Prove that correlation coefficient is independent of origin and scale
Asked by admin @ in Math viewed by 288 People
Prove that correlation coefficient is independent of origin and scale
Answered by admin @
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.
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