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[Σ(f * x^2)] - [Σ(f * x)]^2) / (n(n-1))) Interval 30-36 37-43 44-50 51-57 58-64 65-71 72-78 Frequency 2 2 6 4 9 32 38 Standard deviation = (Round to one decimal place as needed.)

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[Σ(f * x^2)] - [Σ(f * x)]^2) / (n(n-1)))

Interval 30-36 37-43 44-50 51-57 58-64 65-71 72-78
Frequency 2 2 6 4 9 32 38

Standard deviation = (Round to one decimal place as needed.)

Transcript text: Question 30, 3.2.37 Part 1 of 2 points Points: 0 of 1 Save 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. \[ s=\sqrt{\frac{n\left[\sum\left(f \cdot x^{2}\right)\right]-\left[\sum(f \cdot x)\right]^{2}}{n(n-1)}} \] \begin{tabular}{c|c|c|c|c|c|c|c} Interval & $30-36$ & $37-43$ & $44-50$ & $51-57$ & $58-64$ & $65-71$ & $72-78$ \\ \hline Frequency & 2 & 2 & 6 & 4 & 9 & 32 & 38 \end{tabular} Standard deviation $=$ $\square$ (Round to one decimal place as needed.)

Solution

Solution Steps

Step 1: Calculate Class Midpoints

The class midpoints $ x $ for each interval are calculated as follows:

\[ \begin{align_} 30-36 & : \frac{30 + 36}{2} = 33 \\ 37-43 & : \frac{37 + 43}{2} = 40 \\ 44-50 & : \frac{44 + 50}{2} = 47 \\ 51-57 & : \frac{51 + 57}{2} = 54 \\ 58-64 & : \frac{58 + 64}{2} = 61 \\ 65-71 & : \frac{65 + 71}{2} = 68 \\ 72-78 & : \frac{72 + 78}{2} = 75 \\ \end{align_} \]

Thus, the midpoints are $ 33, 40, 47, 54, 61, 68, 75 $.

Step 2: Calculate Total Frequency

The total number of sample values $ n $ is calculated by summing the frequencies:

\[ n = 2 + 2 + 6 + 4 + 9 + 32 + 38 = 93 \]

Step 3: Calculate $ \sum(f \cdot x) $ and $ \sum(f \cdot x^2) $

Next, we compute the sums $ \sum(f \cdot x) $ and $ \sum(f \cdot x^2) $:

\[ \begin{align_} \sum(f \cdot x) & = 2 \cdot 33 + 2 \cdot 40 + 6 \cdot 47 + 4 \cdot 54 + 9 \cdot 61 + 32 \cdot 68 + 38 \cdot 75 \\ & = 66 + 80 + 282 + 216 + 549 + 2176 + 2850 = 5159 \\ \end{align_} \]

\[ \begin{align_} \sum(f \cdot x^2) & = 2 \cdot 33^2 + 2 \cdot 40^2 + 6 \cdot 47^2 + 4 \cdot 54^2 + 9 \cdot 61^2 + 32 \cdot 68^2 + 38 \cdot 75^2 \\ & = 2 \cdot 1089 + 2 \cdot 1600 + 6 \cdot 2209 + 4 \cdot 2916 + 9 \cdot 3721 + 32 \cdot 4624 + 38 \cdot 5625 \\ & = 2178 + 3200 + 13254 + 11664 + 33489 + 147968 + 213750 = 515503 \\ \end{align_} \]

Step 4: Calculate Standard Deviation

Using the formula for the standard deviation $ s $:

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

Substituting the values:

\[ s = \sqrt{\frac{93 \cdot 515503 - (5159)^2}{93 \cdot 92}} \]

Calculating the components:

\[ s = \sqrt{\frac{47962679 - 26617681}{8556}} = \sqrt{\frac{21344998}{8556}} \approx 10.2 \]

Step 5: Compare with Given Standard Deviation

The computed standard deviation is $ 10.2 $ and the given standard deviation is $ 11.1 $. The difference is:

\[ \text{Difference} = |10.2 - 11.1| = 0.9 \]