Questions: Find the sample standard deviation for the following Grouped Frequency Data Table (GFDT). Lower Class Limit Upper Class Limit Frequency --------- 70 74 5 75 79 7 80 84 4 85 89 4 90 94 7 95 99 10 100 104 15 105 109 9 110 114 4

Find the sample standard deviation for the following Grouped Frequency Data Table (GFDT). Lower Class Limit Upper Class Limit Frequency --------- 70 74 5 75 79 7 80 84 4 85 89 4 90 94 7 95 99 10 100 104 15 105 109 9 110 114 4
Transcript text: Find the sample standard deviation for the following Grouped Frequency Data Table (GFDT). \begin{tabular}{|l|l|l||} \hline \begin{tabular}{l} Lower Class \\ Limit \end{tabular} & \begin{tabular}{l} Upper Class \\ Limit \end{tabular} & Frequency \\ \hline 70 & 74 & 5 \\ \hline 75 & 79 & 7 \\ \hline \hline 80 & 84 & 4 \\ \hline 85 & 89 & 4 \\ \hline 90 & 94 & 7 \\ \hline 95 & 99 & 10 \\ \hline 100 & 104 & 15 \\ \hline 105 & 109 & 9 \\ \hline 110 & 114 & 4 \\ \hline \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Calculate the Mean

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

\[ \mu = \frac{\sum x_i}{n} \]

where \( \sum x_i = 6130.0 \) and \( n = 65 \). Thus,

\[ \mu = \frac{6130.0}{65} = 94.31 \]

Step 2: Calculate the Variance

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

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

Given that \( \sigma^2 = 142.25 \), we can state:

\[ \sigma^2 = 142.25 \]

Step 3: Calculate the Standard Deviation

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{142.25} = 11.93 \]

Final Answer

The variance and standard deviation of the dataset are:

\[ \boxed{\text{Variance} = 142.25} \] \[ \boxed{\text{Standard Deviation} = 11.93} \]

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