Suppose all x measurements are changed to x' = 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 lines. In particular, the deviations x - x¯ change to a (x - x¯) and the deviations y - y¯ change to c(y - y¯). Consequently,
and Sxy change to
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. Change the x and y measurements according to x’ = 2x – 3 y’ = -y + 10 Calculate r from the ( x', y' ) measurements. (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?