Topic 2.7 Residuals: calculate and interpret residuals, construct and read residual plots, and use them to assess whether a linear model is appropriate.

What this topic is asking

The College Board (Topic 2.7) wants you to calculate and interpret residuals (observed minus predicted) and to read a residual plot to decide whether a linear model is appropriate. The residual plot is the standard diagnostic for linearity.

What a residual is

The order of subtraction is fixed: observed minus predicted, never the reverse. The sign then carries meaning. A positive residual ($y > \hat{y}$) means the actual value is above the line, so the model underpredicted. A negative residual ($y < \hat{y}$) means the actual value is below the line, so the model overpredicted. A residual of $0$ means the point lies exactly on the line.

Residuals sum to zero for least squares

A property worth knowing is that for a least-squares line the residuals always sum to zero (and so average zero): the line balances the points so that over- and under-predictions cancel overall. This is why you cannot judge fit by adding residuals up; instead you look at their pattern, which is the job of the residual plot. It also connects to the next topic, where the least-squares line is defined precisely as the line that minimizes the sum of squared residuals.

The residual plot

The residual plot magnifies departures from linearity that are hard to see in the original scatterplot. By removing the linear trend, it leaves only what the line failed to capture, so any leftover structure jumps out. The single most important reading is: random scatter means the line fits; a curve means it does not. A U-shaped or arch-shaped residual plot is the classic signature of a relationship that is really curved, telling you to transform the data (Topic 2.9) rather than trust the straight line.

Why the residual plot beats the correlation for checking fit

Students often try to judge a model's appropriateness from $r$ or $r^2$, but those numbers can be large even when a line is the wrong model. A gently curved relationship can have a high correlation while still being non-linear, and a straight-line fit would then systematically over- and under-predict in a pattern, exactly what the residual plot exposes and the correlation hides. So the correct workflow is: fit the line, then look at the residual plot, and only if it shows random scatter do you conclude the linear model is appropriate. This is why the College Board so often pairs a regression with a residual plot and asks "is a linear model appropriate?": the expected reasoning cites the residual plot's pattern (random scatter, yes; curve or fan, no), not the size of $r$. Interpreting residual plots correctly, and resisting the temptation to declare a good fit on the strength of a high $r^2$ alone, is one of the most reliably tested skills in the unit. A strong answer also reads individual large residuals as points the model fits poorly, which links forward to identifying outliers and influential points in Topic 2.9.

Try this

Q1. A point has observed $y = 40$ and predicted $\hat{y} = 46$. Find the residual and state whether the model over- or under-predicted. [2 points]

• Cue. Residual $= 40 - 46 = -6$ (negative), so the model overpredicted; the point lies $6$ below the line.

Q2. A residual plot shows a clear upward-opening curve. What does this say about the linear model? [1 point]

• Cue. The linear model is not appropriate; the curved residual pattern shows a non-linear relationship the straight line fails to capture.