Coefficients of variation of two distribution are 15 and 20 and their means are 20 and 10 respectively. If their standard deviations are `sigma_1` and `sigma_2` then

Options

A. `3sigma_1=2sigma_2`
B. `sigma_1 lt sigma_2`
C. `2sigma_1 =3sigma_2`
D. None of these

Answer

To solve the problem, we need to use the formula for the coefficient of variation (CV), which is given by:

\[ CV = \frac{\sigma}{\bar{x}} \times 100 \]

where \(\sigma\) is the standard deviation and \(\bar{x}\) is the mean.

### Step 1: Write down the given information
We have two distributions with the following information:
- For the first distribution:
- Coefficient of Variation (CV1) = 15
- Mean (\(\bar{x}_1\)) = 20
- For the second distribution:
- Coefficient of Variation (CV2) = 20
- Mean (\(\bar{x}_2\)) = 10

### Step 2: Set up the equations for standard deviations
Using the formula for CV, we can set up the equations for the standard deviations \(\sigma_1\) and \(\sigma_2\):

1. For the first distribution:
\[ CV1 = \frac{\sigma_1}{\bar{x}_1} \times 100 \]
Substituting the values:
\[ 15 = \frac{\sigma_1}{20} \times 100 \]

2. For the second distribution:
\[ CV2 = \frac{\sigma_2}{\bar{x}_2} \times 100 \]
Substituting the values:
\[ 20 = \frac{\sigma_2}{10} \times 100 \]

### Step 3: Solve for \(\sigma_1\)
From the first equation:
\[ 15 = \frac{\sigma_1}{20} \times 100 \]
Rearranging gives:
\[ \sigma_1 = \frac{15 \times 20}{100} \]
Calculating:
\[ \sigma_1 = \frac{300}{100} = 3 \]

### Step 4: Solve for \(\sigma_2\)
From the second equation:
\[ 20 = \frac{\sigma_2}{10} \times 100 \]
Rearranging gives:
\[ \sigma_2 = \frac{20 \times 10}{100} \]
Calculating:
\[ \sigma_2 = \frac{200}{100} = 2 \]

### Step 5: Compare \(\sigma_1\) and \(\sigma_2\)
Now we have:
- \(\sigma_1 = 3\)
- \(\sigma_2 = 2\)

### Step 6: Determine the relationship between \(\sigma_1\) and \(\sigma_2\)
The problem asks if there is a relationship between \(\sigma_1\) and \(\sigma_2\) in the form of options provided. We can check if twice \(\sigma_1\) is equal to thrice \(\sigma_2\):

Calculating:
\[ 2 \times \sigma_1 = 2 \times 3 = 6 \]
\[ 3 \times \sigma_2 = 3 \times 2 = 6 \]

Since \(2 \sigma_1 = 3 \sigma_2\), we confirm that this relationship holds true.

### Conclusion
The correct relationship is:
\[ 2 \sigma_1 = 3 \sigma_2 \]