Suppose all x measurements are changed to x = ax + b and all y measurements to y' = cy + d, where a, b,

Suppose all x measurements are changed to xʹ = ax + b and all y measurements to y' = cy + d, where a, b, c, and d are fixed numbers (a ‰  0, c ‰  0). Then the correlation coefficient remains unchanged if a and c have the same signs; it changes sign but not numerical value if a and c are of opposite signs.
This Property of r can be verified along the line of Exercise 2.74 in Chapter 2. In Particular, the deviations x - Change to a (x - ) and the deviations y - Change to c (y - ). Consequently,
ˆšSxx, ˆšSyy, and Sxy change to |a| ˆšSxx, |c| ˆšSyy, and acSxy,
Respectively (recall that we must take the positive square root of a sum of squares of the deviations). Therefore, r changes to

(a) For a numerical verification of this property of r, consider the data of Exercise 3.18. Change the x and y measurements according to
xʹ = 2x - 3
yʹ = -y + 10
Calculate r from the (xʹ, yʹ) measurements and compare with the result of Exercise 3.18.
(b) Suppose from a data set of height measurements in inches and weight measurements in pounds, the value of r is found to be .86. What would the value of r be if the heights were measured in centimeters and weights in kilograms?

laelr- | r if a and c have the same signs -r if a and c have opposite signs a c