Residual Plots

A scatterplot with a regression line can look fine at a glance, even when the data actually curve. That's because a strong overall trend can hide small, systematic misses — the line tilts through the cloud of points and your eye reads 'close enough.' A residual plot strips away that trend. It plots each residual, which is the observed y minus the predicted y, against x. With the main trend gone, anything left over — a curve, a fan shape, a cluster — stands out because there's nothing else competing for your attention.

To build one, take each data point, find its predicted y from the regression line, and subtract: residual = observed minus predicted. Plot that residual above (positive) or below (negative) the corresponding x value, with a horizontal line at zero marking where residuals of zero would fall. A point above the line means the model underpredicted; below means it overpredicted. Do this for every point and you get the residual plot.

If the true relationship is a parabola but you fit a straight line, the residuals won't be random. Points will be negative in the middle (line overpredicts where the curve dips or peaks) and positive at the ends, or vice versa, tracing a clear curved shape. A random, patternless scatter of residuals around zero is what you want to see — it means the line captured the pattern and only random noise is left. This connects directly to checking the other conditions for regression, like constant spread, since residual plots are also how you spot changing variability (a fan or cone shape) across x.