Residuals
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA residual is $y - \hat{y}$, actual minus predicted, in that order. A positive residual means the point sits above the line, so the model under-predicted; a negative one means it sits below, so the model over-predicted. A student studying 7 hours who scored 86 against a prediction of 83 has a residual of $+3$ points. Residuals are measured vertically and always sum to 0 for a least-squares line. A residual plot graphs them against $x$ with a line at 0, and random scatter supports a linear model while a curve or a funnel does not.
The subtraction gets reversed, so every sign flips and over-prediction trades places with under-prediction, and a point below the line gets a positive residual. The residual plot gets read backwards: a clear U shape is excused because $r = 0.95$, though the residual plot is the tool that answers the shape question and $r$ is not, while random scatter gets reported as a failed model when random is the target. And large residuals get treated as points to delete rather than as information about where the model fits poorly.
The work
3 ways in · any order
Lesson
Residuals
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Fixes the residual as actual minus predicted and reads its sign as above or below the line, interprets one in the response's units, and makes the residual plot the tool that decides whether a linear model is the right shape, overruling a large correlation.
Diagnostic
10-item topic check
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Ten items on residuals: the subtraction reversed, positive residuals read as over-prediction, curved residual plots excused by a large r, and random scatter read as a failed model. Take it cold to find your habit, or after the lesson to check it is gone.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.