Question

Calculate the coefficient of variation of the following data: $ 20 $ , $ 18 $ , $ 32 $ , $ 24 $ , $ 26 $ .
(a) $ 20.41 $
(b) $ 2041 $
(c) $ 204.1 $
(d) None of these

Answer 1

Hint: To find the coefficient of variation, firstly we will arrange the given data in the increasing order. Then we find the mean value of the given data by finding the ratio sum of all the given data to the number of data. Then we will find the standard deviation by writing the data in the tabular form. Next, we will substitute the values of standard deviation and mean in the expression of coefficient of variation to find our answer.
 We can write the expression of mean as:
 $ {\rm{Mean}}\left( {\bar x} \right) = \dfrac{{{\rm{Sum of the data}}}}{{{\rm{Number of data}}}} $
We will also use the expression of standard deviation which can be expressed as:
 $ {\rm{Standard deviation}}\left( \sigma \right) = \sqrt {\dfrac{{{{\sum {\left( {x - \bar x} \right)} }^2}}}{n}} $
Where $ n $ is the number of given data.
Next, we will use the expression of coefficient of variation which can written as:
\[{\rm{Coefficient of variation}} = \dfrac{\sigma }{{\bar x}} \times 100\]

Given: The given data are $ 20 $ , $ 18 $ , $ 32 $ , $ 24 $ , $ 26 $ .

Complete step-by-step answer:
 We will arrange the given data in the increasing order or we can say ascending order.
18, 20, 24, 26, 32
Let us assume $ x $ for the given data.
We know that mean is the ratio of the sum of the given data to the number of data. Hence, we will find the mean of the given data which can be expressed:
 $ \begin{array}{c}
{\rm{Mean}}\left( {\bar x} \right) = \dfrac{{18 + 20 + 24 + 26 + 32}}{5}\\
\bar x = \dfrac{{120}}{5}\\
\bar x = 24
\end{array} $
Now we know that the expression of standard deviation of the given data is:
 $ {\rm{Standard deviation}}\left( \sigma \right) = \sqrt {\dfrac{{{{\sum {\left( {x - \bar x} \right)} }^2}}}{n}} $
Where $ n $ is the number of given data.
To find $ {\left( {x - \bar x} \right)^2} $ we will represent the data in the tabular form.

x	$ x - \bar x $ or $ \left( {x - 24} \right) $	$ {\left( {x - \bar x} \right)^2} $
18	$ - 6 $	36
20	$ - 4 $	16
24	0	0
26	2	4
32	8	64

Now we will find $ {\sum {\left( {x - \bar x} \right)} ^2} $ which can be expressed as:
 $ \begin{array}{l}
{\sum {\left( {x - \bar x} \right)} ^2} = 36 + 16 + 0 + 4 + 64\\
{\sum {\left( {x - \bar x} \right)} ^2} = 120
\end{array} $
 Then we will substitute 120 for $ {\sum {\left( {x - \bar x} \right)} ^2} $ and 5 for $ n $ in the expression of $ \sigma $ .
  $ \begin{array}{l}
\sigma = \sqrt {\dfrac{{120}}{5}} \\
\sigma = \sqrt {24}
\end{array} $
We know that the expression for the coefficient of variation is:
\[{\rm{Coefficient of variation}} = \dfrac{\sigma }{{\bar x}} \times 100\]
Hence, we will substitute 24 for $ \bar x $ and $ \sqrt {24} $ for $ \sigma $ in the above expression.
\[\begin{array}{l}
{\rm{Coefficient of variation}} = \dfrac{{\sqrt {24} }}{{24}} \times 100\\
{\rm{Coefficient of variation}} = 0.2041 \times 100\\
{\rm{Coefficient of variation}} = 20.41
\end{array}\]

 Therefore, the value of coefficient of variation of the given data is $ 20.41 $ . Hence option (a) is the correct answer.


Note: This question is of Statistics and hence to solve this question, we should have prior knowledge about the formulas of mean, standard deviation and coefficient of variation.
The most important step to solve this question is the rearrangement of the given data in ascending order.