Question

If the coefficient of variation and standard deviation are \[{\text{60}}\] and \[{\text{21}}\] respectively, then the arithmetic mean of the distribution is
 \[\left( 1 \right){\text{ }}60\]
 \[\left( 2 \right){\text{ 3}}0\]
 \[\left( 3 \right){\text{ 35}}\]
 \[\left( 4 \right){\text{ 21}}\]

Answer 1

Hint: In this question there is nothing much to do. You just have to know the formula of the coefficient of variation. And the values of coefficient of variation and value of standard deviation are given to us in the question. So, it will be easy for us to calculate their mean. The formula is given below
  \[{\text{coefficient of variation = }}\dfrac{{{\text{standard deviation}}}}{{{\text{mean}}}} \times 100\% \]

Complete step-by-step answer:
It is given to us that the coefficient of variation of a distribution is \[{\text{60}}\] and its standard deviation is \[{\text{21}}\] and we have to find its arithmetic mean.
The coefficient of variation (CV) is a standardized measure of the dispersion of a probability distribution or frequency distribution. The formula for coefficient of variation is given as
 \[{\text{coefficient of variation = }}\dfrac{{{\text{standard deviation}}}}{{{\text{mean}}}} \times 100\% \] --------- (i)
Standard deviation is the square root of variance. It is a measure of the extent to which data varies from the mean. Standard deviation is denoted by \[\sigma \] .
Arithmetic mean or simply the mean or the average. Mean is denoted by \[\overline x \] .
As we have to calculate the arithmetic mean, we have to shift the mean present in the denominator of equation (i) to the left hand side and the coefficient of variation to the right hand side. Therefore, equation (i) becomes
 \[{\text{mean = }}\dfrac{{{\text{standard deviation}}}}{{{\text{coefficient of variation}}}} \times 100\]
Now substitute here the given values of standard deviation and coefficient of variation,
  \[ \Rightarrow {\text{ mean = }}\dfrac{{21}}{{60}} \times 100\]
 \[ \Rightarrow {\text{ mean = }}\dfrac{{21}}{6} \times 10\]
On further solving we get
 \[ \Rightarrow {\text{ }}\overline x {\text{ = }}\dfrac{7}{2} \times 10\]
On dividing ten by two we get,
 \[ \Rightarrow {\text{ }}\overline x {\text{ = 7}} \times 5\]
On multiplying both the numbers we get
 \[ \Rightarrow {\text{ }}\overline x = {\text{ 3}}5\]
Therefore, the required value of arithmetic mean is \[{\text{3}}5\] .
Hence the correct option is \[\left( 3 \right){\text{ 35}}\]
So, the correct answer is “Option 3”.

Note: You have to keep in mind the formula of coefficient of variation. Don’t get confused between arithmetic mean or mean as both are the same. You should know the symbols of these definitions. Avoid mistakes while doing calculations.