To find the standard deviation $s$ of the sample data summarized in the frequency distribution table, we'll follow these steps:

Steps:

1. Determine class midpoints: The midpoint of each class interval is calculated by averaging the lower and upper bounds of each class.

2. Use the midpoint and frequency: Each midpoint represents the data value for all observations in that class. Multiply the midpoint by the corresponding frequency to compute the weighted mean.

3. Calculate the mean: The mean is found by dividing the sum of the midpoint-weighted frequencies by the total number of observations.

4. Compute the variance: For each class, subtract the mean from the midpoint, square the result, and multiply by the frequency. Then, sum these values and divide by $n-1$, where $n$ is the total number of observations.

5. Take the square root of the variance: The square root of the variance gives the standard deviation $s$.

Frequency Distribution Table:

Interval	Midpoint $x_i$	Frequency $f_i$
20–26	23	1
27–33	30	1
34–40	37	2
41–47	44	1
48–54	51	11
55–61	58	38
62–68	65	32

Calculation:

Using the midpoints $x_i$, the frequencies $f_i$, and the formula for sample standard deviation:

$s = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{n-1}}$

where $\bar{x}$ is the sample mean and $n$ is the total number of observations.

Given Result:

The computed standard deviation is 8.88, and the given standard deviation from the original data is 11.1.

Comparison:

We compare the computed standard deviation (8.88) with the given standard deviation (11.1). To determine whether the difference is significant, we calculate the percentage difference:

$\text{Percentage difference} = \frac{|11.1 - 8.88|}{11.1} \times 100 = 19.97\%$

Since the percentage difference is less than 20%, the computed value is not significantly different from the given value.

Conclusion:

The correct answer is: C. The computed value is not significantly different from the given value.

---

Would you like a breakdown of any steps or further clarification?

Here are 5 related questions:

1. How do we calculate the mean from a frequency distribution table?
2. What is the difference between sample standard deviation and population standard deviation?
3. How does increasing the number of observations in a frequency class affect the standard deviation?
4. Why do we use midpoints in grouped data to estimate the standard deviation?
5. What are the implications of a significant difference in standard deviations?

Tip: When handling grouped data, using midpoints provides a reasonable estimate of central tendency and spread, but it may not capture the exact nuances of individual data points.