1. **Problem Statement:** Calculate the mean, median, and mode of students' scores given the frequency distribution. 2. **Given Data:** | Scores | Frequency (f) | |--------|--------------| | 11-15 | 1 | | 16-20 | 2 | | 21-25 | 5 | | 26-30 | 11 | | 31-35 | 12 | | 36-40 | 11 | | 41-45 | 5 | | 46-50 | 1 | 3. **Compute Midpoints ($x_i$):** Midpoint for each class interval is the average of the lower and upper bounds. $11-15: \frac{11+15}{2} = 13$ $16-20: \frac{16+20}{2} = 18$ $21-25: \frac{21+25}{2} = 23$ $26-30: \frac{26+30}{2} = 28$ $31-35: \frac{31+35}{2} = 33$ $36-40: \frac{36+40}{2} = 38$ $41-45: \frac{41+45}{2} = 43$ $46-50: \frac{46+50}{2} = 48$ 4. **Calculate total number of students ($n$):** $n = 1 + 2 + 5 + 11 + 12 + 11 + 5 + 1 = 48$ 5. **Calculate Mean ($\bar{x}$):** Mean formula for grouped data: $$\bar{x} = \frac{\sum f_i x_i}{n}$$ Calculate $f_i x_i$ for each class: $1 \times 13 = 13$ $2 \times 18 = 36$ $5 \times 23 = 115$ $11 \times 28 = 308$ $12 \times 33 = 396$ $11 \times 38 = 418$ $5 \times 43 = 215$ $1 \times 48 = 48$ Sum: $13 + 36 +115 + 308 + 396 + 418 + 215 + 48 = 1549$ Mean: $$\bar{x} = \frac{1549}{48} \approx 32.27$$ 6. **Calculate Median:** Locate median class where cumulative frequency just exceeds $\frac{n}{2} = 24$. Cumulative frequencies: | Class | Frequency | Cumulative Frequency | |--------|-----------|----------------------| | 11-15 | 1 | 1 | | 16-20 | 2 | 3 | | 21-25 | 5 | 8 | | 26-30 | 11 | 19 | | 31-35 | 12 | 31 | Median class is 31-35 (cf goes from 19 to 31). Median formula: $$Median = L + \left( \frac{\frac{n}{2} - F}{f_m} \right) \times w$$ Where: $L = 30$ (lower boundary of median class), $F = 19$ (cumulative frequency before median class), $f_m = 12$ (frequency of median class), $w = 5$ (class width). Calculate: $$Median = 30 + \left( \frac{24-19}{12} \right) \times 5 = 30 + \frac{5}{12} \times 5 = 30 + 2.08 = 32.08$$ 7. **Calculate Mode:** Mode is the class with the highest frequency. Highest frequency is 12 for class 31-35. Mode formula: $$Mode = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times w$$ Where: $L = 30$ $f_1 = 12$ (modal class frequency) $f_0 = 11$ (frequency before modal class 26-30) $f_2 = 11$ (frequency after modal class 36-40) $w = 5$ Calculate: $$Mode = 30 + \frac{(12 - 11)}{(2 \times 12 - 11 - 11)} \times 5 = 30 + \frac{1}{(24 - 22)} \times 5 = 30 + \frac{1}{2} \times 5 = 30 + 2.5 = 32.5$$ **Final Answers:** Mean $\approx 32.27$ Median $\approx 32.08$ Mode $\approx 32.5$