Question

Prove that correlation coefficient is independent of origin and scale

Answer 1

The correlation coefficient is independent of origin and scale:

• Suppose x and y are two variables, rxy is correlation coefficient between x and y.
• Suppose you define u=x-a/h and v=y-b/k for defining change of origin and scale where h,k>0;

          x=a+h u;   y=b+k v;

          X=a+h U;   Y=b+k V;

          (x-X)=h(u-U);   (y-Y)=k(v-V);

• Formula for correlation coefficient is

           r x y= c o v(x,y)/sigma(x).sigma(y)

           r x y=E(x-X)(y-Y)/s q rt(E(x-X)^2)s q rt(E(y-Y)^2)

           r x y=E(h(u-U)k(v-V))/s q rt(Eh^2(u-U)^2))s q rt(Ek^2(v-V)^2)

           r x y=h k/h k.r u v

          Therefore, r x y = r u v.

           Hence proved.