What it shows
A scatter plot of 100 simulated observations, one at each whole number x from 1 to 100, generated as y = 5 + 0.8x plus normal error with a standard deviation of 4 at every x. Because the error variance does not change with x, the data are homoscedastic, and the points form a band of even width along the whole range.
The axes run from 0 to 100 and no line is drawn, so nothing competes with the band. The heteroscedasticity example uses the same trend with an error that grows with x; the two are drawn to be shown together.
How to read it
Compare the vertical spread of the points at the left, in the middle and at the right of the plot. It stays about the same, which is the pattern constant variance predicts.
A plot can fail to show a change in spread; it cannot prove there is none. Here the residuals from the least-squares line have standard deviations of 4.2, 3.9 and 3.7 in the three thirds of the x range, a difference that 33 points per third cannot tell apart from chance.
With constant variance, the usual standard errors of the least-squares line, y = 6.4 + 0.78x, are sound on this count. Linearity, independent observations and nearly normal residuals still have to hold before its tests and confidence intervals can be trusted.
The numbers behind it
- The model is y = 5 + 0.8x + e, with one observation at each x from 1 to 100 and y rounded to a whole number; e is normal with mean 0 and standard deviation 4.
- Least-squares line y = 6.4 + 0.78x. Sample standard deviation of the residuals: 4.2 for x = 1 to 33, 3.9 for 34 to 66 and 3.7 for 67 to 100.
- The data were screened. This is the first draw in sequence in which every y lies between 1 and 100, every point lies within 2.5 standard deviations of the model line (allowing 0.5 for rounding), and the largest third’s residual standard deviation is under 1.15 times the smallest. The screen trims the tails, so an unscreened draw will usually show a few more stray points.
Ways to use it in teaching
Show it unlabeled next to the heteroscedasticity example and ask which plot meets the constant-variability condition, one of the conditions Çetinkaya-Rundel and Hardin (2024, section 24.6.1) set out for inference from a least-squares line.
Have students simulate their own sample from the model in the numbers above and plot it beside this one. The band stays even, but stray points appear, which shows how much noise a real constant-variance plot carries.
Pair it with the random-scatter panel of the residual plot examples, which shows the same kind of even band once the fitted line has been subtracted.
For where regression sits among the statistical tests, 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 November 4, 2026, from https://en.wikipedia.org/wiki/Homoscedasticity_and_heteroscedasticity
