Interpreting Residuals in Context

Every point in a scatterplot has an actual value and a predicted value, the one the least-squares line would give for its x. The residual is observed minus predicted. If that number is positive, the actual value sat above the line, so the model underestimated that case. If it's negative, the actual value sat below the line, so the model overestimated. A residual of -4.2 means: for this one individual, the actual value was 4.2 units less than the line predicted. Always say it in the units and context of the problem, not just as a number.

Interpreting a residual means connecting it back to the real-world variables: not 'the residual is -4.2' but 'this house sold for 4.2 thousand dollars less than the model predicted.' That's the mechanical skill. The trap is thinking a small residual means the model is good. A small residual only describes that one point. A line can nail a handful of points and miss badly everywhere else. Judging the model overall means looking at the whole residual plot: do the residuals scatter randomly around zero with no pattern and roughly constant spread, or do they curve, fan out, or cluster? That pattern, not any single residual, tells you whether a line was the right shape for the data.

This sets up residual plots directly: once you can read one residual, you're ready to read all of them at once, checking for curvature, unequal spread, or outliers that a single point can't reveal.