Question

If the coefficient of variation and standard deviation are 60% and 21 respectively, the arithmetic mean of distribution is
(a) 30
(b) 21
(c) 60
(d) 35

Answer 1

Hint: We have to use the formula of coefficient of variation to the arithmetic mean. The coefficient of variation is given by the formula, \[\text{CV}=\dfrac{\text{Standard deviation}}{\text{Mean}}\times 100\%\] . We have to substitute the given values in this formula and find the value of the mean.

Complete step by step answer:
We are given that the coefficient of variation is 60% and standard deviation is 21. We have to find the arithmetic mean (AM) of distribution. We know that coefficient of variation is given by the formula
\[\begin{align}
  & \Rightarrow \text{CV}=\dfrac{\text{Standard deviation}}{\text{Mean}}\times 100\% \\
 & \Rightarrow \text{CV}=\dfrac{\sigma }{\mu }\times 100\% \\
\end{align}\]

From the given data, we can write $\sigma =21\text{ and CV}=60\%$ . Let us substitute the values in the above formula.
\[\Rightarrow 60=\dfrac{21}{\mu }\times 100\]
We can find the AM by taking $\mu $ to the LHS and 60 to the RHS.
\[\Rightarrow \mu =\dfrac{21}{60}\times 100\]
Let us cancel zeroes from 60 and 100.
\[\Rightarrow \mu =\dfrac{21}{\text{6}\require{cancel}\cancel{\text{0}}}\times 10\require{cancel}\cancel{0}\]
We can write the result of the above simplification as
\[\Rightarrow \mu =\dfrac{21}{\text{6}}\times 10\]
Let us multiply 21 by 10.
\[\Rightarrow \mu =\dfrac{210}{\text{6}}\]
We have to divide 210 by 6.
\[\Rightarrow \mu =35\]
Therefore, the arithmetic mean of the distribution is 35.

So, the correct answer is “Option d”.

Note: Students have a chance of making mistake by the writing the formula for coefficient of variation as \[\text{CV}=\dfrac{\text{Mean}}{\text{Standard deviation}}\times 100\%\] . Students must note that when we substituted the value of CV, the percentage sign on 100% is not included. Even if you substitute 60% without removing the % sign in 100%, we will get the same result after converting the percentages into its number form.