Assessing Fit by Analyzing Residuals

Lesson 1 of 1: Reading the Leftover Misses

In this lesson:

• Compute residuals: observed minus predicted
• Read a residual plot to judge a fit

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Define a residual and compute residuals for points
2. Construct a residual plot with a zero line
3. Interpret the plot — scatter vs pattern
4. Decide whether to keep the model or refit

The Line Looks Fine, But Is It Good?

You fit a line and it looks reasonable. Is it actually good?

• The line sits on the points, hiding the small misses
• We need a tool that shows only how far off each point is

That tool is the residual. Let's define it precisely.

Evaluate One Point, Then Subtract

Using , a student is predicted $25 but paid $26.

Subtract observed minus predicted:

That signed difference is the residual. You measured a miss.

A Residual Is the Signed Vertical Gap

The vertical gap from the point to the line, observed − predicted.

Positive Above the Line, Negative Below

The sign tells you the direction of the miss:

• Above the line → positive residual (model under-predicted)
• Below the line → negative residual (model over-predicted)

Never drop the sign — it's what reveals patterns later.

Compute Residuals for Several Points

Observed minus predicted, keeping the sign:

Your Turn: Compute a Residual

The model predicts 40; the observed value is 37.

• Compute the residual (observed minus predicted)
• What does its sign tell you?

Work it out, then advance.

Answer: . Negative → below the line; the model over-predicted.

You Have the Misses — Now Plot Them

You computed signed residuals: +1, −2, +1, −2.

• A list is hard to read for patterns
• Plotting the residuals against x turns them into a picture

Next: build the residual plot.

The Residual Plot: Misses Against x

Residual vs x, with a bold zero line for a perfect prediction.

Flattening the Fitted Line to Zero

Imagine bending the slanted fitted line flat to horizontal.

• That flattened line becomes the zero line
• Each point's distance from the line → distance from zero

We remove the trend so only the misses remain.

The Residual Plot Is Not the Scatter Plot

Same data, two different pictures — the trend is removed on the right.

Your Turn: Build a Residual Plot

Plot these residuals against x, with a zero line:

• : +2 · : −1 · : +1 · : −2

Sketch the zero line, then plot each point, then advance.

Answer: Two points above zero (at x=1, 3), two below (at x=2, 4), scattered — no pattern.

The Plot Is Built, Now Read Its Shape

Everything so far was setup. Now the plot becomes a diagnostic.

• Its shape tells you if the model captured the relationship
• The key reading will feel backwards at first

Let's read the three cases.

Random Scatter Is the Goal, Not a Problem

Random, patternless scatter means the model captured the structure.

A Curve or U-Shape: Wrong Model Family

A clear curved or U-shaped band of residuals is a pattern.

• It means the data bends but you fit it with a line
• The data is fine — refit with a curve (back to 6.a)

A pattern indicts the model family, not the data.

A Fan: Spread Grows With x

A fan shape — residuals widening as x increases.

• The model fits the center, but the spread grows
• Predictions are less reliable at large x

The model isn't wrong, but trust it more at one end.

The Keep-or-Refit Decision From Residuals

The residual plot gives a clear verdict:

• Random scatter → keep the model
• A pattern (curve or fan) → refit, usually a new family

Informal — the eye reads the shape, no formulas.

Quick Check: Read This Residual Plot

The residuals form a clear U-shape — low middle, high ends.

• What does this shape tell you about the fit?
• What should you do next?

Decide, then advance.

Answer: A pattern → wrong family; the data curves. Refit with a quadratic.

Sort Three Residual Plots by Verdict

Match each residual plot to its verdict:

1. Random scatter
2. A clear curve
3. A fan widening to the right

Assign each a verdict, then advance.

Answer: 1. Good fit — keep. 2. Wrong family — refit. 3. Spread grows — less reliable at large x.

Full Task: Compute, Plot, Judge, Decide

Given data and a fitted line, run the diagnostic:

1. Compute each residual (signed)
2. Sketch the residual plot
3. Judge the fit; decide keep or refit

Do all three, then advance.

Answer: Compute observed − predicted; plot vs x; random → keep, pattern → refit.

Key Takeaways and What's Next

✓ A residual is signed: observed minus predicted

✓ Plot residual vs x — a different picture from the scatter

✓ Random scatter → keep; a pattern → refit

A pattern indicts the model, not the data

Next: the correlation coefficient (C.8).