Question

The standard deviation for the given data is:

• A. \(22.6\)
• B. \(13\)
• C. \(9.6\)
• D. \(6.12\)

Hint:

For grouped or frequency data, always use \(\sigma=\sqrt{\frac{\sum f(x-\bar{x})^2}{\sum f}}\).

Answer 1

The Correct Option is D

Concept:
Standard deviation measures the spread of observations around the mean. For a frequency distribution, it is calculated using: \[ \sigma=\sqrt{\frac{\sum f(x-\bar{x})^2}{\sum f}} \]

Step 1: Find the mean.
First, multiply each observation by its frequency and divide by total frequency: \[ \bar{x}=\frac{\sum fx}{\sum f} \]

Step 2: Find deviations from mean.
For each observation, calculate: \[ x-\bar{x} \]

Step 3: Square the deviations and multiply by frequency. \[ f(x-\bar{x})^2 \] Then add all such values: \[ \sum f(x-\bar{x})^2 \]

Step 4: Apply standard deviation formula. \[ \sigma=\sqrt{\frac{\sum f(x-\bar{x})^2}{\sum f}} \] After substituting the values from the given data table: \[ \sigma=6.12 \] \[ \therefore \text{Correct Answer is (D)} \]