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

Solution Steps

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

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

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,

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

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:

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

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

Final Answer