A residual measures how far off a prediction is: actual y minus predicted ŷ. Click and drag points to watch residuals change on the scatterplot and in the linked residual plot, then decide whether the pattern says a linear model is appropriate.
A residual is how far an actual value is from the value the line predicts. Click a point to see its residual; drag it to change it. The residual plot below the scatterplot shows every residual against x.
Decide what the residual plot says about the linear model, then compare with the verdict below.
A residual is the actual value minus the predicted value. It is the vertical distance from a point to the regression line, with a sign.
A positive residual means the point is above the line, so the model underestimated. A negative residual means the point is below the line, so the model overestimated.
For the least-squares line, the residuals always sum to 0 (and so their mean is 0). Large positive misses are balanced by negative ones.
A residual plot graphs each residual against the explanatory variable (or against ŷ). It magnifies the misses so patterns are easier to see than in the scatterplot.
If the residual plot shows random scatter around 0, a linear model is appropriate. A curved pattern means the relationship is not linear. A fan means the spread changes with x.
s = √(Σ residual² ÷ (n − 2)) estimates the typical size of a prediction error, in the units of y. Smaller s means predictions are closer to the actual values.
What you see, what it means, and what to write.
| Residual plot shows | Meaning | What to write |
|---|---|---|
| Random scatter around 0, no curve, even spread | The line fits the form of the data | "There is no pattern in the residual plot, so a linear model is appropriate." |
| A curve (U shape or upside-down U) | The relationship is nonlinear | "The curved pattern in the residual plot shows a linear model is not appropriate." |
| A fan (spread grows or shrinks across x) | Prediction errors change size with x | "Predictions will be less precise for larger x-values." |
| One point far from 0 | A large residual: an outlier in the regression | Identify the point and its residual. |
| Mostly positive or mostly negative residuals overall | Not possible for the least-squares line | Residuals always sum to 0. |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 5.4. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 5 terminology guide.