1. **Problem Statement:** Given a cumulative frequency table of 500 observations, estimate the median, 20th percentile, and 80th percentile. 2. **Formulas and Important Rules:** - Median position: $\frac{N}{2}$ where $N$ is total observations. - Percentile position: $\frac{p}{100} \times N$ where $p$ is the percentile. - Use the formula for median/percentile in grouped data: $$\text{Value} = L + \left(\frac{P - F}{f}\right) \times c$$ where: - $L$ = lower boundary of the class containing the median/percentile - $P$ = position (median or percentile) - $F$ = cumulative frequency before the class - $f$ = frequency of the class - $c$ = class width 3. **Given Data:** - Total observations $N=500$ - Classes and cumulative frequencies: - 0-500: 41 - 500-1000: 131 - 1000-1500: 258 - 1500-2000: 370 - 2000-2500: 448 - 2500-3000: 500 4. **Median Calculation:** - Median position: $\frac{500}{2} = 250$ - Find class containing 250: cumulative frequency just before 250 is 131 (500-1000 class), next is 258 (1000-1500 class), so median class is 1000-1500. - Parameters: - $L=1000$ - $F=131$ - $f=127$ - $c=500$ - Apply formula: $$\text{Median} = 1000 + \left(\frac{250 - 131}{127}\right) \times 500 = 1000 + \left(\frac{119}{127}\right) \times 500$$ - Calculate: $$\frac{119}{127} \approx 0.937$$ $$0.937 \times 500 = 468.5$$ $$\text{Median} = 1000 + 468.5 = 1468.5$$ 5. **20th Percentile Calculation:** - Position: $\frac{20}{100} \times 500 = 100$ - Class containing 100: cumulative frequency before 100 is 41 (0-500 class), next is 131 (500-1000 class), so class is 500-1000. - Parameters: - $L=500$ - $F=41$ - $f=90$ - $c=500$ - Apply formula: $$P_{20} = 500 + \left(\frac{100 - 41}{90}\right) \times 500 = 500 + \left(\frac{59}{90}\right) \times 500$$ - Calculate: $$\frac{59}{90} \approx 0.6556$$ $$0.6556 \times 500 = 327.8$$ $$P_{20} = 500 + 327.8 = 827.8$$ 6. **80th Percentile Calculation:** - Position: $\frac{80}{100} \times 500 = 400$ - Class containing 400: cumulative frequency before 400 is 370 (1500-2000 class), next is 448 (2000-2500 class), so class is 2000-2500. - Parameters: - $L=2000$ - $F=370$ - $f=78$ - $c=500$ - Apply formula: $$P_{80} = 2000 + \left(\frac{400 - 370}{78}\right) \times 500 = 2000 + \left(\frac{30}{78}\right) \times 500$$ - Calculate: $$\frac{30}{78} \approx 0.3846$$ $$0.3846 \times 500 = 192.3$$ $$P_{80} = 2000 + 192.3 = 2192.3$$ **Final answers:** - Median $= 1468.5$ - 20th percentile $= 827.8$ - 80th percentile $= 2192.3$