1. **Problem Statement:** We have scores of 28 pupils and need to group them into 5 class intervals, then find the frequency distribution, mean, median, and mode. 2. **Step 1: Find the range and class width.** - Minimum score = 9 - Maximum score = 98 - Range = $98 - 9 = 89$ - Number of classes = 5 - Class width = $\frac{89}{5} = 17.8 \approx 18$ 3. **Step 2: Define class intervals starting from 9:** - 9 - 26 - 27 - 44 - 45 - 62 - 63 - 80 - 81 - 98 4. **Step 3: Tally frequencies for each class:** - 9 - 26: 9, 12, 12, 12 (4) - 27 - 44: 35, 35, 37, 43, 43 (5) - 45 - 62: 47, 54, 56, 63 (4) *Note: 63 belongs to next class, so 3 here* - 63 - 80: 63, 65, 65, 65, 67, 67, 76, 78, 78, 80 (10) - 81 - 98: 87, 89, 92, 98 (4) 5. **Step 4: Frequency distribution table:** | Class Interval | Frequency (f) | |----------------|--------------| | 9 - 26 | 4 | | 27 - 44 | 5 | | 45 - 62 | 3 | | 63 - 80 | 10 | | 81 - 98 | 4 | 6. **Step 5: Calculate midpoints (x) for each class:** - 9 - 26: $\frac{9+26}{2} = 17.5$ - 27 - 44: $\frac{27+44}{2} = 35.5$ - 45 - 62: $\frac{45+62}{2} = 53.5$ - 63 - 80: $\frac{63+80}{2} = 71.5$ - 81 - 98: $\frac{81+98}{2} = 89.5$ 7. **Step 6: Calculate $f \times x$ for each class:** - 4 * 17.5 = 70 - 5 * 35.5 = 177.5 - 3 * 53.5 = 160.5 - 10 * 71.5 = 715 - 4 * 89.5 = 358 8. **Step 7: Calculate mean:** $$\text{Mean} = \frac{\sum f x}{\sum f} = \frac{70 + 177.5 + 160.5 + 715 + 358}{28} = \frac{1481}{28} \approx 52.89$$ 9. **Step 8: Calculate median:** - Total frequency $n=28$ - Median class is where cumulative frequency $\geq \frac{n}{2} = 14$ - Cumulative frequencies: - 9-26: 4 - 27-44: 9 - 45-62: 12 - 63-80: 22 (median class) - Median class = 63-80, $l=63$, $f_m=10$, $F=12$ (cumulative before median class), $h=18$ - Median formula: $$\text{Median} = l + \left(\frac{\frac{n}{2} - F}{f_m}\right) \times h = 63 + \left(\frac{14 - 12}{10}\right) \times 18 = 63 + 0.2 \times 18 = 63 + 3.6 = 66.6$$ 10. **Step 9: Find mode:** - Mode class is the class with highest frequency = 63-80 (frequency 10) - Using mode formula: $$\text{Mode} = l + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ Where: - $l=63$ - $f_1=10$ (modal class frequency) - $f_0=3$ (previous class frequency) - $f_2=4$ (next class frequency) - $h=18$ $$\text{Mode} = 63 + \frac{10 - 3}{2 \times 10 - 3 - 4} \times 18 = 63 + \frac{7}{20 - 7} \times 18 = 63 + \frac{7}{13} \times 18 \approx 63 + 9.69 = 72.69$$ **Final answers:** - Mean $\approx 52.89$ - Median $\approx 66.6$ - Mode $\approx 72.69$