1. **Problem Statement:** Given the frequency distribution of marks obtained by students in an exam with full marks 50, and the median of the data is 29, we need to find: - (1) The median class - (2) The value of $x$ - (3) The mean mark of the data - (4) The ratio of students scoring less than 20 and those scoring 20 or more 2. **Given Data:** | Obtained Marks | Number of Students | |----------------|--------------------| | 0-10 | 3 | | 10-20 | 7 | | 20-30 | 10 | | 30-40 | $x$ | | 40-50 | 10 | Median = 29 3. **Step 1: Find total number of students and median class** Total students $= 3 + 7 + 10 + x + 10 = 30 + x$ Median position $= \frac{30 + x}{2}$ Cumulative frequencies: - Up to 10: 3 - Up to 20: $3 + 7 = 10$ - Up to 30: $10 + 10 = 20$ - Up to 40: $20 + x$ - Up to 50: $20 + x + 10 = 30 + x$ Since median is 29, it lies in the class where cumulative frequency just exceeds median position. Median position $= \frac{30 + x}{2}$ must lie in the 20-30 class or 30-40 class. Given median is 29, which lies in 20-30 class range, so median class is **20-30**. 4. **Step 2: Find $x$ using median formula** Median formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times h$$ Where: - $L$ = lower boundary of median class = 20 - $N$ = total frequency = $30 + x$ - $F$ = cumulative frequency before median class = 10 - $f$ = frequency of median class = 10 - $h$ = class width = 10 Substitute median = 29: $$29 = 20 + \left(\frac{\frac{30 + x}{2} - 10}{10}\right) \times 10$$ Simplify: $$29 - 20 = \left(\frac{15 + \frac{x}{2} - 10}{10}\right) \times 10$$ $$9 = 15 + \frac{x}{2} - 10$$ $$9 = 5 + \frac{x}{2}$$ $$9 - 5 = \frac{x}{2}$$ $$4 = \frac{x}{2}$$ $$x = 8$$ 5. **Step 3: Find the mean mark** Calculate total students: $$N = 30 + x = 30 + 8 = 38$$ Calculate midpoints of classes: - 0-10: 5 - 10-20: 15 - 20-30: 25 - 30-40: 35 - 40-50: 45 Calculate sum of marks: $$\text{Sum} = 3 \times 5 + 7 \times 15 + 10 \times 25 + 8 \times 35 + 10 \times 45$$ $$= 15 + 105 + 250 + 280 + 450 = 1100$$ Mean mark: $$\bar{x} = \frac{\text{Sum}}{N} = \frac{1100}{38} \approx 28.95$$ 6. **Step 4: Find ratio of students scoring less than 20 and 20 or more** Students scoring less than 20: $$3 + 7 = 10$$ Students scoring 20 or more: $$10 + 8 + 10 = 28$$ Ratio: $$10 : 28 = 5 : 14$$ **Final answers:** - (1) Median class = 20-30 - (2) $x = 8$ - (3) Mean mark $\approx 28.95$ - (4) Ratio = 5 : 14