Calculate residuals, construct and interpret a residual plot, and use it to judge whether a linear model is appropriate.

How to calculate a residual as observed minus predicted, build a residual plot, and read it to decide whether a straight line fits or whether the data needs transforming.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
What a residual is
The residual plot
Why we trust the residual plot over r
What to do with a curved residual plot

What this dot point is asking

You must compute residuals, draw or read a residual plot, and use its shape to decide whether a straight line was an appropriate model.

What a residual is

A residual measures how far one data point sits above or below the fitted line, measured vertically.

The least-squares line is fitted so that the residuals sum to zero, so some are positive and some negative by design.

The residual plot

A residual plot graphs each residual on the vertical axis against the explanatory variable on the horizontal axis, with a horizontal line at zero for reference. Its job is to reveal patterns that the original scatterplot hides.

Why we trust the residual plot over r

A reasonably high correlation coefficient can occur even when the true relationship is curved. The residual plot magnifies the leftover pattern after the line is removed, so it is the more reliable test of linearity.

Worked example

Calculating residuals and reading the plot

The least-squares line for a small data set is $y = 3 + 2x$ . Three observed points are $(1, 6)$ , $(2, 7)$ and $(3, 9)$ .

Predicted values: at $x = 1$ , $\hat{y} = 3 + 2(1) = 5$ ; at $x = 2$ , $\hat{y} = 7$ ; at $x = 3$ , $\hat{y} = 9$ .

Residuals:

$x = 1$ : $6 - 5 = +1$ (above the line).
$x = 2$ : $7 - 7 = 0$ (on the line).
$x = 3$ : $9 - 9 = 0$ (on the line).

Plotting these residuals against $x$ shows them scattered close to zero with no curve, so over this small set a straight line looks appropriate. If instead the residuals had run negative, positive, negative as $x$ increased, that wave shape would signal a non-linear relationship and call for a transformation.

What to do with a curved residual plot

If the residual plot is curved, the linear model is rejected. The standard next step is to transform one variable (for example using $x^2$ , $\log x$ or $1/x$ ) and refit, which is the subject of the data transformation dot point.