1. **Problem Statement:** Given a frequency distribution with class intervals and frequencies, find the First Quartile (Q1), Third Quartile (Q3), 5th Decile (D5), 9th Decile (D9), 5th Percentile (P5), and 88th Percentile (P88). 2. **Data and Total Frequency:** Class Intervals and Frequencies: - 29 – 32: 3 - 25 – 28: 7 - 21 – 24: 10 - 17 – 20: 9 - 13 – 16: 5 - 9 – 12: 2 Total frequency $N = 36$ 3. **Step 1: Arrange classes in ascending order:** | Class Interval | Frequency | Cumulative Frequency | |---------------|-----------|----------------------| | 9 – 12 | 2 | 2 | | 13 – 16 | 5 | 7 | | 17 – 20 | 9 | 16 | | 21 – 24 | 10 | 26 | | 25 – 28 | 7 | 33 | | 29 – 32 | 3 | 36 | 4. **Step 2: Formula for quartiles, deciles, and percentiles:** - Quartile $Q_k$ is the value at position $\frac{kN}{4}$ - Decile $D_k$ is the value at position $\frac{kN}{10}$ - Percentile $P_k$ is the value at position $\frac{kN}{100}$ 5. **Step 3: Calculate positions:** - $Q_1$ position = $\frac{1 \times 36}{4} = 9$ - $Q_3$ position = $\frac{3 \times 36}{4} = 27$ - $D_5$ position = $\frac{5 \times 36}{10} = 18$ - $D_9$ position = $\frac{9 \times 36}{10} = 32.4$ - $P_5$ position = $\frac{5 \times 36}{100} = 1.8$ - $P_{88}$ position = $\frac{88 \times 36}{100} = 31.68$ 6. **Step 4: Find class intervals containing these positions:** - Position 1.8: in 9–12 (CF before = 0, freq = 2) - Position 9: in 17–20 (CF before = 7, freq = 9) - Position 18: in 17–20 (CF before = 7, freq = 9) - Position 27: in 25–28 (CF before = 26, freq = 7) - Position 31.68: in 25–28 (CF before = 26, freq = 7) - Position 32.4: in 29–32 (CF before = 33, freq = 3) 7. **Step 5: Use the formula for grouped data to find the value:** $$\text{Value} = L + \left(\frac{\text{Position} - F}{f}\right) \times h$$ Where: - $L$ = lower boundary of class - $F$ = cumulative frequency before class - $f$ = frequency of class - $h$ = class width 8. **Step 6: Calculate each value:** - For $P_5$ (position 1.8) in 9–12: - $L=8.5$, $F=0$, $f=2$, $h=4$ - $P_5 = 8.5 + \frac{1.8 - 0}{2} \times 4 = 8.5 + 3.6 = 12.1$ - For $Q_1$ (position 9) in 17–20: - $L=16.5$, $F=7$, $f=9$, $h=4$ - $Q_1 = 16.5 + \frac{9 - 7}{9} \times 4 = 16.5 + \frac{2}{9} \times 4 = 16.5 + 0.89 = 17.39$ - For $D_5$ (position 18) in 17–20: - $L=16.5$, $F=7$, $f=9$, $h=4$ - $D_5 = 16.5 + \frac{18 - 7}{9} \times 4 = 16.5 + \frac{11}{9} \times 4 = 16.5 + 4.89 = 21.39$ - For $Q_3$ (position 27) in 25–28: - $L=24.5$, $F=26$, $f=7$, $h=4$ - $Q_3 = 24.5 + \frac{27 - 26}{7} \times 4 = 24.5 + \frac{1}{7} \times 4 = 24.5 + 0.57 = 25.07$ - For $P_{88}$ (position 31.68) in 25–28: - $L=24.5$, $F=26$, $f=7$, $h=4$ - $P_{88} = 24.5 + \frac{31.68 - 26}{7} \times 4 = 24.5 + \frac{5.68}{7} \times 4 = 24.5 + 3.25 = 27.75$ - For $D_9$ (position 32.4) in 29–32: - $L=28.5$, $F=33$, $f=3$, $h=4$ - $D_9 = 28.5 + \frac{32.4 - 33}{3} \times 4 = 28.5 + \frac{-0.6}{3} \times 4 = 28.5 - 0.8 = 27.7$ 9. **Final answers:** - First Quartile $Q_1 = 17.39$ - Third Quartile $Q_3 = 25.07$ - 5th Decile $D_5 = 21.39$ - 9th Decile $D_9 = 27.7$ - 5th Percentile $P_5 = 12.1$ - 88th Percentile $P_{88} = 27.75$