If 2 data sets having 10 and 20 observation have coefficients of variation 50 and 60 respectively and arithmetic means 30 and 25 respectively, then the combined variance of those 30 observation is

Options

A. `(2075)/(3)`
B. `(2075)/(9)`
C. `(1000)/(9)`
D. `(1075)/(3)`

Answer

To find the combined variance of two data sets with given coefficients of variation and means, we will follow these steps:

### Step 1: Gather the given information
- For the first data set:
- Number of observations (n1) = 10
- Mean (x̄1) = 30
- Coefficient of Variation (CV1) = 50%

- For the second data set:
- Number of observations (n2) = 20
- Mean (x̄2) = 25
- Coefficient of Variation (CV2) = 60%

### Step 2: Calculate the standard deviations
The formula for the coefficient of variation is given by:
\[ CV = \frac{\sigma}{\bar{x}} \times 100 \]
Where:
- \( \sigma \) is the standard deviation
- \( \bar{x} \) is the mean

For the first data set:
\[ 50 = \frac{\sigma_1}{30} \times 100 \]
\[ \sigma_1 = \frac{50 \times 30}{100} = 15 \]

For the second data set:
\[ 60 = \frac{\sigma_2}{25} \times 100 \]
\[ \sigma_2 = \frac{60 \times 25}{100} = 15 \]

### Step 3: Calculate the combined mean
The combined mean \( \bar{x} \) can be calculated using the formula:
\[ \bar{x} = \frac{n_1 \cdot x̄_1 + n_2 \cdot x̄_2}{n_1 + n_2} \]
Substituting the values:
\[ \bar{x} = \frac{10 \cdot 30 + 20 \cdot 25}{10 + 20} = \frac{300 + 500}{30} = \frac{800}{30} = \frac{80}{3} \]

### Step 4: Calculate deviations from the combined mean
For the first data set:
\[ D_1 = \bar{x}_1 - \bar{x} = 30 - \frac{80}{3} = \frac{90 - 80}{3} = \frac{10}{3} \]

For the second data set:
\[ D_2 = \bar{x}_2 - \bar{x} = 25 - \frac{80}{3} = \frac{75 - 80}{3} = -\frac{5}{3} \]

### Step 5: Calculate the combined variance
The formula for combined variance is:
\[ \sigma^2_{combined} = \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2 + n_1 D_1^2 + n_2 D_2^2}{n_1 + n_2} \]

Calculating each term:
- \( \sigma_1^2 = 15^2 = 225 \)
- \( \sigma_2^2 = 15^2 = 225 \)
- \( D_1^2 = \left(\frac{10}{3}\right)^2 = \frac{100}{9} \)
- \( D_2^2 = \left(-\frac{5}{3}\right)^2 = \frac{25}{9} \)

Now substituting into the variance formula:
\[ \sigma^2_{combined} = \frac{10 \cdot 225 + 20 \cdot 225 + 10 \cdot \frac{100}{9} + 20 \cdot \frac{25}{9}}{30} \]
\[ = \frac{2250 + 4500 + \frac{1000}{9} + \frac{500}{9}}{30} \]
\[ = \frac{6750 + \frac{1500}{9}}{30} \]
\[ = \frac{6750 \cdot 9 + 1500}{270} \]
\[ = \frac{60750 + 1500}{270} \]
\[ = \frac{62250}{270} \]
\[ = \frac{2075}{9} \]

### Final Answer
The combined variance of the 30 observations is \( \frac{2075}{9} \).