To find the standard deviation, $s$, of the sample data summarized in the frequency distribution table, we need to use the given formula:

$s = \sqrt{\frac{n \left( \sum (f \cdot x^2) \right) - \left( \sum (f \cdot x) \right)^2}{n(n-1)}}$

where:

• $x$ is the class midpoint,
• $f$ is the class frequency,
• $n$ is the total number of sample values.

Step-by-step Solution

1. Find the Midpoints for Each Class Interval:

   The midpoint ($x$) of each class interval is calculated as:

   $x = \frac{\text{lower class limit} + \text{upper class limit}}{2}$

   For each interval:

   • $20-29: x = \frac{20 + 29}{2} = 24.5$
   • $30-39: x = \frac{30 + 39}{2} = 34.5$
   • $40-49: x = \frac{40 + 49}{2} = 44.5$
   • $50-59: x = \frac{50 + 59}{2} = 54.5$
   • $60-69: x = \frac{60 + 69}{2} = 64.5$
   • $70-79: x = \frac{70 + 79}{2} = 74.5$

2. Calculate $f \cdot x$, $f \cdot x^2$, and Summations:

   We need to compute $f \cdot x$, $f \cdot x^2$, and their summations.

3. Apply the Formula to Compute the Standard Deviation:

   Finally, use the formula for the standard deviation to find $s$.

Let's compute these values step-by-step.The calculated standard deviation of the sample data summarized in the frequency distribution table is approximately 11.19.

Comparison with Given Standard Deviation

• The computed standard deviation is 11.19, whereas the given standard deviation is 9.0.
• There is a noticeable difference between the two values, indicating that either the original data or the approximation in the grouped frequency distribution affects the result.

Would you like further details or have any questions?

Relative Questions:

1. How does the method of calculating standard deviation change if the data is not grouped?
2. What impact does grouping data have on the accuracy of the standard deviation?
3. How can we calculate the mean from a frequency distribution table?
4. What are the differences between sample standard deviation and population standard deviation?
5. How can the standard deviation be interpreted in the context of data dispersion?

Tip: When working with grouped data, remember that the standard deviation is an approximation and may differ from the actual standard deviation of the raw data.