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 find the relationship between the standard deviations \( \sigma_1 \) and \( \sigma_2 \) of two distributions given their coefficients of variation and means.

### Step-by-Step Solution:

1. **Understand the Coefficient of Variation (CV)**:
The coefficient of variation is defined as:
\[ CV = \frac{\sigma}{\text{mean}} \times 100 \]
where \( \sigma \) is the standard deviation and mean is the average of the distribution.

2. **Set Up Equations for Each Distribution**:
For the first distribution:
\[ CV_1 = \frac{\sigma_1}{\text{mean}_1} \times 100 \]
Given \( CV_1 = 15 \) and \( \text{mean}_1 = 20 \):
\[ 15 = \frac{\sigma_1}{20} \times 100 \]
Rearranging gives:
\[ \sigma_1 = \frac{15 \times 20}{100} = 3 \]

For the second distribution:
\[ CV_2 = \frac{\sigma_2}{\text{mean}_2} \times 100 \]
Given \( CV_2 = 20 \) and \( \text{mean}_2 = 10 \):
\[ 20 = \frac{\sigma_2}{10} \times 100 \]
Rearranging gives:
\[ \sigma_2 = \frac{20 \times 10}{100} = 2 \]

3. **Establish the Relationship Between \( \sigma_1 \) and \( \sigma_2 \)**:
Now we have:
\[ \sigma_1 = 3 \quad \text{and} \quad \sigma_2 = 2 \]
To find the relationship, we can express it as:
\[ \frac{\sigma_1}{\sigma_2} = \frac{3}{2} \]
This can be rearranged to:
\[ 2\sigma_1 = 3\sigma_2 \]

4. **Final Relation**:
Thus, the relationship between the standard deviations is:
\[ 2\sigma_1 = 3\sigma_2 \]

### Conclusion:
The relationship derived from the coefficients of variation and means is:
\[ 2\sigma_1 = 3\sigma_2 \]