1. **Problem Statement:** We have a set of examination scores: 55, 65, 72, 40, 70, 65, 69, 50, 45, 80, 48, 59, 76, 77, 55, 90. We need to create a frequency table with grouped data using an interval size of 15. We will list the true limits, midpoints, cumulative frequency, group percentage, and degrees for each group. 2. **Step 1: Determine the class intervals** The minimum score is 40 and the maximum is 90. Using an interval size of 15, the classes are: 40-54, 55-69, 70-84, 85-99. 3. **Step 2: True limits** True limits account for the smallest and largest possible values in each class, usually by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. So the true limits are: $$ \text{Class} \quad \text{True Limits}\\ 40-54 \quad 39.5 - 54.5\\ 55-69 \quad 54.5 - 69.5\\ 70-84 \quad 69.5 - 84.5\\ 85-99 \quad 84.5 - 99.5 $$ 4. **Step 3: Midpoints** Midpoint is the average of the lower and upper class limits: $$ \text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2} $$ Calculations: $$ \frac{40 + 54}{2} = 47, \quad \frac{55 + 69}{2} = 62, \quad \frac{70 + 84}{2} = 77, \quad \frac{85 + 99}{2} = 92 $$ 5. **Step 4: Frequency count** Count how many scores fall into each class: - 40-54: 40, 45, 48, 50 (4 scores) - 55-69: 55, 55, 59, 65, 65, 69 (6 scores) - 70-84: 70, 72, 76, 77, 80 (5 scores) - 85-99: 90 (1 score) 6. **Step 5: Cumulative frequency** Add frequencies cumulatively: - 40-54: 4 - 55-69: 4 + 6 = 10 - 70-84: 10 + 5 = 15 - 85-99: 15 + 1 = 16 7. **Step 6: Group percentage** Calculate percentage for each group: $$ \text{Percentage} = \frac{\text{Frequency}}{\text{Total}} \times 100 $$ Total scores = 16 Calculations: $$ \frac{4}{16} \times 100 = 25\%, \quad \frac{6}{16} \times 100 = 37.5\%, \quad \frac{5}{16} \times 100 = 31.25\%, \quad \frac{1}{16} \times 100 = 6.25\% $$ 8. **Step 7: Degrees** Degrees for pie chart representation: $$ \text{Degrees} = \frac{\text{Frequency}}{\text{Total}} \times 360 $$ Calculations: $$ \frac{4}{16} \times 360 = 90^\circ, \quad \frac{6}{16} \times 360 = 135^\circ, \quad \frac{5}{16} \times 360 = 112.5^\circ, \quad \frac{1}{16} \times 360 = 22.5^\circ $$ 9. **Final frequency table:** | Class | True Limits | Midpoint | Frequency | Cumulative Frequency | Percentage | Degrees | |-------|-------------|----------|-----------|----------------------|------------|---------| | 40-54 | 39.5-54.5 | 47 | 4 | 4 | 25% | 90° | | 55-69 | 54.5-69.5 | 62 | 6 | 10 | 37.5% | 135° | | 70-84 | 69.5-84.5 | 77 | 5 | 15 | 31.25% | 112.5° | | 85-99 | 84.5-99.5 | 92 | 1 | 16 | 6.25% | 22.5° | This completes the grouped frequency table with all requested values.