The geometric mean G of the product of n sets of data with geometric means G1,G2,………,Gn respectively, then

Let x1,x2,……,xn be the variates corresponding to n sets of data, each having the same number of observations say Kand x be their product. Then x=x1,x2,……,xn log⁡x=log⁡x1+log⁡x2+…+log⁡xn or Σlog⁡xK=Σlog⁡x1K+Σlog⁡x2K+……+Σlog⁡xnK or  logG=logG1+logG2+……+logGn⇒ G=G1G2……Gn