# Understanding the Formula slope = r (Sy/Sx) Al Coons Buckingham Browne & Nichols School Cambridge, MA

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

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.

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?

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.

slope = r (Sy/Sx)=? S y =? S x =2.16 S y =8.64 r = ? 34567893456789 S x =? S y =8.64 r = 1 r =.214 S x =2.16 10 34 10 34 slope = r (Sy/Sx)=? slope = r (Sy/Sx) =.214(8.64/2.16) ~.856 slope = r (Sy/Sx) =.214(8.64/2.16) ~.856 slope = r (Sy/Sx) =1(8.64/2.16) = 4

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.

slope = r (Sy/Sx)=? Sy=?Sy=? S x =2.16 r = ? 3456789 S x =? S y =8.64 r =.214 S x =2.16 5 34 10 slope = r (Sy/Sx)=? slope = r (Sy/Sx) =.214(4.32/2.16) ~.428 slope = r (Sy/Sx) =.214(4.32/2.16) ~.428 slope = r (Sy/Sx) =.214(8.64/2.16) =.856 17 S y =4.32 What is the transformation? 3456789

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