Understanding the Formula slope = r (Sy/Sx)

1. Understanding the Formulaslope = r (Sy/Sx) Al Coons Buckingham Browne & Nichols School Cambridge, MA al_coons@bbns.org

2. The Confusion • In linear regression: m =r(Sy/Sx). • So the slope of a LSR model depends on three variables: r, Sy & Sx. Typical Explanation “In general, moving any number of standard deviations in x moves r times that number of standard deviations in y.” BVD. Ed 3. P. 174.

3. The Confusion • In linear regression: m =r(Sy/Sx). • So the slope of a LSR model depends on three variables: r, Sy & Sx. A Question Can we understand the interactions of r, Sy, & Sx?

4. First Look: Keeping Sy and Sx Constant • Task • In linear regression: m =r(Sy/Sx). Draw two different data sets in which Sy and Sx are unchanged yet r is different.

5. 3 3 4 4 5 5 6 6 7 7 8 8 9 9 34 34 Sy =8.64 Sy=? Sy=8.64 10 10 Sx=? Sx=2.16 Sx=2.16 r = ? r = 1 r = .214 r = ? slope = r (Sy/Sx)=? slope = r (Sy/Sx)=? slope = r (Sy/Sx) =1(8.64/2.16) = 4 slope = r (Sy/Sx) =.214(8.64/2.16) ~ .856

6. Second Look: Keeping r and Sx Constant • Task • In linear regression: m =r(Sy/Sx). Draw two different data sets in which r and Sx each are unchanged yet r is different.

7. 3 3 4 4 5 5 6 6 7 7 8 8 9 9 What is the transformation? 34 Sy=8.64 17 Sy=? Sy=4.32 10 5 Sx=? Sx=2.16 Sx=2.16 r = ? r = ? r = .214 r = .214 slope = r (Sy/Sx)=? slope = r (Sy/Sx) =.214(8.64/2.16) =.856 slope = r (Sy/Sx) =.214(4.32/2.16) ~ .428 slope = r (Sy/Sx)=?