To calculate the standard deviation from the given frequency distribution table, we can follow these steps:

Step 1: Find the Midpoint for Each Class

The midpoint ($x_i$) for each class interval is found by averaging the lower and upper limits of each class.

$\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

Step 2: Calculate the Deviation and Squared Deviation

For each class, subtract the midpoint from the mean ($\overline{x}$) and square the result to get the squared deviation.

Step 3: Multiply the Squared Deviation by the Frequency

Multiply each squared deviation by the corresponding frequency.

Step 4: Compute the Variance

To compute the variance, sum up all the values from Step 3 and divide by the total number of data points (total frequency).

$\text{Variance} = \frac{\sum f \cdot (x_i - \overline{x})^2}{N}$

Step 5: Take the Square Root

The standard deviation ($\sigma$) is the square root of the variance.

I'll perform the calculations step by step to find the standard deviation based on the data from the image:

Next, I'll calculate the mean, variance, and standard deviation.The calculated standard deviation for the given frequency table is approximately 11.89, which does not match any of the provided options. This might indicate a misunderstanding in interpreting the class intervals or options provided.

Would you like a breakdown of each step or have further questions about the process?

Here are 5 related questions for further practice:

1. How do you calculate the mean of a frequency distribution?
2. What is the formula for variance in grouped data?
3. Why do we square deviations when calculating variance?
4. How does the choice of class interval affect the standard deviation?
5. What is the difference between population and sample standard deviation?

Tip: Always ensure that class intervals are interpreted correctly, as they affect the calculation of midpoints and other statistics.