Questions: Calculate the sample standard deviation and sample variance for the following frequency distribution of final exam scores in class. If necessary, round to one more decimal place than the largest number of decimal places given in the data. Final Exam Scores Class Frequency 51-60 10 61-70 5 71-80 5 81-90 6

Calculate the sample standard deviation and sample variance for the following frequency distribution of final exam scores in class. If necessary, round to one more decimal place than the largest number of decimal places given in the data.

Final Exam Scores

Class Frequency

51-60 10

61-70 5

71-80 5

81-90 6

Transcript text: Calculate the sample standard deviation and sample variance for the following frequency distribution of final exam scores in class. If necessary, round to one more decimal place than the largest number of decimal places given in the data. \begin{tabular}{|c|c|} \hline \multicolumn{2}{|c|}{ Final Exam Scores } \\ \hline Class & Frequency \\ \hline $51-60$ & 10 \\ \hline $61-70$ & 5 \\ \hline $71-80$ & 5 \\ \hline $81-90$ & 6 \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Construct the Dataset

The frequency distribution of final exam scores is given as follows:

\[ \begin{array}{|c|c|} \hline \text{Class} & \text{Frequency} \\ \hline 51-60 & 10 \\ \hline 61-70 & 5 \\ \hline 71-80 & 5 \\ \hline 81-90 & 6 \\ \hline \end{array} \]

To construct the dataset, we calculate the midpoint for each class interval and replicate it according to its frequency:

For the class $51-60$, the midpoint is $55.5$ and it appears $10$ times.
For the class $61-70$, the midpoint is $65.5$ and it appears $5$ times.
For the class $71-80$, the midpoint is $75.5$ and it appears $5$ times.
For the class $81-90$, the midpoint is $85.5$ and it appears $6$ times.

Thus, the constructed dataset is:

\[ \text{Dataset} = [55.5, 55.5, 55.5, 55.5, 55.5, 55.5, 55.5, 55.5, 55.5, 55.5, 65.5, 65.5, 65.5, 65.5, 65.5, 75.5, 75.5, 75.5, 75.5, 75.5, 85.5, 85.5, 85.5, 85.5, 85.5, 85.5] \]

Step 2: Calculate the Mean

The mean $\mu$ of the dataset is calculated as follows:

\[ \mu = \frac{\sum x_i}{n} = \frac{1773.0}{26} = 68.2 \]

where $n$ is the total number of observations, which is $26$.

Step 3: Calculate the Variance

The sample variance $\sigma^2$ is calculated using the formula:

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

Substituting the values, we find:

\[ \sigma^2 = 148.5 \]

Step 4: Calculate the Standard Deviation

The sample standard deviation $\sigma$ is the square root of the variance:

\[ \sigma = \sqrt{148.5} = 12.2 \]