1. **Problem Statement:** We have grouped data with marks intervals and their frequencies. We need to find the first quartile (Q1), second quartile (Q2), and third quartile (Q3). 2. **Data:** Marks intervals: 0–10, 10–20, 20–30, 30–40, 40–50 Frequencies: 3, 7, 15, 10, 5 3. **Step 1: Calculate cumulative frequencies (CF):** - CF for 0–10 = 3 - CF for 10–20 = 3 + 7 = 10 - CF for 20–30 = 10 + 15 = 25 - CF for 30–40 = 25 + 10 = 35 - CF for 40–50 = 35 + 5 = 40 4. **Step 2: Total number of observations, $n = 40$** 5. **Step 3: Find positions of quartiles:** - $Q1$ position = $\frac{n}{4} = \frac{40}{4} = 10$ - $Q2$ position = $\frac{n}{2} = \frac{40}{2} = 20$ - $Q3$ position = $\frac{3n}{4} = \frac{3 \times 40}{4} = 30$ 6. **Step 4: Use the quartile formula for grouped data:** $$Q_k = L + \left(\frac{\frac{k n}{4} - F}{f}\right) \times h$$ Where: - $L$ = lower boundary of the quartile class - $F$ = cumulative frequency before the quartile class - $f$ = frequency of the quartile class - $h$ = class width - $k$ = quartile number (1, 2, or 3) 7. **Step 5: Find class width $h$:** All classes have width $10$ (e.g., 0–10, 10–20, etc.) 8. **Step 6: Find quartile classes:** - For $Q1$ (position 10): CF just before 10 is 3 (for 0–10), so quartile class is 10–20 - For $Q2$ (position 20): CF just before 20 is 10 (for 10–20), so quartile class is 20–30 - For $Q3$ (position 30): CF just before 30 is 25 (for 20–30), so quartile class is 30–40 9. **Step 7: Calculate $Q1$:** - $L = 10$, $F = 3$, $f = 7$, $h = 10$, position = 10 $$Q1 = 10 + \left(\frac{10 - 3}{7}\right) \times 10 = 10 + \left(\frac{7}{7}\right) \times 10 = 10 + 10 = 20$$ 10. **Step 8: Calculate $Q2$ (Median):** - $L = 20$, $F = 10$, $f = 15$, $h = 10$, position = 20 $$Q2 = 20 + \left(\frac{20 - 10}{15}\right) \times 10 = 20 + \left(\frac{10}{15}\right) \times 10 = 20 + 6.67 = 26.67$$ 11. **Step 9: Calculate $Q3$:** - $L = 30$, $F = 25$, $f = 10$, $h = 10$, position = 30 $$Q3 = 30 + \left(\frac{30 - 25}{10}\right) \times 10 = 30 + \left(\frac{5}{10}\right) \times 10 = 30 + 5 = 35$$ **Final answers:** - First quartile $Q1 = 20$ - Second quartile $Q2 = 26.67$ - Third quartile $Q3 = 35$