Question

The coefficient of variation of two distributions are 60% and 70% and their S.D are 21 and 16 respectively. What is their arithmetic means

Answer 1

Hint: The relation between the standard deviation and the coefficient of variation of a distribution is given as $ V=\dfrac{\sigma }{\overline{X}}\times 100 $ , where V is the coefficient of variation, $ \sigma $ is the standard deviation and $ \overline{X} $ is the mean of the distribution.
Formula used:
 $ V=\dfrac{\sigma }{\overline{X}}\times 100 $

Complete step-by-step answer:
The coefficient of variation of a distribution is denoted by V.
The standard deviation of a distribution is denoted by $ \sigma $ .
The relation between the standard deviation and the coefficient of variation of a distribution is given as $ V=\dfrac{\sigma }{\overline{X}}\times 100 $ , where $ \overline{X} $ is the mean of the distribution.
It is given that the coefficient for the first distribution is 60% and the standard deviation (S.D) for this distribution is 21.
This means that $ {{V}_{1}}=60 $ and $ {{\sigma }_{1}}=21 $ .
Let the mean of the first distribution be $ {{\overline{X}}_{1}} $ .
Then we can write that $ {{V}_{1}}=\dfrac{{{\sigma }_{1}}}{{{\overline{X}}_{1}}}\times 100 $ .
Substitute the values of the known variables.
 $ \Rightarrow 60=\dfrac{21}{{{\overline{X}}_{1}}}\times 100 $
 $ \Rightarrow {{\overline{X}}_{1}}=\dfrac{21}{60}\times 100=35 $ .
This means that the mean of the first distribution is 35.
So, the correct answer is “35.”.

It is given that the coefficient for the second distribution is 70% and the standard deviation (S.D) for this distribution is 16.
This means that $ {{V}_{2}}=70 $ and $ {{\sigma }_{2}}=16 $ .
Let the mean of the second distribution be $ {{\overline{X}}_{2}} $ .
Then we can write that $ {{V}_{2}}=\dfrac{{{\sigma }_{2}}}{{{\overline{X}}_{2}}}\times 100 $ .
Substitute the values of the known variables.
 $ \Rightarrow 70=\dfrac{16}{{{\overline{X}}_{2}}}\times 100 $
 $ \Rightarrow {{\overline{X}}_{2}}=\dfrac{16}{70}\times 100=22.85 $ .
This means that the mean of the second distribution is 22.85.
So, the correct answer is “22.85.”.

Note: Arithmetic mean of a distribution is equal to the sum of all the frequencies divided by the total numbers of observations. Arithmetic means is also called average value.
It is one of the central values of a distribution, which is analysing the data.