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Residuals & residual plots

Same idea, four ways to study it. Tap a style and find the one that clicks for you. 📺 Prefer to watch? Videos are on the lesson page.

🔍 A residual is just actual − predicted (y − ŷ): how far off the line was for that point. A random residual plot = the line fits. A patterned one = a straight line isn't the right model.

What it is

Once you have a regression line, how do you know it actually fits? A residual is the gap between an actual data point and what the line predicted, how wrong the model was for that point.

residual = actual − predicted = y − ŷ

Positive, negative, and the zero rule

Positive residual: the point is above the line, the model under-predicted (e.g. predicted 70, scored 75 → +5).

Negative residual: the point is below the line, the model over-predicted (e.g. predicted 80, scored 74 → −6).

For a least-squares line, the sum of all residuals is always 0, the over- and under-predictions balance out. Handy as a check.

The residual plot rule

Plot the residuals against x, with a zero line through the middle:

Random scatter around zero → a linear model is appropriate. ✓
A clear pattern (curve, fan shape) → the relationship is non-linear, a straight line isn't the best model.

Worked example

Using ŷ = 4.5x + 38, find Cal's residual. Cal studied x = 6 hours and scored 67.

Predict: ŷ = 4.5(6) + 38 = 27 + 38 = 65.
Residual = actual − predicted = 67 − 65 = +2.
Positive, so Cal scored 2 marks above what the line predicted.

Watch out

• Residual = actual − predicted, in that order (not predicted − actual).
• Read the residual plot's shape, not the original scatter, to judge the fit.
• Sum of residuals near 0 is a sign your line is right.
• A point far from the zero line is an outlier, mention it.

Good fit vs bad fit

Left: residuals scattered randomly around the zero line, the straight-line model is fine. Right: residuals make a clear curve, the data is really non-linear.

Where a residual comes from

The residual is the vertical gap from the actual point to the line: actual minus predicted.

Warm up first

Don't read yet, just have a go in your head:

Predicted 70, actual 75. Residual?

75 − 70 = +5. Positive, so the point is above the line.

Predicted 80, actual 74. Residual?

74 − 80 = −6. Negative, so the point is below the line.

A residual plot makes a clear U-shape. Good fit?

No. A pattern means the relationship is non-linear, a straight line isn't the best model.

Faded example: residuals from ŷ = 4.5x + 38

Rung 1 · watch one done fully

Cal: x = 6, scored 67. Predicted = 4.5(6) + 38 = 65. Residual = 67 − 65 = +2.

Rung 2 · you fill the gaps

Brooke: x = 4, scored 54. Predicted = 4.5(4) + 38 = ?. Residual = 54 − ? = ?

Check my gaps

Predicted 56, so residual = 54 − 56 = −2 (below the line).

Rung 3 · all you

Dana: x = 8, scored 72. Find Dana's residual, and say if Dana is above or below the line. Check below.

Check my answer

Predicted = 4.5(8) + 38 = 74. Residual = 72 − 74 = −2. Negative, so Dana is below the line.

Exam-style stretch: read the plot

A residual plot shows the points forming a clear downward-then-upward curve. What does this tell you about the model?

Show the answer

The curved pattern means the residuals are not random, so a linear model is not appropriate. The true relationship between the variables is non-linear, and a different (curved) model would fit better.

Say it back

In one sentence, out loud: what does a residual plot tell you that the correlation alone doesn't?

⚡ Residuals, one look

Residualactual − predicted = y − ŷ

Positivepoint above the line (model under-predicted)

Negativepoint below the line (model over-predicted)

Sumall residuals add to 0 (least-squares check)

Plot: random→ linear model fits ✓

Plot: pattern→ non-linear, line not the best model ✗

Examplex=6 scored 67, ŷ=65 → residual +2

Trapactual − predicted (that order) · read the plot's shape