Q. If $G$ is the GM of the product of $r$ set of observation with geometric means $G_{1},\ G_{2},\ldots G_{r}$ respectively, then $G$ is equal to
A) $\log G_{1} + \log G_{2} + ... + \log G_{n}$
B) $G_{1},G_{2},\ldots.,G_{r}$
C) $\log G_{1},\log G_{2},\ldots.,\log G_{n}$
D) None of the above
Explanation
(b)
Taking $x$ as the product of variates $x_{1},x_{2},\ldots.,x_{r}$ corresponding to $r$ set of observations $ie,\ x = x_{1}x_{2}\ldots\ldots,x_{r}$, we have
$\log x = \log x_{1} + \log x_{2} + \ldots + \log x_{r}$
$\Rightarrow \sum_{}^{}{\log x} = \sum_{}^{}{\log x_{1}} + \sum_{}^{}{\log x_{2}} + \ldots + \sum_{}^{}{\log x_{r}}$
$\Rightarrow \frac{1}{n}\sum_{}^{}{\log x} = \frac{1}{n}\sum_{}^{}{\log x_{1}} + \frac{1}{n}\sum_{}^{}{\log x_{2}} + \ldots + \frac{1}{n}\sum_{}^{}{\log x_{r}}$
$\Rightarrow \ \log G = \log G_{1} + \log G_{2} + \ldots + \log G_{r}$
$\Rightarrow \ G = G_{1}G_{2}\ldots G_{r}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.