Questions: For the following set of data, find the sample standard deviation, to the nearest hundredth. Data Frequency --- --- 2 3 3 8 6 2 8 4 16 6 17 2 19 1 20 9

For the following set of data, find the sample standard deviation, to the nearest hundredth.

Data  Frequency 
---  --- 
2  3 
3  8 
6  2 
8  4 
16  6 
17  2 
19  1 
20  9
Transcript text: Solution For the following set of data, find the sample standard deviation, to the nearest hundredth. \begin{tabular}{|c|c|} \hline Data & Frequency \\ \hline 2 & 3 \\ \hline 3 & 8 \\ \hline 6 & 2 \\ \hline 8 & 4 \\ \hline 16 & 6 \\ \hline 17 & 2 \\ \hline 19 & 1 \\ \hline 20 & 9 \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Expand the Data

The given data and their frequencies are expanded to create a complete dataset. The expanded data is:

\[ \text{Expanded Data} = [2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 17, 17, 19, 20, 20, 20, 20, 20, 20, 20, 20, 20] \]

Step 2: Calculate the Mean

The mean \( \mu \) is calculated using the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} \]

Where \( \sum_{i=1}^N x_i = 403 \) and \( N = 35 \). Thus,

\[ \mu = \frac{403}{35} = 11.51 \]

Step 3: Calculate the Sample Standard Deviation

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \]

Where the calculated variance is \( 54.32 \). The sample standard deviation \( \sigma \) is then:

\[ \sigma = \sqrt{54.32} = 7.37 \]

Conclusion

The sample standard deviation of the dataset is \( 7.37 \).

Final Answer

\(\boxed{7.37}\)

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