1. **Problem Statement:** Given the data of marks and the number of students scoring more than those marks, we need to: i) Convert the distribution into class intervals and draw a histogram. ii) Calculate the mean mark. iii) Find the median mark. iv) Find the range. 2. **Convert to Class Intervals:** The data shows marks more than a certain value. We can form class intervals by considering the marks as lower boundaries: - 0-9, 10-19, 20-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89 3. **Frequency Calculation:** Frequency for each class = Number of students scoring more than lower limit - Number scoring more than upper limit: - 0-9: 50 - 46 = 4 - 10-19: 46 - 40 = 6 - 20-29: 40 - 33 = 7 - 30-39: 33 - 25 = 8 - 40-49: 25 - 15 = 10 - 50-59: 15 - 8 = 7 - 60-69: 8 - 3 = 5 - 70-79: 3 - 0 = 3 - 80-89: 0 - 0 = 0 4. **Mean Calculation:** Mean formula: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ Where $f_i$ is frequency and $x_i$ is class midpoint. Class midpoints: - 0-9: 4.5 - 10-19: 14.5 - 20-29: 24.5 - 30-39: 34.5 - 40-49: 44.5 - 50-59: 54.5 - 60-69: 64.5 - 70-79: 74.5 - 80-89: 84.5 Calculate $\sum f_i x_i$: - $4 \times 4.5 = 18$ - $6 \times 14.5 = 87$ - $7 \times 24.5 = 171.5$ - $8 \times 34.5 = 276$ - $10 \times 44.5 = 445$ - $7 \times 54.5 = 381.5$ - $5 \times 64.5 = 322.5$ - $3 \times 74.5 = 223.5$ - $0 \times 84.5 = 0$ Sum of frequencies $\sum f_i = 4+6+7+8+10+7+5+3+0 = 50$ Sum of $f_i x_i = 18 + 87 + 171.5 + 276 + 445 + 381.5 + 322.5 + 223.5 + 0 = 1925$ Mean: $$\bar{x} = \frac{1925}{50} = 38.5$$ 5. **Median Calculation:** Median class is where cumulative frequency reaches half of total frequency (25). Cumulative frequencies: - 0-9: 4 - 10-19: 10 - 20-29: 17 - 30-39: 25 Median class is 30-39. Median formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times h$$ Where: - $L = 30$ (lower boundary of median class) - $N = 50$ (total frequency) - $F = 17$ (cumulative frequency before median class) - $f = 8$ (frequency of median class) - $h = 10$ (class width) Calculate: $$\text{Median} = 30 + \left(\frac{25 - 17}{8}\right) \times 10 = 30 + \frac{8}{8} \times 10 = 30 + 10 = 40$$ 6. **Range Calculation:** Range = Highest mark - Lowest mark From data, lowest mark = 0, highest mark = 80 Range = $80 - 0 = 80$ **Final answers:** - Mean mark = 38.5 - Median mark = 40 - Range = 80 Histogram can be drawn using the class intervals and frequencies calculated.