leverage is about the \(x\)-position; influence requires a with-versus-without comparison of the fitted model.
- Leaving out context and units. A full-credit explanation says what the residual means, such as “the plant’s yield was 5.5 kg above the fitted prediction,” rather than reporting only “the residual is 5.5.”
A strong AP response makes the comparison explicit: “At 4 liters per week, the fitted yield is 9.6 kg, so the plant’s residual is \(15.1-9.6=5.5\) kg. This is much larger than the other residuals, so the plant is an outlier in the y-direction; its positive residual places it above the line.” This connects the calculation, direction, context, and reason for the outlier description.
Check Your Understanding
Use residual size and sign to answer each question. Compare a residual with the stated pattern of other residuals rather than applying a universal cutoff.
- A fitted model predicts a river depth of 42 cm at a particular location, where the observed depth is 37 cm. Find the residual and describe the point’s position relative to the line.
- A residual is \(4.2\) seconds, while most residuals are between \(-0.6\) and \(0.7\) second. What evidence supports calling the observation a y-direction outlier?
- In a residual plot, one point lies far below the zero line. What does that say about the residual’s sign and the observation’s position compared with its prediction?
- Why is a point with an unusual explanatory-variable value not automatically an outlier in the y-direction?
- A point has a large residual from a fitted line. What additional comparison would be needed to decide whether the point is influential?