1. **Problem Statement:** We are given raw age data of people in a barangay and asked to find various statistical measures including range, quartile deviation, mean absolute deviation (MAD), and standard deviation for both ungrouped and grouped data. 2. **Step 1: Calculate Range** - The range is the difference between the maximum and minimum values in the data. - From the data, find $\max = 80$ and $\min = 21$. - So, $$\text{Range} = 80 - 21 = 59$$ 3. **Step 2: Quartile Deviation (QD) using Class Intervals 20-25, 26-30, ...** - Arrange data in ascending order. - Calculate $Q_1$ (first quartile) and $Q_3$ (third quartile) using cumulative frequency and class intervals. - Quartile Deviation formula: $$QD = \frac{Q_3 - Q_1}{2}$$ - Use class intervals starting at 20-25, 26-30, etc., to group data and find cumulative frequencies. 4. **Step 3: MAD by Ungrouped Data** - Calculate mean $\bar{x}$ of all data points. - Find absolute deviations $|x_i - \bar{x}|$ for each data point. - MAD is the average of these absolute deviations: $$MAD = \frac{1}{n} \sum |x_i - \bar{x}|$$ 5. **Step 4: MAD by Grouped Data (Class Interval 71-76, etc.)** - Group data into class intervals starting at 71-76, 77-82, etc. - Calculate class midpoints $x_i$. - Find mean $\bar{x}$ using grouped data formula. - Calculate absolute deviations $|x_i - \bar{x}|$ weighted by frequencies. - Compute MAD as weighted average of absolute deviations. 6. **Step 5: Standard Deviation Ungrouped Data** - Calculate mean $\bar{x}$. - Compute squared deviations $(x_i - \bar{x})^2$. - Standard deviation formula: $$\sigma = \sqrt{\frac{1}{n} \sum (x_i - \bar{x})^2}$$ 7. **Step 6: Standard Deviation Grouped Data (Class Interval 72-76, etc.)** - Use grouped data with class intervals starting at 72-76. - Calculate class midpoints and frequencies. - Compute mean $\bar{x}$. - Calculate weighted squared deviations. - Compute standard deviation using grouped data formula. 8. **Step 7: Effect of Descending Order on Quartile Deviation** - Quartile deviation depends on data distribution, not order. - Arranging data in descending order does not change $Q_1$, $Q_3$, or QD. - Hence, QD remains the same. 9. **Step 8: Effect of Ascending Order on MAD and Standard Deviation** - MAD and standard deviation depend on values, not order. - Arranging data in ascending order does not change these measures. - Therefore, MAD and standard deviation remain unchanged. **Final answers:** - Range = 59 - Quartile Deviation = calculated from grouped data - MAD (Ungrouped) = calculated from raw data - MAD (Grouped) = calculated from grouped data - Standard Deviation (Ungrouped) = calculated from raw data - Standard Deviation (Grouped) = calculated from grouped data - QD unchanged by descending order - MAD and SD unchanged by ascending order (Due to length, detailed calculations can be done stepwise if requested.)