Residuals on a Scatterplot With the Line

Finding the Residual Segment on a Scatterplot

In “Interpreting a Residual in Context,” you used a residual to describe how far a model’s prediction was from an observed response. On a scatterplot that shows the fitted line, this difference has a visible representation: it is the vertical segment between an observed point and the line at the same predictor value.

To find that segment, start at the observed point and move straight up or down until you reach the fitted line. The segment is vertical because the comparison holds the predictor value \(x\) fixed: it compares the observed response \(y\) with the line’s predicted response \(\hat{y}\) for that same \(x\). A horizontal segment would change the predictor value instead, so it would not show the residual.

As established in “Defining a Residual as Observed Minus Predicted” and “Sign of a Residual: Over- and Underprediction,” the residual is \(y-\hat{y}\). If the point is above the line, its observed response is greater than the prediction, so the residual is positive. If the point is below the line, its observed response is less than the prediction, so the residual is negative. A point on the line has a residual of zero.

The segment’s length shows the size of the prediction error, but its direction supplies information too. For example, a segment extending 4 response units above the line represents a residual of \(+4\); one extending 4 units below represents a residual of \(-4\). These residuals have equal magnitudes but opposite signs.

Read the Gap Along the Response Axis

A scatterplot may not display a separate segment for every point. Even so, you can identify a residual by locating where the fitted line passes above or below a plotted point at the same horizontal position. The line’s height there is the predicted response; the point’s height is the observed response. The vertical gap between those heights is the residual segment.

To judge the gap’s size, read the response-axis scale. If each grid interval represents 2 response units and the point is one-and-a-half intervals from the line, the vertical distance is about 3 response units. If the scale is labeled in tens, a gap of half an interval is about 5 response units. Estimate from the axis labels, not just from how many pixels the segment seems to occupy.

A residual is measured in the response variable’s units. If the response is plant height in centimeters, a vertical segment represents centimeters; if the response is travel time in minutes, it represents minutes. The segment does not represent a change in the predictor. Also, its length is not generally the shortest distance from the point to the line: the residual uses a vertical, not perpendicular, comparison.

Comparisons are clearest when the segments are on the same plot and share the same response axis. A segment that is visibly longer represents a larger residual magnitude. Across separate plots with different vertical scales or aspect ratios, apparent pixel lengths can be misleading; compare the values in response units instead.

Worked Example: A Greenhouse Point Above the Line

Notice that finding the segment does not mean moving left or right from the point. Both the point and its prediction use \(x=4\); only the response value changes along the vertical direction.

Worked Example: A Commute Point Below the Line

The segment has a positive length of 3 minutes, even though the residual is \(-3\) minutes. The negative sign records the direction of the difference \(y-\hat{y}\); it does not make a geometric distance negative.

Compare Segments Without Losing Their Signs

When judging several plotted residuals, keep two questions separate: Which segment is longer, and which points are above or below the line? The first compares magnitudes; the second identifies signs. A long segment below the line has a large negative residual, while a shorter segment above the line has a smaller positive residual.

For example, suppose two vertical segments on the same plot have lengths of 5 and 2 response units. If the first extends below the fitted line and the second extends above it, their residuals are \(-5\) and \(+2\), respectively. The first has the greater magnitude because \(5>2\), but its sign is negative. Calling it “the largest residual” can be ambiguous: say “the greatest residual magnitude” when comparing distances, and state signs separately.

A point exactly on the fitted line has no visible gap, so its residual is zero. Two points can also have equal-length segments on opposite sides of the line. Those residuals are equal in magnitude but not equal as signed values.

Worked Example: Compare Three Plotted Points

This comparison illustrates why position and length should not be confused. Point \(A\) is above the line, but its segment is not the longest. Point \(B\) is below the line and has the greatest distance from the line among these three.

Common Mistakes and AP Exam Tips

A complete plot-based answer identifies the vertical segment at the point’s own \(x\)-value, uses the response-axis scale to describe its size, and states whether the point is above or below the line. If the question asks for a residual, include its sign; if it asks for distance or magnitude, give the nonnegative length.

• Drawing or imagining a horizontal gap. A residual compares observed and predicted responses for the same predictor value. The correct segment is vertical.
• Using the nearest point on the line. The residual is measured vertically to the line at the observed point’s \(x\)-value, not along the shortest perpendicular route.
• Reversing the sign. A point above the line has a positive residual; a point below the line has a negative residual. This follows from observed minus predicted.
• Confusing signed residual with segment length. A \(-3\)-unit residual has a segment of length 3 units. The sign describes direction, not negative distance.
• Reading only the picture’s apparent size. Use tick marks and response-axis units. If the axis scale is not uniform or the scale is unclear, do not claim a precise numerical length from appearance alone.
• Comparing residuals without clarifying magnitude. When one question asks which gap is largest, compare absolute values or vertical lengths. Then mention whether the corresponding point is above or below the line.
• Using predictor units. The gap is measured in the response’s units. A height residual is in centimeters, not weeks; a travel-time residual is in minutes, not kilometers.

A useful visual check is to imagine a vertical guide through the observed point. Where that guide crosses the fitted line is the prediction for the point. The gap from the crossing to the point gives the residual segment; its position and the response-axis scale tell you the sign and size.