One Question, Two Kinds of Evidence
A regression free response may ask you to do more than read a fitted line. You might need to diagnose a residual plot and decide whether an unusual observation changes the model. These are related questions, but they use different evidence: the residual plot shows the fitted line’s errors, while a with-and-without comparison shows the unusual observation’s effect on regression output.
In “Reading a Residual Plot for Model Fit,” you learned to look for residuals scattered around zero without a clear pattern or changing spread. In “Testing Influence by Removing a Point,” you learned that influence is assessed by comparing fits with and without the candidate observation. This tutorial brings those skills together in a free-response format and adds a point-by-point practice scoring guide. The guide is for practice; it is not an official College Board rubric.
A strong response makes a claim, gives relevant evidence, and explains what that evidence means in context. Avoid writing only “the plot is bad” or “the point is influential.” Name the pattern or regression values that support your conclusion.
Worked Example: Diagnose the Plot and Assess Influence
Worked Example: A Far-Out Observation in an Environmental Model
Original AP-style question. An invented environmental exercise relates \(x\), distance in kilometers from a monitoring station, to \(y\), a water-quality index. Six observations are \((1,12),(2,14),(3,16),(4,18),(5,20),(12,45)\). A residual plot from the fit using all six observations shows these approximate points: \((1,1.8925),(2,0.8280),(3,-0.2366),(4,-1.3011),(5,-2.3656),(12,1.1828)\), where each ordered pair is \((x,\text{residual})\). Describe what the residual plot suggests about the linear fit, then assess whether the observation at \(x=12\) is influential using the computer output.
| Fit | Intercept \(a\) | Slope \(b\) | \(r\) | \(r^2\) | \(s\) | \(n\) |
|---|---|---|---|---|---|---|
| All six observations | 7.0430 | 3.0645 | 0.9912 | 0.9824 | 1.8035 | 6 |
| Without \((12,45)\) | 10 | 2 | 1 | 1 | 0 | 5 |
State. We will describe the pattern in the full-data residual plot and then compare the regression results with and without \((12,45)\) to assess influence.
Plan. For the plot, check whether the residuals form a pattern rather than a roughly random band around zero. For influence, compare the slope, intercept, \(r\), \(r^2\), and \(s\). The candidate observation’s \(x\)-value is far from the other five, but that position suggests high leverage; it does not, by itself, settle the influence question.
Do. The residuals move from positive at \(x=1\) to negative at \(x=5\), then are positive again at \(x=12\). This downward-then-upward pattern is not random scatter around zero. It suggests that the fitted line may miss a systematic pattern in these data. The residual plot does not, on its own, prove that the relationship is curved; it gives evidence to investigate alongside the scatterplot and the unusual observation.
The five observations without the candidate point lie exactly on \(\hat{y}=10+2x\). For the fit using all six, the means are \(\bar{x}=27/6=4.5\) and \(\bar{y}=125/6\approx20.8333\). The centered sums are \(S_{xx}=77.5\) and \(S_{xy}=237.5\), so the slope and intercept are
For a check on the scatter statistics, \(S_{yy}=740.8333\). Thus the residual sum of squares is \(SSE=S_{yy}-S_{xy}^2/S_{xx}\approx740.8333-727.8226=13.0108\). With \(n=6\), the residual standard deviation is \(s=\sqrt{13.0108/(6-2)}\approx1.8035\) index points. Also, \(r^2=S_{xy}^2/(S_{xx}S_{yy})\approx0.9824\), so \(r\approx0.9912\), positive because the slope is positive.
Conclude. The residuals show a pattern rather than random scatter, so the plot raises concern about the linear fit. Removing \((12,45)\) changes the slope from about 3.0645 to 2 index points per kilometer and the intercept from about 7.0430 to 10 index points. It also changes \(r\), \(r^2\), and \(s\). These changes support calling the observation influential for this invented data set. Do not conclude that it must be deleted: as discussed in “What to Do With an Unusual Point,” first investigate whether it is a valid observation.
A Scoring Guide for a Combined Response
For practice, divide the response into separate scoring targets. This makes it easier to notice when an answer gives a correct conclusion but does not support it, or when it identifies a plot pattern but does not explain its meaning. The six-point guide below is one useful way to check a response to the question in the worked example.
- 1 point: Describes the residual-plot pattern accurately, including the change from positive to negative and back to positive residuals.
- 1 point: Explains that the pattern is evidence against residuals being randomly scattered around zero, so the linear fit deserves concern or further investigation.
- 1 point: Uses the with-and-without comparison to assess influence rather than relying only on the candidate point’s unusual \(x\)-value.
- 1 point: Identifies a meaningful change in the fitted line, such as the slope changing from about 3.0645 to 2 index points per kilometer.
- 1 point: Correctly interprets that slope change in context and with units.
- 1 point: Gives a qualified conclusion: the comparison supports calling the observation influential, but unusualness alone does not establish that it is an error or should be removed.
This guide rewards evidence and interpretation separately. For example, merely stating “the point is influential” does not show how the fits differ. Likewise, reporting that the slope changes does not earn the context-and-units part unless you say what the slope represents.
Describe the residuals’ direction or spread across \(x\); do not substitute a vague label such as “bad fit.”
Connect the visible pattern to the behavior of the line’s errors. A curve or changing spread is more informative than simply saying that some residuals are not zero.
Name at least one regression feature that changes substantially, with its values before and after removal.
State what the change means for the model, and avoid treating influence as proof that the observation is invalid.
Worked Example: A Residual Plot Without a Clear Pattern
Worked Example: Check the Errors Before Criticizing the Line
Original AP-style question. In a separate invented environmental exercise, \(x\) is depth in meters and \(y\) is a measured soil reading. The fitted line is \(\hat{y}=20+3x\). For depths \(x=1,2,3,4,5,6\), the observed readings are \(24,24,30,33,33,39\). Calculate the residuals and describe what their plot suggests about the linear model.
State. We will calculate observed minus predicted readings and look for a systematic pattern or changing spread in the residuals.
Plan. For each case, use \(y-\hat{y}\). Then assess the residuals against depth: a band around zero without an obvious pattern supports the linear model’s fit, while a curve or fan-shaped spread would raise concern.
Do. The fitted readings at depths 1 through 6 are 23, 26, 29, 32, 35, and 38. Subtracting each fitted reading from its observed reading gives
The residuals are therefore \(1,-2,1,1,-2,1\) reading units. They fall on both sides of zero, with no steady increase, decrease, or curved sequence across the six depths. Their vertical spread also appears fairly similar across the observed range.
Conclude. This residual plot would show a roughly pattern-free band around zero with reasonably even spread. That supports using a linear model to describe these invented observations. It does not prove that a linear model is correct in every respect; it means the residual plot provides no obvious pattern contradicting the fit.
Worked Example: Score a Brief Student Answer
Worked Example: Find What a Response Supports—and What It Misses
Original AP-style question. Use the monitoring-station example and the six-point practice scoring guide. A student writes: “The point at 12 km is influential because it is far away. The line is not a good fit because the residuals are not all zero. The slope gets smaller when the point is removed.”
State. We will check whether each claim is supported by the evidence and whether it is explained in context.
Plan. Award credit only for the scoring targets the response actually meets. In particular, look for the residual pattern, a meaningful with-and-without comparison, slope units, and a qualified conclusion.
Do. The answer does not describe the residuals’ positive-to-negative-to-positive pattern, so it misses the pattern point. Saying the residuals are “not all zero” is not a sufficient reason to reject the fit: residuals in a regression fit are generally not all zero, and the issue is whether they show a systematic pattern. The student also uses the point’s distant \(x\)-value as the evidence of influence, rather than comparing the two fits. The answer does identify that the slope decreases when the point is removed, but gives no values, units, or contextual meaning. It does not earn the conclusion point because it does not distinguish influence from whether the observation is an error.
Conclude. Under the six-point guide, this response earns at most partial credit for noticing a slope change, but it misses most of the required evidence and explanation. A stronger version would say: “The residuals decrease from positive to negative and then become positive, so they show a pattern rather than random scatter around zero. Removing the observation at 12 km changes the slope from about 3.0645 to 2 index points per kilometer, a substantial change in the predicted index increase for each additional kilometer. The with-and-without comparison supports calling the observation influential, although it does not show that the observation is an error.”
Common Mistakes and AP Exam Tip
- Calling every nonzero residual a model failure. Residuals are observed minus predicted values and commonly differ from zero. Describe a pattern or changing spread, not merely the fact that residuals exist.
- Calling a point influential because its \(x\)-value is extreme. An extreme \(x\)-value can indicate high leverage. To assess influence, compare regression results with and without the candidate point.
- Combining two conclusions into one. Say separately what the residual plot suggests about fit and what the with-and-without comparison says about influence. One observation may affect the plot and the fitted line, but those claims use different evidence.
- Leaving out units and context. A slope is measured in response units per explanatory-variable unit. Name both variables when explaining what a slope change means.
- Recommending removal without checking the observation. Influence is a description of an observation’s effect on the model, not proof that its recorded value is wrong. Investigate it before deciding how to handle it.
- Using a vague conclusion. Replace “the regression changes a lot” with a specific comparison, such as “the slope changes from about 3.0645 to 2 index points per kilometer.”
On an AP-style question, a reliable answer links each conclusion to its evidence. For fit, name the residual pattern and explain its implication. For influence, give a with-and-without comparison and interpret a meaningful change in context. Keep those parts distinct, then finish with a qualified conclusion.
Check Your Understanding
Use the examples and scoring guide to answer each question.
- A residual plot shows residuals that are positive at low \(x\)-values, negative in the middle, and positive at high \(x\)-values. What pattern should you describe, and what does it suggest about the linear fit?
- A point has an \(x\)-value far from the others. What comparison is needed before calling it influential?
- Why is “the residuals are not all zero” not enough to show that a linear model fits poorly?
- A slope changes from 4 response units per kilometer to 2 response units per kilometer after removing a point. Write one sentence interpreting this change in context.
- Does evidence that a point is influential prove that it should be removed? Explain what should happen before deciding.