To calculate the standard deviation $s$ of the sample data using the provided frequency distribution, we'll follow the steps described in the formula:

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

Step-by-Step Calculation

1. Calculate the midpoint $x$ for each class interval: The midpoint for each class interval is found by averaging the lower and upper bounds of the interval.

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

For each class interval:

• 20–29: $\frac{20 + 29}{2} = 24.5$
• 30–39: $\frac{30 + 39}{2} = 34.5$
• 40–49: $\frac{40 + 49}{2} = 44.5$
• 50–59: $\frac{50 + 59}{2} = 54.5$
• 60–69: $\frac{60 + 69}{2} = 64.5$
• 70–79: $\frac{70 + 79}{2} = 74.5$
• 80–89: $\frac{80 + 89}{2} = 84.5$

2. Create a table with midpoints $x$, frequencies $f$, and calculate $f \cdot x$ and $f \cdot x^2$ for each class interval:

\hline \text{Class Interval} & \text{Frequency } f & \text{Midpoint } x & f \cdot x & f \cdot x^2 \\ \hline 20–29 & 2 & 24.5 & 2 \cdot 24.5 = 49 & 2 \cdot 24.5^2 = 2 \cdot 600.25 = 1200.5 \\ 30–39 & 1 & 34.5 & 1 \cdot 34.5 = 34.5 & 1 \cdot 34.5^2 = 1 \cdot 1190.25 = 1190.25 \\ 40–49 & 6 & 44.5 & 6 \cdot 44.5 = 267 & 6 \cdot 44.5^2 = 6 \cdot 1980.25 = 11881.5 \\ 50–59 & 3 & 54.5 & 3 \cdot 54.5 = 163.5 & 3 \cdot 54.5^2 = 3 \cdot 2967.25 = 8901.75 \\ 60–69 & 9 & 64.5 & 9 \cdot 64.5 = 580.5 & 9 \cdot 64.5^2 = 9 \cdot 4160.25 = 37442.25 \\ 70–79 & 34 & 74.5 & 34 \cdot 74.5 = 2533 & 34 \cdot 74.5^2 = 34 \cdot 5550.25 = 188708.5 \\ 80–89 & 35 & 84.5 & 35 \cdot 84.5 = 2957.5 & 35 \cdot 84.5^2 = 35 \cdot 7140.25 = 249908.75 \\ \hline \end{array}$$ 3. **Compute the sums $$\sum (f \cdot x)$$ and $$\sum (f \cdot x^2)$$:** - $$\sum (f \cdot x) = 49 + 34.5 + 267 + 163.5 + 580.5 + 2533 + 2957.5 = 6585$$ - $$\sum (f \cdot x^2) = 1200.5 + 1190.25 + 11881.5 + 8901.75 + 37442.25 + 188708.5 + 249908.75 = 498233.5$$ 4. **Calculate $$n$$, the total number of sample values:** $$n = 2 + 1 + 6 + 3 + 9 + 34 + 35 = 90$$ 5. **Apply the standard deviation formula:** $$s = \sqrt{\frac{90 \times 498233.5 - (6585)^2}{90 \times 89}}$$ Let's calculate the expression inside the square root: $$90 \times 498233.5 = 44841015$$ $$(6585)^2 = 43372025$$ $$44841015 - 43372025 = 1468990$$ $$90 \times 89 = 8010$$ $$s = \sqrt{\frac{1468990}{8010}} = \sqrt{183.43} = 13.54$$ ### Final Answer: The computed standard deviation $$s$$ is approximately 13.54. ### Comparison: The given standard deviation from the original list of data values is 11.1. The computed standard deviation (13.54) is slightly higher than the original standard deviation (11.1). Would you like any additional details or further explanation? Here are five related questions you might find helpful: 1. How do you calculate the variance from the standard deviation? 2. What is the difference between population and sample standard deviation? 3. Why is the standard deviation important in statistical analysis? 4. How does changing class intervals affect the standard deviation? 5. What are the limitations of using grouped data to compute standard deviation? **Tip**: The standard deviation is sensitive to outliers—extreme values can significantly affect it!