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Statistics 9-12 / Week 04 / Thursday
4/6
Week 04 · Scatter Plots and Lines of Fit

Thursday

Residuals judge the fit
// Weeks of practice against laps run
⏱ about 20 min

Thursday: Residuals Judge the Fit

"Why squares?" Comet asks. "Why not just add the gaps?"

"Add the plain gaps and the positives cancel the negatives," Wren says. "A terrible line could score zero."

"Squares are all positive," Comet says. "And a big gap counts a lot more than two small ones."

Nova projects a second chart: minutes into the practice run against runners who had finished. "Would you like a hint? Fit a line, then look at the residuals."

Wren fits one. "Slope 2.1. But look. The first dots sit above the line, the middle ones below, the last one far above."

"A U-shape in the residuals," Comet says. "The line is missing a curve. What do you notice about the practice chart?"

"Its residuals scatter with no pattern," Wren says. "That line fits. This one does not."

How the least-squares line is chosen

Read the five steps, then put them in order below.

  1. Pick any line. For each dot, find its residual: observed y minus the line's y.
  2. Adding the plain residuals is no good. Positives above the line cancel negatives below it.
  3. Square each residual. Every square is positive, and a large gap counts far more than a small one.
  4. Add the squares. That sum measures how badly the line misses the dots.
  5. Choose the line with the smallest sum. That one line is the least-squares line.
CHOOSING THE LEAST-SQUARES LINE, IN ORDER
  • ?Square each residual so every gap counts and big gaps count more
  • ?Notice that plain residuals cancel, above against below
  • ?Choose the line with the smallest score
  • ?Add the squared residuals to score the line
  • ?Pick a line and find every residual, observed minus predicted
WHY THIS EXERCISEThe word "least-squares" is the whole method: the least sum of squared gaps.

A residual plot

A residual plot puts the explanatory variable across and each dot's residual up, with a zero line through the middle.

If the dots scatter above and below zero with no pattern, the line fits the data well.

If the dots make a curve, a U or an arch, the line is missing a pattern. A straight line is the wrong model.

Residual plot for weeks of practice and laps: eight dots scattered above and below zero with no pattern.
Minutes into the run against runners finished: six dots that curve upward, with a straight line fitted through them.
Residual plot for minutes and finishers: dots above zero, then below, then above, a U-shape.
MinutesFinished observedFinished predictedResidual
10-1.21.2
2110
323.1-1.1
445.2-1.2
577.4-0.4
6119.51.5
Observed y minus predicted y is called the this. Type one word.
The least-squares line makes the sum of the what of the residuals smallest? Type one word.
RESIDUALS AND THE FIT
  • Read the question.
  • Tap your answer.
Ash ran 3 laps after 1 week. The line predicts 2.8. What is Ash's residual? (Round to 1 place.)
Gus ran 9 laps after 7 weeks. The line predicts 8.5. What is Gus's residual? (Round to 1 place.)
Which residual plot says the straight line fits well?
The minutes-and-finishers residuals form a U. Which conclusion is justified?
StatementTrue or false?
Adding plain residuals lets positives and negatives cancel.?
A residual plot with a clear U-shape means the line fits well.?
9 - 3 = 6?
A dot below the line has a negative residual.?
WHY THIS EXERCISEResiduals are how the data talks back to the line. Learn to listen for a pattern.
Try it
On your lab plot from Wednesday, measure each dot's gap from your by-eye line in squares of graph paper.
Square the gaps and add them. Then try a slightly tilted line and see if the sum shrinks.
Draw two residual plots side by side: one with no pattern and one with a U-shape. Label which fits.

Careful reasoning. Tomorrow a falling line: warm-up minutes against lap times, and the week in review.

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