Residuals

Unit 2 · Exploring Two-Variable Data

What AP Stats asks here

A residual is observed minus predicted — the 'leftover' the regression line did not explain. The recurring AP question: does the residual plot look like random scatter? If yes, a linear model is appropriate. If not, the leftover pattern (curvature, funnel, outliers) tells you exactly what is wrong with the linear fit.

Residual definition

Sign convention

e_i = y_i - \hat{y}_i

Positive residual

y_i > \hat{y}_i \;\;(\text{observation above the line})

Sum on the LSRL

\sum_{i=1}^{n} e_i = 0

Residual plot patterns

Random scatter around 0

\text{linear model appropriate}

Curved pattern

\text{linear inappropriate; try transformation}

Funnel shape

\text{non-constant variance (heteroscedasticity)}

AP Tip: Always observed − predicted. Predicted − observed gives the wrong sign and contradicts the standard interpretation 'positive residual = above the line'.

Caution: A high R² does not rescue a curved residual plot. Diagnose the residual plot before reporting any prediction as trustworthy.

Type 1

Compute a residual

Plug $x$ into the LSRL to get $\hat{y}$ , then subtract from the observed $y$ . Sign tells you whether the observation sits above or below the regression line.

Example 1

For the regression

\hat{y} = 5 + 2x

and the observed point

(x = 4, y = 15)

, what is the residual? (A)

-2

(B)

0

(C)

+2

(D)

+13

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 2

Read a residual plot

Random scatter ⇒ linear is appropriate. A clear curve (U-shape) ⇒ try a transformation. A funnel that widens with $\hat{y}$ ⇒ non-constant variance.

Example 2

A residual plot shows residuals that are negative at low

\hat{y}

, positive in the middle, and negative again at high

\hat{y}

— a clear upside-down U. What does this indicate? (A) The linear model is appropriate; the pattern is random. (B) The linear model is inappropriate — the curved residual pattern signals leftover curvature; try a transformation or non-linear model. (C) The data is heteroscedastic — try a log transform of

x

. (D) The data is fine; the curve is a sampling artifact.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 3

Interpret s (SD of residuals)

$s$ has the units of the response variable. It is the typical magnitude of the prediction error.

Example 3

An LSRL predicting exam score (out of 100) from study hours has

s = 2.4

. Which interpretation is correct? (A) The model explains 2.4% of the variation in exam score. (B) Predictions from the regression line are typically off by about 2.4 points (in exam-score units). (C) Every prediction is exactly 2.4 points high. (D) The slope is 2.4 points per study hour.

Practice more of this type— AI-generated · always-new problems

Generate Problems →