Residual vs Residual Plot

A residual belongs to one data point and one fitted line, and the order of the subtraction is fixed: $\text{residual} = y - \hat{y}$, observed minus predicted, where $\hat{y}$ (read y-hat) is the height of the line above that point's $x$. The sign is where the meaning lives. A positive residual puts the observation above the line, so the model underpredicted it, and a negative residual puts it below, so the model overpredicted.

Six students study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86. The least-squares line for them is $\hat{y} = 58 + 4x$. For the student who studied 5 hours the line gives $\hat{y} = 58 + 4(5) = 78$, and that student scored 71, so the residual is $71 - 78 = -7$. The line overpredicted that score by 7 points. Doing the same at every $x$ gives residuals of -2, 6, -5, 4, -7 and 4.

The error that costs points is the reversed subtraction. "The residual is 7, so the line missed by 7" keeps the size and throws the direction away, and computing predicted minus observed flips the sign of every residual in the table, turning each overprediction into an underprediction. Say observed minus predicted every time, and read a negative residual as the line sitting above the data point.

Two things bound what a single residual can tell you. It is measured against one particular line, so change the line and every residual changes with it. And for a least-squares line fitted with an intercept the residuals add to exactly zero: here $-2 + 6 - 5 + 4 - 7 + 4 = 0$. Their mean is zero for a good fit and for a hopeless one alike, which is why fit gets judged from the squared residuals and from the shape of the residual plot, never from their average.

Residuals are topic 5.4 in Unit 5, Regression Analysis, of the Fall 2026 AP Statistics course.

A residual plot puts $y - \hat{y}$ on the vertical axis against either $x$ or $\hat{y}$ on the horizontal axis, with a reference line at 0. Both horizontal choices are standard: plotting against $\hat{y}$ rescales the horizontal axis, and reverses it when the slope is negative. You read the plot for shape, not for size: a bend says a straight line is the wrong model for the trend, and a fan says the scatter is not constant across the data.

Six students study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86. Their least-squares line is $\hat{y} = 58 + 4x$, so the residuals plotted above $x = 1$ through $x = 6$ are -2, 6, -5, 4, -7 and 4. They land on both sides of zero, they do not bend, and they do not widen from left to right. With six points that is about as much as such a plot can support, so read it beside the scatterplot rather than on its own.

Here is the sentence to retire: "the residual plot has no upward trend, so hours and score are not associated." A residual plot from a least-squares fit can never show a linear trend. The correlation between the residuals and $x$ is exactly 0 by construction, for these six points and for every other data set, because the line already absorbed the straight-line part of the pattern. The plot answers whether the line has the right shape, not whether the variables are related.

A flat, patternless residual plot is also not proof that the model is correct. It only means nothing obvious is left over, and a small sample can hide a real curve. Curvature is still evidence that a linear model does not belong on the data, and the Fall 2026 course has no topic on transforming data to achieve linearity, so the expected response to a bend is to say a line is not appropriate rather than to re-express the variables.