1. **State the problem:** We are given a grouped frequency distribution of marks scored by 100 students in a Biology exam. We need to find: (a) The number of students who scored between 75 and 79. (b) The median mark of the distribution to the nearest whole number. 2. **Given data:** | Marks | Number of Students | |--------|--------------------| | 90-94 | 8 | | 85-89 | 5 | | 80-84 | 10 | | 75-79 | 55 | | 70-74 | 12 | | 65-69 | 10 | 3. **Part (a):** Number of students scoring 75-79 is directly given as 55. 4. **Part (b): Find the median** - Total number of students, $N = 100$ - Median position is at $\frac{N}{2} = 50^{th}$ student. 5. **Calculate cumulative frequencies:** | Marks | Frequency ($f$) | Cumulative Frequency ($CF$) | |--------|-----------------|-----------------------------| | 65-69 | 10 | 10 | | 70-74 | 12 | 10 + 12 = 22 | | 75-79 | 55 | 22 + 55 = 77 | | 80-84 | 10 | 77 + 10 = 87 | | 85-89 | 5 | 87 + 5 = 92 | | 90-94 | 8 | 92 + 8 = 100 | 6. The median class is the class where the $50^{th}$ student lies. Since $CF$ reaches 77 at 75-79, the median class is 75-79. 7. **Median formula for grouped data:** $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times h$$ where: - $L$ = lower boundary of median class = 74.5 (assuming continuous classes) - $N$ = total frequency = 100 - $F$ = cumulative frequency before median class = 22 - $f$ = frequency of median class = 55 - $h$ = class width = 5 8. **Calculate median:** $$\text{Median} = 74.5 + \left(\frac{50 - 22}{55}\right) \times 5$$ $$= 74.5 + \left(\frac{28}{55}\right) \times 5$$ $$= 74.5 + \frac{140}{55}$$ $$= 74.5 + 2.5454...$$ $$= 77.0454...$$ 9. **Final answer:** Median $\approx 77$ (nearest whole number).