Consider the unfinished frequency distribution table representing quiz scores in a math class.
(a) (4 points) Complete the frequency distribution table below:
| Letter Grade | Class Limits | Class Midpoint | Class Frequency | Relative Frequency | Percentage Frequency |
|---|---|---|---|---|---|
| D | 0-6 | | 5 | | |
| C | 7-13 | | 7 | | |
| B | 14-20 | | 9 | | |
| A | | | 4 | | |
(b) (1 point) What is the sample size for this data?
(c) (1 point) What is the class width for this grouped data?
(d) (2 points) Find [tex]$\bar{x}$[/tex]. Round your answer to a whole number.
(e) (2 points) Find [tex]$s$[/tex]. Round your answer to a whole number.
(f) (2 points) Find the exact value in reduced fraction for [tex]$s^2$[/tex].
(g) (3 points) Draw the bar chart for this grouped data. Clearly label and mark your graph with letter grade and class frequency.
(h) (3 points) Draw pie chart. Clearly label and mark your graph. Show the percentage for each slice of the pie chart and letter grade to name corresponding slice.
Answer (2)
The completed frequency distribution table includes letter grades, class limits, midpoints, frequencies, relative frequencies, and percentage frequencies. The sample size is 25, the class width is 7, the mean is approximately 13, the standard deviation is approximately 7, and the variance is 49. Bar and pie charts can be constructed to visually represent the frequency distribution and its percentage frequencies.
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Complete the frequency table.
Calculate the sample size: 25 .
Determine the mean: 13 .
Calculate the standard deviation: 7 .
Determine the variance: 100 4949 .
Explanation
Analyze the problem We are given an incomplete frequency distribution table representing quiz scores in a math class and asked to complete it, then calculate several statistical measures and create visualizations.
Complete the table First, let's complete the frequency distribution table. We need to find the class limits for letter grade A, the class midpoints for all grades, the relative frequencies, and the percentage frequencies.
Determine class limits for A The class width is the difference between the lower limits of consecutive classes. Here, the class width is 7 − 0 = 7 . Since the class limits for B are 14-20, the class limits for A are 14 + 7 = 21 to 20 + 7 = 27 . So, the class limits for A are 21-27.
Calculate class midpoints The class midpoint is the average of the lower and upper class limits. For D: 2 0 + 6 = 3 For C: 2 7 + 13 = 10 For B: 2 14 + 20 = 17 For A: 2 21 + 27 = 24
Calculate relative frequencies The relative frequency is the class frequency divided by the total frequency. The total frequency is 5 + 7 + 9 + 4 = 25 .
For D: 25 5 = 0.2 For C: 25 7 = 0.28 For B: 25 9 = 0.36 For A: 25 4 = 0.16
Calculate percentage frequencies The percentage frequency is the relative frequency multiplied by 100. For D: 0.2 ∗ 100 = 20% For C: 0.28 ∗ 100 = 28% For B: 0.36 ∗ 100 = 36% For A: 0.16 ∗ 100 = 16%
Completed Frequency Table The completed frequency distribution table is:
Letter Grade
Class Limits
Class Midpoint
Class Frequency
Relative Frequency
Percentage Frequency
D
0-6
3
5
0.2
20%
C
7-13
10
7
0.28
28%
B
14-20
17
9
0.36
36%
A
21-27
24
4
0.16
16%
Find the sample size The sample size is the sum of the class frequencies, which is 5 + 7 + 9 + 4 = 25 .
Find the class width The class width is the difference between the lower limits of consecutive classes, which is 7 − 0 = 7 .
Calculate the mean To find the mean x ˉ , we use the formula x ˉ = S am pl e S i ze ∑ ( M i d p o in t ∗ F re q u e n cy ) .
x ˉ = 25 ( 3 ∗ 5 + 10 ∗ 7 + 17 ∗ 9 + 24 ∗ 4 ) = 25 15 + 70 + 153 + 96 = 25 334 = 13.36 . Rounding to the nearest whole number, x ˉ = 13 .
Calculate the standard deviation To find the standard deviation s , we use the formula s = n − 1 ∑ F re q u e n cy ∗ ( M i d p o in t − x ˉ ) 2 .
s = 25 − 1 5 ∗ ( 3 − 13.36 ) 2 + 7 ∗ ( 10 − 13.36 ) 2 + 9 ∗ ( 17 − 13.36 ) 2 + 4 ∗ ( 24 − 13.36 ) 2 = 24 5 ∗ ( 10.36 ) 2 + 7 ∗ ( − 3.36 ) 2 + 9 ∗ ( 3.64 ) 2 + 4 ∗ ( 10.64 ) 2 = 24 5 ∗ 107.3296 + 7 ∗ 11.2896 + 9 ∗ 13.2496 + 4 ∗ 113.2096 = 24 536.648 + 79.0272 + 119.2464 + 452.8384 = 24 1187.76 = 49.49 ≈ 7.03 . Rounding to the nearest whole number, s = 7 .
Calculate the variance The variance s 2 is the square of the standard deviation. s 2 = ( 7.03 ) 2 = 49.4209 . The exact value in reduced fraction form is 24 1187.76 = 100 4949 .
Draw the bar chart For the bar chart, the x-axis represents the letter grades (D, C, B, A), and the y-axis represents the class frequencies (5, 7, 9, 4). Each bar's height corresponds to the frequency of the respective letter grade.
Draw the pie chart For the pie chart, each slice represents the percentage frequency for each letter grade. The slices are: D: 20%, C: 28%, B: 36%, A: 16%. Each slice is labeled with the letter grade and its corresponding percentage.
Final Answer (a) Completed table is in step 7. (b) The sample size is 25. (c) The class width is 7. (d) The mean is 13. (e) The standard deviation is 7. (f) The variance is 100 4949 . The bar chart and pie chart are described in steps 13 and 14.
Examples
Understanding frequency distributions and calculating statistics like mean and standard deviation are crucial in many real-world scenarios. For instance, a teacher can use this to analyze student performance on exams. A business can use it to analyze customer satisfaction scores. A researcher can use it to analyze survey responses. In each case, the frequency distribution provides a clear picture of the data, and the statistics provide a way to summarize and compare different datasets.
