Residuals: Actual Values Minus Model Values

Every line or curve fit to data is a guess machine: plug in x, get a predicted y. The actual data point almost never lands exactly on the line, so there's a leftover gap. That gap is the residual, and it's measured straight up or down, actual y minus predicted y. A positive residual means the real point sits above the line, the model underestimated it. A negative residual means the point sits below, the model overestimated it. One residual alone doesn't tell you much. The real payoff comes from looking at all of them together.

To build a residual plot, calculate the residual at every x-value, then plot those residuals against x on their own graph, with a horizontal line at zero marking where a perfect prediction would sit. If the residuals scatter randomly above and below zero with no pattern, a straight line was a reasonable model for the data. If the residuals curve, say going negative, then positive, then negative again, that curve is the shape the straight line missed, and a different model fits better.

This sits right next to scatterplots: a scatterplot shows the raw relationship, a line of best fit summarizes it, and the residual plot is the diagnostic check on whether that summary was honest. Center and spread describe a single list of numbers; residuals describe how well a model describes a relationship between two lists. Together they cover how data gets summarized and how that summary gets judged.