What it shows
Three simulated datasets, each fitted with a least-squares straight line. The top row shows the data and the fitted line; the bottom row shows each fit’s residuals against the fitted values, with a dashed line at zero.
In the first dataset a straight line is the right model and the residuals scatter evenly. In the second the true relationship is curved; in the third the spread grows with the fitted value. Axis values are omitted, because the shapes are what matter.
How to read it
An even band of residuals around zero, with no pattern, suggests the straight line fits.
Residuals positive at both ends and negative in the middle, a U shape, point to a curved relationship the line has missed.
Residuals that spread out as the fitted value grows, a funnel, indicate heteroscedasticity: the variance of the errors is not constant.
The numbers behind it
- Random scatter: y = 2 + 1.5x + normal noise (SD 1.5); fitted line y = 2.92 + 1.33x.
- Curved: y = 1 + 0.4x² + normal noise (SD 1.5); fitted line y = −4.79 + 3.93x.
- Funnel: y = 2 + 1.5x + normal noise with SD 0.4x; fitted line y = 2.11 + 1.44x.
- 50 points per dataset, x drawn uniformly from 0 to 10, lines fitted by ordinary least squares.
Ways to use it in teaching
Show only the bottom row and ask which fit is adequate before revealing the data.
Ask what each pattern suggests doing next: a curved term for the second dataset, a transformation, robust standard errors or weighted least squares for the third.
Before students run their own regressions, use it to show that a high R² does not make the residual plot unnecessary.
For regression and the other techniques residual plots check, see our guide to inferential statistics.
Sources
- Çetinkaya-Rundel, M., & Hardin, J. (2024). Introduction to modern statistics (2nd ed.). OpenIntro. https://openintro-ims.netlify.app/
- Homoscedasticity and heteroscedasticity. (n.d.). In Wikipedia. Retrieved October 18, 2026, from https://en.wikipedia.org/wiki/Homoscedasticity_and_heteroscedasticity
