Math 1680 Spring 2008
Chapter 10
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Chapter 10: Regression
A Quick Recall: For a study of 1,078 fathers and sons;
Average fathers’ height =68 inches
SD = 2.7 inches
Average sons’ height = 69 inches
SD = 2.7 inches
r  0.5
Question 1: Suppose a father is 72 inches tall. How tall would you
predict his son to be?
Wrong Answer: The father is 72  68  1.5 SDs taller than average.
2.7
Therefore, his son should also be 1.5 SDs taller than average, or
69  1.5( 2.7)  73 inches
tall.
A father 72 inches tall with a son 73 inches tall would lie on the SD
line, denoted by the dashed line in the diagram.
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Math 1680 Spring 2008
Chapter 10
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As seen in the diagram, the average height of the son with a 72”-tall
father is NOT 73”.
Correct Answer: Since r  0.5 , a father 1.5 SDs above average may be
predicted to have a son 0.5(1.5)  0.75 SDs above average, so
69  (0.5)(1.5)( 2.7)  71 inches
Notice that this regression estimate is at roughly the average of the
values in the 72”strip.
Ex. #1: Estimate the height of a son with a 64” tall father.
Notice that tall fathers tend to have tall sons – though sons who are not
as tall. Likewise, short fathers on average will have short sons – just
not as short. Hence the term, “regression.” There is no biological
cause to this effect – it is strictly statistical.
Ex. #2: A preschool program attempts to boost students’ IQs. The
children are tested when they enter the program (pretest), and again
when they leave the program (post-test). On both occasions, the
average IQ score was 100, with an SD of 15. Also, students with
below-average IQs on the pretest had scores that went up by 5 points,
while students with above average scores of the pretest had their scores
drop by an average of 5 points.
What is going on? Does the program equalize intelligence?
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Math 1680 Spring 2008
Chapter 10
Page 3 of 4
Answer: No. (If the program equalized intelligence, the post-test SD
would be less than 15 points.) There is no explanation for this
phenomenon other than the regression effect.
Thinking that the regression effect is due to something important is
called the regression fallacy.
Understanding the Regression Effect.
Question 2: Suppose someone gets a score of 140 on the pretest. Does
this mean that the student has an IQ of exactly 140?
Answer: No. There will always be chance error associated with the
measurement. For the sake of argument, let’s assume that the chance
error is on the order of 5 points.
Question 3: If the student gets a 140 on the pretest, there are two likely
explanations, they are:


IQ of 135, with a chance error of
IQ of 145, with a chance error of
5
5
Which of the above two choices is the likely explanation?
Answer: The first – there are more people with IQs of 135 than 145.
This explains the regression effect. If someone scores above average on
the first test, we would estimate that the true score is probably a bit
lower than the observed score.
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Math 1680 Spring 2008
Chapter 10
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Ex. #3: An instructor gives a midterm. She asks the students who
score 20 points below average to see her regularly during her office
hours for special tutoring. They all score at least average on the final.
Can this improvement be attributed to the regression effect?
Ex. #4: In a study of 1,000 families,
Husbands’ average height = 68 inches SD = 2.7 inches
Wives’ average height = 63 inches SD = 2.5 inches
r  0.25
Predict the height of the wife when the height of her husband is
A) 64 inches tall
B) 76 inches tall
C) unknown
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