The variable `x` takes two values `x_(1)` and `x_(2)` with frequencies `f_(1)` and `f_(2)`, respectively. If `sigma` denotes the standard deviation of x, then

Options

A. `sigma^(2)=(f_(1)x_(1)^(2)+f_(2)x_(2)^(2))/(f_(1)+f_(2))-((f_(1)x_(1)+f_(2)x_(2))/(f_(1)+f_(2)))^(2)`
B. `sigma^(2)=(f_(1)f_(2))/((f_(1)+f_(2))^(2))(x_(1)-x_(2))^(2)`
C. `sigma^(2)=((x_(1)-x_(2))^(2))/((f_(1)+f_(2))^(2))`
D. none of these

Answer

To find the variance (σ²) of the variable x that takes two values x₁ and x₂ with frequencies f₁ and f₂, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total frequency (N)**:
\[ N = f_1 + f_2 \]

2. **Calculate the mean (μ)**:
The mean of the variable x can be calculated using the formula:
\[ \mu = \frac{x_1 f_1 + x_2 f_2}{N} \]

3. **Calculate the sum of squares of x values weighted by their frequencies**:
We need to calculate the weighted sum of squares:
\[ \text{Weighted Sum of Squares} = \frac{x_1^2 f_1 + x_2^2 f_2}{N} \]

4. **Use the formula for variance (σ²)**:
The variance can be calculated using the formula:
\[ \sigma^2 = \text{Weighted Sum of Squares} - \mu^2 \]
Substituting the values we calculated:
\[ \sigma^2 = \frac{x_1^2 f_1 + x_2^2 f_2}{N} - \left(\frac{x_1 f_1 + x_2 f_2}{N}\right)^2 \]

5. **Simplify the expression**:
To simplify, we can express it as:
\[ \sigma^2 = \frac{x_1^2 f_1 + x_2^2 f_2}{N} - \frac{(x_1 f_1 + x_2 f_2)^2}{N^2} \]
This leads to:
\[ \sigma^2 = \frac{(x_1^2 f_1 + x_2^2 f_2)N - (x_1 f_1 + x_2 f_2)^2}{N^2} \]

6. **Factor out common terms**:
The numerator can be factored to show the relationship:
\[ \sigma^2 = \frac{f_1 f_2 (x_1 - x_2)^2}{(f_1 + f_2)^2} \]

### Final Result:
Thus, the variance (σ²) of the variable x is given by:
\[ \sigma^2 = \frac{f_1 f_2 (x_1 - x_2)^2}{(f_1 + f_2)^2} \]