For a frequency distribution standard deviation is computed by

Options

A. `sigma=(Sigmaf(x-barx))/(Sigmaf)`
B. `sigma=sqrt(Sigmaf(x -barx)^2)/(Sigmaf)`
C. `sigma=sqrt((Sigmaf(x-barx)^2)/(Sigmaf))`
D. `sigma=sqrt((Sigma(x-barx))/(Sigmaf))`

Answer

To find the standard deviation for a frequency distribution, we can follow these steps:

### Step 1: Understand the Components
- **n**: Total number of observations (sum of frequencies).
- **fi**: Frequency of each observation.
- **xi**: Each observation value.
- **x̄**: Mean of the distribution.

### Step 2: Calculate the Mean (x̄)
The mean for a frequency distribution is calculated using the formula:
\[ x̄ = \frac{\sum (f_i \cdot x_i)}{n} \]
Where:
- \( \sum (f_i \cdot x_i) \) is the sum of the product of frequencies and observations.
- \( n = \sum f_i \) is the total number of observations.

### Step 3: Calculate the Variance (σ²)
The variance for a frequency distribution is given by:
\[ \sigma^2 = \frac{\sum f_i (x_i - x̄)^2}{n} \]
This can also be expressed as:
\[ \sigma^2 = \frac{\sum f_i x_i^2 - n \cdot (x̄)^2}{n} \]
Where:
- \( \sum f_i x_i^2 \) is the sum of the squares of the observations multiplied by their frequencies.

### Step 4: Calculate the Standard Deviation (σ)
The standard deviation is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} \]
Thus, substituting the variance formula, we get:
\[ \sigma = \sqrt{\frac{\sum f_i (x_i - x̄)^2}{n}} \]
or
\[ \sigma = \sqrt{\frac{\sum f_i x_i^2 - n \cdot (x̄)^2}{n}} \]

### Conclusion
The correct formula to compute the standard deviation for a frequency distribution is:
\[ \sigma = \sqrt{\frac{\sum f_i (x_i - x̄)^2}{n}} \]