Three Labels for Different Ways a Point Can Stand Out
A scatterplot can contain a point that sits far above the general pattern, a point whose explanatory-variable value is far from the others, or a point that changes the fitted line noticeably. These features are related, but they are not the same. In regression, the terms outlier, high-leverage point, and influential point describe different properties of an individual observation.
As covered in “Variables, Units, and Meaning in a Regression Model,” each plotted point represents an individual with a pair of measurements. Here, \(x\) is the explanatory variable and \(y\) is the response variable. The fitted line gives a predicted response \(\hat{y}\) for each \(x\). A point’s residual is its observed response minus its predicted response, \(y-\hat{y}\), so it measures the point’s vertical distance from the fitted line, including whether the point is above or below it.
These definitions suggest three separate questions: Is the point vertically unusual? Is its \(x\)-value unusual? Does it substantially affect the fitted model? A point may meet more than one description, or only one. Do not decide that a point is influential just because it looks unusual, or call it an outlier just because its \(x\)-value is far from the rest.
Where Each Type Appears on a Scatterplot
The following small schematic plots use a rising relationship. The columns represent increasing \(x\)-values, and the rows represent increasing \(y\)-values. A dot marks ordinary points following the pattern, a star marks the point being discussed, and the dashed mark indicates the approximate line or trend. These sketches are conceptual, not numerical data.
| Outlier in the y-direction | Small \(x\) | Middle \(x\) | Large \(x\) |
|---|---|---|---|
| High \(y\) | ★ | ||
| Near the trend | · | · | · |
| Low \(y\) |
The star is far above the trend in the vertical direction, even though its \(x\)-value is among the central values. It represents an outlier in the y-direction. A point far below the trend could also be a y-direction outlier. The essential feature is the large vertical departure, not whether the point is high or low on the page.
| High-leverage point | Small \(x\) | Middle \(x\) | Large \(x\) |
|---|---|---|---|
| High \(y\) | ★ | ||
| Near the trend | · | · | |
| Low \(y\) |
Here the star is far out in the \(x\)-direction, beyond the cluster of ordinary explanatory-variable values. It lies near the continuation of the rising trend, so it need not be an outlier in the y-direction. Its unusual \(x\)-position gives it high leverage: it has the potential to pull or tilt the fitted line.
| Potentially influential point | Small \(x\) | Middle \(x\) | Large \(x\) |
|---|---|---|---|
| High \(y\) | |||
| Near the original trend | · | · | — |
| Low \(y\) | ★ |
The last sketch shows a point far out in \(x\) and away from the continuation of the main trend. It could pull the fitted line toward itself. The sketch signals possible influence, but the definition of influence is about the change in the fitted line when the point is removed. To confirm influence, compare the two fits rather than relying only on the point’s appearance.
Outliers: Unusual Vertical Distance
To describe a point as a y-direction outlier, focus on the response compared with what the line predicts at that point’s \(x\)-value. The residual gives a signed measure of that vertical difference: a positive residual places the point above the line, and a negative residual places it below. The size of the residual, in response units, indicates how far vertically the observation lies from the fitted value.
“Unusually far” is judged in relation to the scatter in the data and the overall pattern; there is no single residual cutoff that applies in every setting. A residual of 5 response units might be large when other points are within 1 unit of the line, but not especially unusual when residuals commonly vary by 10 units. A y-direction outlier may also affect the fitted line, but its vertical distance alone does not determine how influential it is.
Worked Example: A Point Far Above a Height Trend
Original AP-style question. A greenhouse records seedling height, \(y\), in centimeters, after \(x\) days. At day 5, the fitted line predicts 13 cm. The observed seedling is 23 cm tall, while the other seedlings’ residuals are between \(-2\) cm and \(2\) cm. Describe the point’s vertical position.
Calculate the residual. Subtract the predicted height from the observed height:
Interpret. The residual is positive, so this seedling is 10 cm above the fitted line at day 5. The other residuals are no more than 2 cm from zero, so a 10 cm residual is unusually large in this comparison. The point is an outlier in the y-direction relative to the pattern described. This calculation supports the vertical-outlier description; it does not by itself establish that the point has high leverage or substantially changes the fitted line.
High Leverage: An Unusual Explanatory-Variable Value
Leverage concerns the horizontal position of a point: how unusual its \(x\)-value is compared with the other \(x\)-values. A point near an end of the observed \(x\)-range may have high leverage, especially when the rest of the observations are clustered in the middle. A point at a typical \(x\)-value usually has lower leverage. This is a comparison with the distribution of \(x\)-values in the data, not a comparison of the point’s \(y\)-value with the fitted line.
High leverage means the point can have the capacity to affect the fitted line; it does not guarantee that it does so substantially. If a high-leverage point follows the same linear pattern as the other observations, adding it may extend support for that pattern without changing the line much. If it sits away from the pattern, it may pull the line toward itself. Those possibilities are why leverage and influence must be kept distinct.
Worked Example: An Observation Far Out in \(x\)
Original AP-style question. A horticulture class studies plant height, \(y\), in centimeters, against hours of supplemental light, \(x\), per day. Most plants received between 2 and 6 hours. One plant received 12 hours. The line fitted to the main cluster predicts 27 cm at 12 hours, and that plant’s observed height is 28 cm. Which description is supported?
Compare the \(x\)-values. The point’s 12 hours is well above the 2-to-6-hour cluster. Its explanatory-variable value is unusual compared with most of the data, so it is a high-leverage point.
Check its vertical position. At 12 hours, the line predicts 27 cm. The residual is
Interpret. The observed height is only 1 cm above the line’s prediction. Based on this comparison, the point is close to the trend in the y-direction, not far from it like the outlier in the previous example. The evidence supports calling it high leverage because of its unusual \(x\)-value. It does not, on its own, establish that the point is influential; that requires comparing fitted lines with and without it.
Influence: Does the Fitted Line Change?
Influence is determined by what happens to the fitted model when an observation is omitted. Fit or consider the regression line with all the points, then compare it with the line fitted without the point in question. If removing that observation substantially changes the slope, intercept, or predictions in the context of interest, the point is influential. “Substantially” is a judgment about the size and practical meaning of the change; it is not a universal numerical threshold.
A point far out in \(x\) can strongly affect the slope because it is distant from the center of the explanatory-variable values. A point with a large vertical residual can also affect the fitted line. But neither label alone settles influence. Conversely, a point can be influential even if its residual from the line fitted with that point included is not especially large: the point may have pulled the line toward itself. Comparing fits is the key idea.
Worked Example: A High-Leverage Point Changes the Slope
Original AP-style question. Four observations follow the exact pattern \((1,2)\), \((2,4)\), \((3,6)\), and \((4,8)\), where \(x\) is a device setting and \(y\) is an output measurement. A fifth observation is \((10,10)\). Compare the fitted line without the fifth point to the line with it, and assess whether the fifth point is influential.
Fit without the fifth point. The first four observations satisfy \(y=2x\) exactly, so the line fitted to them is \(\hat{y}=2x\), with slope 2 and intercept 0.
Fit with all five points. For all five observations, \(\bar{x}=4\) and \(\bar{y}=6\). The slope is the sum of the cross-products of deviations divided by the sum of squared \(x\)-deviations:
The intercept is \(a=\bar{y}-b\bar{x}=6-(0.8)(4)=2.8\). Thus the line fitted to all five observations is \(\hat{y}=2.8+0.8x\).
Compare and conclude. Adding the fifth point changes the slope from 2 to 0.8 and the intercept from 0 to 2.8. At \(x=4\), for example, the line without the point predicts \(2(4)=8\), while the line with the point predicts \(2.8+0.8(4)=6\). The fifth observation is far out in \(x\) compared with the first four and substantially changes the fitted line, so it is a high-leverage and influential point.
The observed response of 10 at \(x=10\) is below the original line’s prediction of \(2(10)=20\). Yet its residual from the line fitted with all five points is \(10-[2.8+0.8(10)]=10-10.8=-0.8\). That small residual does not show the point is unimportant: the point has already pulled the fitted line toward itself. This illustrates why influence is assessed by comparing fits, not by looking only at the residual from the line that includes the point.
Keep the Labels Separate
The three terms describe different evidence. An outlier is unusual vertically; a high-leverage point is unusual horizontally; an influential point substantially changes the fit when included. A point can be both an outlier and high leverage, and such a point may be influential. A point can also be an outlier at a central \(x\)-value, or high leverage while staying close to the trend. Influence must be checked through the change in the model.
| Term | What to examine | Question to ask |
|---|---|---|
| Outlier in the y-direction | Vertical distance or residual | Is the response far from the fitted pattern at this \(x\)? |
| High leverage | Position of the \(x\)-value | Is this explanatory-variable value unusual among the observations? |
| Influential | Fitted model with and without the point | Does removing it substantially change the line or relevant predictions? |
Common Mistakes and AP Exam Tip
- Calling a point an outlier because its \(x\)-value is extreme. That describes possible high leverage. To support a y-direction outlier claim, discuss the point’s vertical distance from the pattern.
- Calling every high-leverage point influential. High leverage gives a point the potential to affect the line. Influence requires evidence that the fitted line or predictions change substantially when the point is removed.
- Using a small residual to rule out influence. A point may pull the line toward itself, leaving a small residual in the fit that includes it. Compare the fits with and without the observation.
- Treating “far” as a universal cutoff. Describe unusual position in relation to the other \(x\)-values or the typical vertical scatter. The relevant comparison depends on the data and context.
- Leaving out units and context. When using a residual, state what it means in response units—for example, “10 cm above the predicted seedling height.”
For a strong AP response, name the feature you are judging and point to the evidence for it. For example: “The observation is a y-direction outlier because its residual is 10 cm, much larger than the other residuals. Its \(x\)-value is not unusual, so that does not support calling it high leverage.” Or: “The point is high leverage because its \(x\)-value is far beyond the cluster; it is influential because removing it changes the fitted slope from 2 to 0.8.” These statements connect each label to its own defining comparison.
Check Your Understanding
For each item, identify which property is supported and state what additional evidence, if any, would be needed.
- A point’s residual is \(-7\) minutes, while the other residuals are mostly between \(-1\) and \(1\) minute. What does the negative sign say about its position, and which label may be appropriate?
- Most observations have \(x\)-values from 10 to 20. One point has \(x=45\) and lies close to the fitted pattern. Which label does its \(x\)-position support? Does that alone establish influence?
- A point has a small residual in the fitted model, but removing it changes the slope noticeably. Which term describes this property, and why is the small residual not decisive?
- Explain the difference between an outlier in the y-direction and a high-leverage point without using the word “unusual” as the only explanation.
- What comparison would you make to decide whether a point is influential?