Questions: Find the standard deviation, s, of sample data summarized in the frequency distribution table below by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviation to the standard deviation obtained from the original list of data values, 11.1. s = sqrt((n*sum(fx^2) - [sum(fx)]^2)/(n*(n-1))) Interval 20-29 30-39 40-49 50-59 60-69 70-79 80-89 Frequency 2 5 18 38 39 7 Standard deviation = (Round to one decimal place as needed.)

Solution Steps

The class midpoints for the given intervals are calculated as follows:

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

For the intervals:

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

Thus, the class midpoints are: \[ \text{Class midpoints} = [24.5, 34.5, 44.5, 54.5, 64.5, 74.5, 84.5] \]

The total number of sample values \( n \) is calculated by summing the frequencies:

\[ n = 2 + 5 + 18 + 38 + 39 + 7 = 109 \]

Next, we calculate \( \sum(fx) \) and \( \sum(fx^2) \):

\[ \sum(fx) = 2 \cdot 24.5 + 5 \cdot 34.5 + 18 \cdot 44.5 + 38 \cdot 54.5 + 39 \cdot 64.5 + 7 \cdot 74.5 = 6130.5 \]

\[ \sum(fx^2) = 2 \cdot (24.5)^2 + 5 \cdot (34.5)^2 + 18 \cdot (44.5)^2 + 38 \cdot (54.5)^2 + 39 \cdot (64.5)^2 + 7 \cdot (74.5)^2 = 356767.25 \]

Using the formula for the standard deviation \( s \):

\[ s = \sqrt{\frac{n \sum (fx^2) - [\sum(fx)]^2}{n(n-1)}} \]

Substituting the values:

\[ s = \sqrt{\frac{109 \cdot 356767.25 - (6130.5)^2}{109 \cdot (109 - 1)}} \]

Calculating this gives:

\[ s \approx 10.5 \]

The computed standard deviation is \( 10.5 \) and the given standard deviation is \( 11.1 \). The difference between the two is:

\[ \text{Difference} = |10.5 - 11.1| = 0.6 \]

Final Answer