Research Methods · Simplified
Does the Spread Stay the Same?
A regression line tells you the average relationship. It tells you nothing about whether the scatter around it stays the same width all the way across.
The relationship
Sixty people, and one line through them
The line below is the best summary of the average relationship between the two variables: as the predictor goes up, the outcome goes up by about the same amount each time. That is what the slope means, and it is a single number for the whole range.
Key terms
- Residual
- The vertical distance from one person's dot to the line: how far the line missed them.
- Homoscedasticity
- The residuals are spread about equally all the way across.
- Heteroscedasticity
- The residuals are spread more in some parts of the range than in others.
- Robust standard error
- A measure of how uncertain the slope is, worked out without assuming the spread is even.
Before you go on, look at the plot properly. Follow the line from left to right. Does the band of dots around it stay about the same width the whole way, or does it get wider or narrower in places? Try each of the three groups and decide for yourself. It is harder to judge than it looks, which is the point.
What this shows
Two plots, two different questions
The scatter plot answers "what is the relationship?" The residual plot answers "how well does that relationship hold up across the range?" You need both, and the second one is the one people skip.
Key idea: The slope is a single number describing the average relationship, so it cannot say anything about whether the fit is equally good everywhere. Subtracting the line pulls the trend flat and leaves only the misses, and once the trend is out of the way an uneven band is easy to see. When the spread stays roughly constant across the range, that is homoscedasticity, and the usual formulas for how uncertain the slope is are doing what they claim. When the spread changes across the range, that is heteroscedasticity. The slope itself stays trustworthy either way. What stops being trustworthy is the stated uncertainty around it, and a slope on its own looks exactly the same in both cases.
Going further: how much does it actually matter?
Once you can see uneven spread, the natural next question is how badly it damages the numbers you would report. There are two ways to work out how uncertain the slope is: the ordinary way, which assumes the spread is even, and the robust way, which assumes nothing about it. When they agree, the assumption was harmless here. When they disagree, it was not.
Four cautions. A slope is pinned down mostly by the people at the two ends of the range, so what matters is not how alarming the residual plot looks but whether the extra spread sits at those ends: Group B looks dramatic and barely moves the numbers, while Group C looks no worse and moves them by around a fifth. Uneven spread is also not the worst thing a residual plot can show, and a residual plot with a clear curve in it means the straight line was the wrong model altogether, which no adjustment to a standard error repairs. Judging these plots by eye is genuinely hard at sixty people, where an even band can look uneven by chance, so change the seed a few times before concluding anything. And this is one sample: the two figures differing tells you the assumption is doing work, not which of them is nearer the truth on any particular occasion.
The longer version adds a sample-size control, a repeated-sampling check on whether the reported intervals contain the true slope as often as they claim, and a diagnose-four-plots exercise. It is at Homoscedasticity and Residual Diagnostics in the main collection.