1. **Problem Statement:** We have a frequency distribution table of Subjective Happiness scores for 200 students. We need to complete the missing columns: Real limits, Median, Frequency percentage, Cumulative frequency, and Cumulative frequency percentage. 2. **Step 1: Calculate Real Limits** Real limits are the boundaries of each class interval. For example, for class 120-124, the real limits are 119.5 to 124.5 (subtract 0.5 from lower limit and add 0.5 to upper limit). 3. **Step 2: Calculate Median of each class** Median is the midpoint of each class interval, calculated as $$\text{Median} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$$. 4. **Step 3: Calculate Frequency Percentage** Frequency percentage for each class is $$\frac{\text{Frequency}}{N} \times 100 = \frac{\text{Frequency}}{200} \times 100$$. 5. **Step 4: Calculate Cumulative Frequency** Cumulative frequency is the running total of frequencies from the first class up to the current class. 6. **Step 5: Calculate Cumulative Frequency Percentage** Cumulative frequency percentage is $$\frac{\text{Cumulative frequency}}{N} \times 100$$. 7. **Step 6: Complete the table:** | Class | Real limits | Median | Frequency | Frequency % | Cumulative Frequency | Cumulative Frequency % | |-------|-------------|--------|-----------|-------------|----------------------|------------------------| |120-124|119.5-124.5 |122 |8 |$\frac{8}{200} \times 100=4$|200|100| |115-119|114.5-119.5 |117 |12 |6 |192 |96 | |110-114|109.5-114.5 |112 |25 |12.5 |180 |90 | |105-109|104.5-109.5 |107 |33 |16.5 |155 |77.5 | |100-104|99.5-104.5 |102 |98 |49 |122 |61 | |95-99 |94.5-99.5 |97 |18 |9 |24 |12 | |90-94 |89.5-94.5 |92 |4 |2 |6 |3 | |85-89 |84.5-89.5 |87 |1 |0.5 |2 |1 | |80-84 |79.5-84.5 |82 |1 |0.5 |1 |0.5 | (Note: Cumulative frequencies and percentages are calculated from bottom to top to match the descending order of classes.) 8. **Step 7: Calculate Percentile 85 score** Percentile 85 means the score below which 85% of the data fall. Formula for percentile position: $$L = \text{Lower real limit of percentile class}$$ $$N = 200$$ $$CF = \text{Cumulative frequency before percentile class}$$ $$f = \text{Frequency of percentile class}$$ $$P = 85$$ $$\text{Percentile} = L + \left(\frac{P \times N/100 - CF}{f}\right) \times \text{class width}$$ 9. **Step 8: Identify percentile class** From cumulative frequency percentages, 85% lies between 77.5% and 90%, so percentile class is 110-114. 10. **Step 9: Calculate values:** $$L = 109.5$$ $$CF = 122$$ (cumulative frequency before 110-114) $$f = 25$$ $$\text{class width} = 5$$ 11. **Step 10: Calculate percentile 85 score:** $$\text{Percentile 85} = 109.5 + \left(\frac{0.85 \times 200 - 122}{25}\right) \times 5 = 109.5 + \left(\frac{170 - 122}{25}\right) \times 5 = 109.5 + \left(\frac{48}{25}\right) \times 5 = 109.5 + 1.92 \times 5 = 109.5 + 9.6 = 119.1$$ **Final answer:** The score at the 85th percentile is approximately **119.1**.