Is a residual observed minus predicted or predicted minus observed?

Always observed minus predicted: y - y. Reversing the order flips every sign, so a point above the line would wrongly show a negative residual. Memorize the correct order to avoid losing points.

What does a good residual plot look like?

A good residual plot shows random scatter of points above and below the zero line with no pattern — no curve, no fan shape, no clusters. Random scatter is your evidence that a linear model is appropriate.

What's the difference between a high-leverage point and an influential point?

A high-leverage point has an extreme x-value, giving it the potential to affect the line. An influential point is one whose removal actually changes the slope, intercept, or correlation noticeably. High-leverage points are often influential, but not always.

Can a linear model have a high correlation but still be inappropriate?

Yes. Correlation only measures linear strength and can be high even when data follow a curve. That is exactly why you check the residual plot — a curved residual pattern tells you the linear model is wrong despite a high r or r^2.

AP-STATS-2.7-2.9

U2.7 Residuals and Residual Plots

Learn to compute residuals (y − ŷ), build and read residual plots, and judge whether a linear model fits — plus spot influential and high-leverage points.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on U2.7 Residuals and Residual Plots, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

You already know how to fit a least-squares regression line. But a line can be drawn through any cloud of points — the real question is whether a line is the right model. That's what residuals answer. A residual measures how far each actual data point falls from your prediction, and a residual plot magnifies those errors so patterns you can't see in a scatterplot jump out.

In this lesson you'll compute residuals as

y - \hat{y}

, construct residual plots, and interpret them the way AP graders expect: is the pattern random scatter (linear model is appropriate) or does it curve, fan out, or contain unusual points? You'll also learn to distinguish influential points from high-leverage points — a distinction that shows up in both multiple-choice and FRQ questions.

What a Residual Is and How to Compute It

A residual is the difference between an observed response and the value the model predicts:

\text{residual} = y - \hat{y}

Here

y

is the actual data value and

\hat{y}

is the predicted value from the regression equation. A positive residual means the model underpredicted (the point sits above the line); a negative residual means the model overpredicted (the point sits below the line). A residual of zero means the point lands exactly on the line.

To compute a residual: plug the explanatory value

x

into the least-squares equation to get

\hat{y}

, then subtract that from the observed

y

.

Suppose

\hat{y} = 3 + 2x

and a data point is

(4, 12)

. Then

\hat{y} = 3 + 2(4) = 11

, so the residual is

12 - 11 = 1

. The actual value is 1 unit higher than predicted.

A key property of least-squares regression: the residuals always sum to zero (and therefore have mean zero). That's a direct consequence of how the line is fit. On the exam, be careful with the order — it is always observed minus predicted, never the reverse. Reversing the subtraction flips every sign and is a common point-losing error.

Constructing and Reading a Residual Plot

A residual plot graphs the residuals on the vertical axis against the explanatory variable

x

(or sometimes against

\hat{y}

) on the horizontal axis. A horizontal reference line at zero represents perfect prediction.

The purpose of a residual plot is to check the single most important condition for using a linear model: is the relationship actually linear? A scatterplot can look roughly linear even when a subtle curve is present. The residual plot strips away the linear trend so any leftover structure becomes obvious.

The interpretation rule is simple and testable:

Residual plot shows	Conclusion
Random scatter with no pattern	Linear model is appropriate
A curved (U-shaped) pattern	Relationship is nonlinear; line is not appropriate
Fanning / changing spread	Variability isn't constant; predictions less reliable

When you write your conclusion on the AP exam, name the evidence: "The residual plot shows no leftover pattern (random scatter around zero), so a linear model is appropriate." Or: "The residual plot shows a clear curved pattern, so a linear model is not appropriate for these data." Avoid vague phrases like "it looks fine" — always tie the verdict to the presence or absence of a pattern.

Departures from Linearity and Non-Constant Spread

Two departures from a good linear fit show up repeatedly on the exam.

First, curvature. If the residual plot bends — for example residuals are positive at the ends and negative in the middle, forming a U — the data have a nonlinear relationship the line failed to capture. Even a strong correlation like

r = 0.95

can hide curvature, which is exactly why you must check the residual plot rather than trust

r

alone.

Second, non-constant variability (sometimes called changing spread). If the residuals form a fan or megaphone shape — small spread on one side, large spread on the other — the model's prediction error grows across the range of

x

. The line may still be the correct form, but predictions are much less precise where the spread is wide.

A common misconception is that large residuals automatically mean the model is bad. Not so — one or two big residuals with otherwise random scatter is normal. What matters is a systematic pattern across the whole plot. Another misconception: a high

r^2

guarantees linearity. It does not. Always let the residual plot, not the correlation, be the deciding evidence about whether the linear form is appropriate.

Influential Points and High-Leverage Points

Unusual points can distort a regression, and the AP exam expects you to tell two types apart.

A high-leverage point has an

x

-value far from the mean of the other

x

-values. Because leverage comes from horizontal distance, such a point has the potential to pull the line strongly.

An influential point is any point whose removal would noticeably change the slope, intercept, or correlation. High-leverage points are often influential, but influence is about the actual effect on the fit.

Feature	High-leverage point	Influential point
Defined by	Extreme $x$ -value	Its removal changes the model a lot
Effect	Potential to affect the line	Actually affects slope/intercept/ $r$
Relationship	May or may not be influential	Often (not always) has high leverage

An outlier in regression is a point with a large residual — it lies far from the line vertically. An outlier sitting near the center of the

x

-values usually has little influence on the slope, while a point with extreme

x

can swing the slope even with a modest residual. On the exam, justify claims: "Removing this point would increase the slope substantially, so it is influential," rather than just labeling it.

Key terms

Residual.: The difference between an observed and predicted response, computed as $y - \hat{y}$ ; positive means the model underpredicted, negative means it overpredicted.
Predicted value ( $\hat{y}$ ).: The response value produced by the regression equation for a given $x$ .
Residual plot.: A scatterplot of residuals versus the explanatory variable (or $\hat{y}$ ), used to check whether a linear model is appropriate.
Random scatter.: A residual plot with no leftover pattern around zero, indicating the linear model fits well.
High-leverage point.: A point with an $x$ -value far from the mean of the other $x$ -values, giving it the potential to strongly affect the regression line.
Influential point.: A point whose removal would substantially change the slope, intercept, or correlation of the regression.
Regression outlier.: A point with an unusually large residual, lying far from the regression line in the vertical direction.

Worked example

A least-squares regression line predicting a plant's height (cm) from days since planting is

\hat{y} = 2.5 + 1.8x

. On day 10 the plant was measured at 24 cm. (a) Compute the residual. (b) The full residual plot shows residuals that are negative for small and large

x

but positive in the middle. What does this indicate about the linear model?

Part (a): First find the predicted height at

x = 10

. Substitute into the equation:

\hat{y} = 2.5 + 1.8(10) = 2.5 + 18 = 20.5 \text{ cm}

Now apply the residual formula, observed minus predicted:

\text{residual} = y - \hat{y} = 24 - 20.5 = 3.5 \text{ cm}

The residual is positive, so the model underpredicted the plant's height by 3.5 cm on day 10 — the actual point lies above the line.

Part (b): Residuals that are negative at the extremes and positive in the middle form a curved (U-shaped) pattern in the residual plot. This is a systematic pattern, not random scatter. It tells us the true relationship is nonlinear, so a linear model is not appropriate for these data even if the correlation is high. A better model would follow the curve — for example a transformation of the data, which you'll study later in the unit.

Practice questions

A residual plot for a set of data displays a clear fan shape, with residuals tightly clustered near zero for small

x

but widely spread for large

x

. What is the most appropriate conclusion?

The relationship is nonlinear, so a linear model should not be used
The variability of the response is not constant across values of $x$
There is an influential point pulling the slope upward
The correlation must be negative

Answer: The variability of the response is not constant across values of $x$

A fan or megaphone shape indicates changing spread of the residuals — the prediction error grows as

x

increases. A curved pattern (not a fan) would signal nonlinearity, so the first choice is wrong. Nothing about a fan shape indicates a single influential point or the sign of the correlation. The key idea: fan shape = non-constant variability.

A regression line is

\hat{y} = 12 - 0.5x

. For the data point

(8, 5)

, compute the residual and state whether the model over- or underpredicted.

Answer: The residual is $-3$ ; the model overpredicted.

Predict first:

\hat{y} = 12 - 0.5(8) = 12 - 4 = 8

. Then residual

= y - \hat{y} = 5 - 8 = -3

. Because the residual is negative, the predicted value (8) was larger than the observed value (5), so the model overpredicted, and the point lies below the line.

A scatterplot of two variables has

r = 0.97

. A student concludes a linear model is appropriate based on this strong correlation. Explain why this reasoning is incomplete and what the student should examine.

Answer: A high correlation does not confirm linearity; the student should examine the residual plot for a pattern.

Correlation measures the strength of a linear association but can be high even when the true relationship is curved. The definitive check is the residual plot: if it shows random scatter around zero, the linear model is appropriate; if it shows a curved pattern, the model is not appropriate despite the high

r

. The student must inspect the residual plot before concluding.

FAQ

Is a residual observed minus predicted or predicted minus observed?: Always observed minus predicted: $y - \hat{y}$ . Reversing the order flips every sign, so a point above the line would wrongly show a negative residual. Memorize the correct order to avoid losing points.
What does a good residual plot look like?: A good residual plot shows random scatter of points above and below the zero line with no pattern — no curve, no fan shape, no clusters. Random scatter is your evidence that a linear model is appropriate.
What's the difference between a high-leverage point and an influential point?: A high-leverage point has an extreme $x$ -value, giving it the potential to affect the line. An influential point is one whose removal actually changes the slope, intercept, or correlation noticeably. High-leverage points are often influential, but not always.
Can a linear model have a high correlation but still be inappropriate?: Yes. Correlation only measures linear strength and can be high even when data follow a curve. That is exactly why you check the residual plot — a curved residual pattern tells you the linear model is wrong despite a high $r$ or $r^2$ .

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