Question

If $G$ is the $GM$ of the product of r sets of observations with geometric means ${G_1}$​,${G_2}$​,...,${G_r}$​ respectively, then $G$ is equal to
A. \[\log ({G_1}) + \log ({G_2}) + ....... + \log ({G_r})\]
B. \[{G_1}{G_2}.....{G_r}\]
C. \[\log ({G_1}).\log ({G_2}).........\log ({G_r})\]
D. None of these

Answer 1

Hint: Mean is the most commonly used measure of central tendency. The average value or mean which indicates the central tendency by using the product of the values of the asset of numbers is called a geometric mean (GM). Formula for GM = (product of all numbers in the set)1/nth , where n is total numbers in the set. In other words, GM is defined as the nth root of the product of n numbers.

Complete step by step answer:
Let $x$ be the product of numbers \[{x_1},{x_2},....,{x_r}\] corresponding to r sets of observations.
\[x = ({x_1} \times {x_2} \times .... \times {x_r})\]
\[ \Rightarrow x = ({x_1}.{x_2}....{x_r})\]
We know that, the geometric mean formula can also be represented as below:
\[\log x = \log {x_1} + \log {x_2} + ..... + \log {x_r}\]
\[ \Rightarrow \sum \log x = \sum \log {x_1} + \sum \log {x_2} + ..... + \sum \log {x_r}\]
\[ \Rightarrow \dfrac{{\sum \log x}}{n} = \dfrac{{\sum \log {x_1}}}{n} + \dfrac{{\sum \log {x_2}}}{n} + ..... + \dfrac{{\sum \log {x_r}}}{n}\]
\[ \Rightarrow \log G = \log ({G_1}) + \log ({G_2}) + ....... + \log ({G_r})\]
\[ \Rightarrow \log G = \log ({G_1}{G_2}.....{G_r})\]
\[ \therefore G = {G_1}.{G_2}......{G_r}\]

Note: The different types of means are Arithmetic mean (AM), GM and Harmonic mean (HM). The relation between all three is $G{M^2} = AM \times HM$. GM is applied only to positive values meaning there will be no zero and negative values. The difference between the arithmetic mean and geometric mean is that, we add numbers in arithmetic mean whereas we calculate the product of numbers in geometric mean. The harmonic mean is used to average ratios.