Outliers, leverage and influential points

About this lesson

Statistics (AP-aligned) › Exploring two-variable data

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Outliers, leverage and influential points

Warm-up

Three quick questions from earlier lessons.

Discover

A scatterplot of 20 points has a line through them. One more point is added far to the right, well below the trend. What happens to the line?

  1. It tilts towards the new point
  2. Nothing: one point cannot change a line fitted to 20

Key idea

Some unusual points matter more than others.

An outlier in regression has a large residual. A high-leverage point has an xx-value far from xˉ\bar{x}. An influential point changes the slope (or intercept) a lot when removed; high-leverage points are often influential.

Worked example

Three unusual points.

Outlier in yy: lowers rr, barely moves the slope
x near x̄, large residual.
High leverage, not influential: it strengthens rr
x far away, on the trend.
High leverage and influential: tilts the line
x far away, off the trend.

Your turn, guided

Your turn: removing a point changes the slope from 2.1 to 0.4. By how much did the slope change?

1.71.7
2.1 − 0.4: influential.

Practice ★

  1. In a scatterplot, the xx-values lie between 10 and 51, except one point at x=153x = 153. What is that point called?
    1. A high-leverage point: its x is far from the others
    2. A point with a large residual
    3. The median point
  1. Removing one of 52 points changes the regression slope a lot. What is that point?
    1. An influential point
    2. Any outlier: all outliers change the slope a lot
    3. A typical point near the center of the data

Watch out!

Practice ★★

  1. Among 46 points, one has xx close to xˉ\bar{x} but a very large residual. Removing it mostly changes what?
    1. The correlation (it rises); the slope barely changes
    2. The slope a lot
    3. Nothing at all
  2. Among 77 points, one has xx close to xˉ\bar{x} but a very large residual. Removing it mostly changes what?
    1. The correlation (it rises); the slope barely changes
    2. The slope a lot
    3. Nothing at all

Exit ticket

Three questions to check you have got it.

  1. In a scatterplot, the xx-values lie between 10 and 51, except one point at x=153x = 153. What is that point called?
    1. A high-leverage point: its x is far from the others
    2. A point with a large residual
    3. The median point
  2. In a scatterplot, the xx-values lie between 10 and 36, except one point at x=108x = 108. What is that point called?
    1. A high-leverage point: its x is far from the others
    2. A point with a large residual
    3. The median point
  3. In a scatterplot, the xx-values lie between 10 and 30, except one point at x=90x = 90. What is that point called?
    1. A high-leverage point: its x is far from the others
    2. A point with a large residual
    3. The median point

Summary card

Unusual points

  • Outlier: large residual. Leverage: extreme xx.
  • Influential: removing it changes the line.

Watch out: leverage, not residual size, drives influence.

End of the lesson

Back to the chapter

Answers: Outliers, leverage and influential points

A

  1. A high-leverage point: its x is far from the others

B

  1. An influential point

C

  1. The correlation (it rises); the slope barely changes
  2. The correlation (it rises); the slope barely changes

D

  1. A high-leverage point: its x is far from the others
  2. A high-leverage point: its x is far from the others
  3. A high-leverage point: its x is far from the others

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