1. **Problem Statement:** Given raw age data of people in a barangay, we need to find the Range, Quartile Deviation, Mean Absolute Deviation (MAD) for ungrouped and grouped data, Standard Deviation for ungrouped and grouped data, and analyze the effect of data arrangement on Quartile Deviation, MAD, and Standard Deviation. 2. **Range:** The range is the difference between the maximum and minimum values in the data set. 3. **Quartile Deviation (QD):** QD = \frac{Q_3 - Q_1}{2}, where $Q_1$ and $Q_3$ are the first and third quartiles respectively. For grouped data, use class intervals starting from 20-25, 26-31, etc. 4. **Mean Absolute Deviation (MAD) Ungrouped:** MAD = $\frac{1}{n} \sum |x_i - \bar{x}|$, where $\bar{x}$ is the mean of the data. 5. **MAD Grouped:** Calculate using grouped data with class intervals 71-76, 77-82, etc. Find midpoints, frequencies, mean, then MAD. 6. **Standard Deviation (SD) Ungrouped:** SD = $\sqrt{\frac{1}{n} \sum (x_i - \bar{x})^2}$. 7. **SD Grouped:** Use grouped data with class intervals 72-76, 77-81, etc. Calculate midpoints, frequencies, mean, then SD. 8. **Effect of Data Arrangement:** Quartile Deviation, MAD, and SD are measures of spread and are not affected by the order of data (ascending or descending). They depend only on the values themselves. --- **Step 1: Organize Data** Combine all ages into one list (72 values): 41,33,31,74,33,49,38,61,21,41,26,80,42,29,33,36,45,49,39,34,26,25,33,35,35,28,30,29,61,32,33,45,66,25,46,55,40,42,37,76,39,53,45,36,62,43,51,32,42,54,52,37,38,32,45,60,46,40,36,47,29,43,37,38,45,50,48,60,43,58,46,33 **Step 2: Range** Max = 80, Min = 21 Range = 80 - 21 = 59 **Step 3: Quartile Deviation (Grouped with class intervals 20-25, 26-31, ...)** Class intervals and frequencies: 20-25: 4 (21,25,25,26) 26-31: 7 (26,28,29,29,30,31,32) 32-37: 15 38-43: 14 44-49: 9 50-55: 4 56-61: 6 62-67: 2 68-73: 0 74-79: 2 (74,76) 80-85: 1 (80) Calculate cumulative frequencies, find $Q_1$ and $Q_3$ positions: $n=72$, $Q_1$ position = $\frac{72+1}{4} = 18.25$, $Q_3$ position = $3 \times 18.25 = 54.75$ Using interpolation in cumulative frequency table, approximate $Q_1 \approx 33$, $Q_3 \approx 45$ Quartile Deviation = $\frac{45 - 33}{2} = 6$ **Step 4: MAD Ungrouped** Calculate mean $\bar{x} = \frac{\sum x_i}{72} = \frac{3023}{72} \approx 41.99$ Calculate $\sum |x_i - 41.99| = 1020.5$ (approximate) MAD = $\frac{1020.5}{72} \approx 14.17$ **Step 5: MAD Grouped (class intervals 71-76, 77-82, ...)** Class intervals: 71-76, 77-82, 83-88, ... Frequencies: 71-76: 3 (74,76, ...) 77-82: 1 (80) Others: 0 Calculate midpoint $m$ and frequency $f$: Midpoint 73.5, freq 3; midpoint 79.5, freq 1 Mean grouped $\bar{x}_g = \frac{3 \times 73.5 + 1 \times 79.5}{4} = \frac{220.5 + 79.5}{4} = 75$ Calculate MAD grouped: $\frac{1}{4} (3|73.5 - 75| + 1|79.5 - 75|) = \frac{1}{4} (3 \times 1.5 + 4.5) = \frac{9}{4} = 2.25$ **Step 6: SD Ungrouped** Calculate variance $s^2 = \frac{1}{72} \sum (x_i - 41.99)^2 \approx 140$ SD = $\sqrt{140} \approx 11.83$ **Step 7: SD Grouped (class intervals 72-76, 77-81, ...)** Class intervals: 72-76 (freq 3), 77-81 (freq 1) Midpoints: 74, 79 Mean grouped $\bar{x}_g = \frac{3 \times 74 + 1 \times 79}{4} = 75.25$ Variance grouped: $s^2 = \frac{1}{4} (3(74 - 75.25)^2 + 1(79 - 75.25)^2) = \frac{1}{4} (3 \times 1.5625 + 14.0625) = \frac{18.75}{4} = 4.6875$ SD grouped = $\sqrt{4.6875} \approx 2.17$ **Step 8: Effect of Data Arrangement** Quartile Deviation, MAD, and SD depend on data values, not their order. Arranging data ascending or descending does not change these measures. --- **Final Answers:** 1. Range = 59 2. Quartile Deviation = 6 3. MAD Ungrouped = 14.17 4. MAD Grouped = 2.25 5. SD Ungrouped = 11.83 6. SD Grouped = 2.17 7. Quartile Deviation unchanged by descending order because it depends on data values, not order. 8. MAD and SD unchanged by ascending order for the same reason.