1. **Problem Statement:** We have scores of 50 students and need to find the mean, median, mode, first quartile (Q1), fourth decile (D4), 60th percentile (P60), and range. 2. **Step 1: Organize the data** Sort the scores in ascending order: $$34, 37, 39, 42, 44, 46, 49, 49, 50, 50, 52, 53, 54, 56, 57, 58, 59, 60, 60, 61, 64, 64, 65, 65, 66, 68, 68, 68, 70, 71, 71, 71, 73, 74, 74, 75, 75, 75, 76, 77, 77, 77, 78, 78, 84, 85, 87, 88, 90, 90, 95$$ 3. **Step 2: Calculate the Mean** Mean formula: $$\text{Mean} = \frac{\sum \text{scores}}{n}$$ Sum of scores = 53+75+52+...+60 = 3333 Number of scores $n=50$ $$\text{Mean} = \frac{3333}{50} = 66.66$$ 4. **Step 3: Calculate the Median** Median is the middle value. For even $n$, median is average of $\frac{n}{2}$th and $\left(\frac{n}{2}+1\right)$th values. Positions: 25th and 26th scores 25th score = 66, 26th score = 68 $$\text{Median} = \frac{66 + 68}{2} = 67$$ 5. **Step 4: Calculate the Mode** Mode is the most frequent score. Frequencies: 75 appears 3 times, 71 appears 3 times, 77 appears 3 times, 68 appears 3 times Modes = 68, 71, 75, 77 (multimodal) 6. **Step 5: Calculate Q1 (First Quartile)** Q1 is the 25th percentile. Position: $\frac{25}{100} \times (50+1) = 12.75$th score Interpolate between 12th (54) and 13th (56) scores: $$Q1 = 54 + 0.75 \times (56 - 54) = 54 + 1.5 = 55.5$$ 7. **Step 6: Calculate D4 (Fourth Decile)** D4 is the 40th percentile. Position: $\frac{40}{100} \times (50+1) = 20.4$th score Interpolate between 20th (61) and 21st (64) scores: $$D4 = 61 + 0.4 \times (64 - 61) = 61 + 1.2 = 62.2$$ 8. **Step 7: Calculate P60 (60th Percentile)** Position: $\frac{60}{100} \times (50+1) = 30.6$th score Interpolate between 30th (71) and 31st (71) scores: $$P60 = 71 + 0.6 \times (71 - 71) = 71$$ 9. **Step 8: Calculate Range** Range = Maximum score - Minimum score $$\text{Range} = 95 - 34 = 61$$ **Final answers:** - Mean = 66.66 - Median = 67 - Mode = 68, 71, 75, 77 - Q1 = 55.5 - D4 = 62.2 - P60 = 71 - Range = 61