1. **Problem statement:** Calculate mean (2 methods), median, mode (2 methods), third quartile $Q_3$, seventh decile $D_7$, eighty-fifth percentile $P_{85}$, and construct Frequency Distribution Table (FDT), Histogram, and Bargraph from the given grouped data. Given data: Rental rate intervals and frequencies (number of market stalls): $150-179:16$, $180-209:9$, $210-239:12$, $240-269:15$, $270-299:35$, $300-329:45$, $330-359:37$, $360-389:35$, $390-419:20$, $420-449:16$ Total frequency $N = 16+9+12+15+35+45+37+35+20+16=240$ 2. **Mean calculation:** - Method 1 (Direct method): Calculate midpoints $x_i$ of each class and use $\bar{x} = \frac{\sum f_i x_i}{N}$ Midpoints: $164.5, 194.5, 224.5, 254.5, 284.5, 314.5, 344.5, 374.5, 404.5, 434.5$ Calculate $f_i x_i$ and sum: $16\times164.5=2632$, $9\times194.5=1750.5$, $12\times224.5=2694$, $15\times254.5=3817.5$, $35\times284.5=9957.5$, $45\times314.5=14152.5$, $37\times344.5=12746.5$, $35\times374.5=13107.5$, $20\times404.5=8090$, $16\times434.5=6952$ Sum $= 2632+1750.5+2694+3817.5+9957.5+14152.5+12746.5+13107.5+8090+6952=71800$ approximately Mean $= \frac{71800}{240} = 299.17$ - Method 2 (Assumed mean method): Choose assumed mean $A=314.5$ (midpoint of $300-329$) Calculate deviations $d_i = x_i - A$ and $f_i d_i$: $16\times(-150)= -2400$, $9\times(-120)= -1080$, $12\times(-90)= -1080$, $15\times(-60)= -900$, $35\times(-30)= -1050$, $45\times0=0$, $37\times30=1110$, $35\times60=2100$, $20\times90=1800$, $16\times120=1920$ Sum $f_i d_i = -2400 -1080 -1080 -900 -1050 +0 +1110 +2100 +1800 +1920 = 420$ Mean $= A + \frac{\sum f_i d_i}{N} = 314.5 + \frac{420}{240} = 314.5 + 1.75 = 316.25$ 3. **Median calculation:** Median class is where cumulative frequency $\geq \frac{N}{2} = 120$ Cumulative frequencies: $16, 25, 37, 52, 87, 132, 169, 204, 224, 240$ Median class: $300-329$ (6th class) Median formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ Where $L=300$, $F=87$ (cumulative freq before median class), $f_m=45$, $h=30$ $$\text{Median} = 300 + \left(\frac{120 - 87}{45}\right) \times 30 = 300 + \frac{33}{45} \times 30 = 300 + 22 = 322$$ 4. **Mode calculation:** - Method 1 (Using modal formula): Modal class is class with highest frequency: $300-329$ with $45$ $$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ Where $L=300$, $f_1=45$, $f_0=35$ (previous freq), $f_2=37$ (next freq), $h=30$ $$\text{Mode} = 300 + \frac{(45 - 35)}{(2\times45 - 35 - 37)} \times 30 = 300 + \frac{10}{18} \times 30 = 300 + 16.67 = 316.67$$ - Method 2 (Using histogram peak): The mode corresponds to the highest bar in histogram, which is $300-329$ interval, so mode is approximately midpoint $314.5$ 5. **Third quartile $Q_3$ calculation:** $Q_3$ is the value at $\frac{3N}{4} = 180$th observation Cumulative frequencies show $Q_3$ class is $330-359$ (7th class) since cumulative freq at 6th class is 132 and at 7th is 169, at 8th is 204 $$Q_3 = L + \left(\frac{\frac{3N}{4} - F}{f} \right) \times h = 330 + \left(\frac{180 - 132}{37} \right) \times 30 = 330 + \frac{48}{37} \times 30 = 330 + 38.92 = 368.92$$ 6. **Seventh decile $D_7$ calculation:** $D_7$ is the value at $\frac{7N}{10} = 168$th observation From cumulative freq, $D_7$ class is $330-359$ (7th class) $$D_7 = 330 + \left(\frac{168 - 132}{37} \right) \times 30 = 330 + \frac{36}{37} \times 30 = 330 + 29.19 = 359.19$$ 7. **Eighty-fifth percentile $P_{85}$ calculation:** $P_{85}$ is the value at $0.85N = 204$th observation From cumulative freq, $P_{85}$ class is $360-389$ (8th class) $$P_{85} = 360 + \left(\frac{204 - 169}{35} \right) \times 30 = 360 + \frac{35}{35} \times 30 = 360 + 30 = 390$$ 8. **Frequency Distribution Table (FDT):** Already given as rental rate intervals and frequencies. 9. **Histogram and Bargraph:** - Histogram: Rental rate intervals on x-axis, frequencies on y-axis, bars with heights 16, 9, 12, 15, 35, 45, 37, 35, 20, 16. - Bargraph: Same as histogram but bars represent discrete intervals with frequencies as heights. Final answers: - Mean (method 1): $299.17$ - Mean (method 2): $316.25$ - Median: $322$ - Mode (method 1): $316.67$ - Mode (method 2): $314.5$ - $Q_3 = 368.92$ - $D_7 = 359.19$ - $P_{85} = 390$