Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A residual is the prediction error for one data point: how far the observed value is from the line, measured vertically.
- It has the units of the response variable.
- Model sentence: “The residual of −$1.67 means the actual fare for the 6.8 km ride was $1.67 less than the model predicted, so the model overpredicted it.”
- A residual plot shows residuals against x (or ŷ) and is the main check of whether a line is the right model.
Key relationships
| Idea | What to remember |
|---|---|
| Residual | y − ŷ = observed − predicted |
| Observed value | y = ŷ + residual |
| Positive residual | point above the line; model underpredicts |
| Negative residual | point below the line; model overpredicts |
| Residual of 0 | point on the line |
| Worst prediction | residual furthest from 0, either sign |
| Residual plot | residuals (vertical) against x or ŷ (horizontal), with a line at 0 |
| Random scatter | linear form; a linear model is appropriate |
| Curvature (U or arch) | non-linear form; a linear model is not the most appropriate |
Assumptions and conventions
- Residuals come from a stated model, ŷ = a + bx. Calculate ŷ first, then subtract it from y.
- With one explanatory variable, plotting residuals against x or against ŷ shows the same pattern (mirrored if the slope is negative).
- A high r does not replace the residual plot: curved data can have r close to −1 or 1.
Mistakes to avoid
- Predicted − observed: the wrong order flips the sign.
- Swapping the meanings: negative means overprediction.
- No units or no context in an interpretation.
- Choosing the most negative residual as the worst prediction instead of the one furthest from 0.
- Trusting r alone: always look at the residual plot.
- Thinking random scatter means small residuals: it only shows the form is linear.
Quick self-check
- A model predicts 47.5 g and the observed value is 52 g. Find the residual. (52 − 47.5 = +4.5 g; the model underpredicted)
- ŷ = 30 minutes and the residual is −2.4 minutes. What was observed? (30 + (−2.4) = 27.6 minutes)
- A residual plot is positive at both ends and negative in the middle. Is a linear model appropriate? (No: the U-shape shows curvature, so the form is not linear)
Next: practice questions.