1. **Problem Statement:** We have a frequency distribution of boys' weights in kg and their corresponding number of students. We need to compute the median and mean weight. 2. **Given Data:** | Weight Interval (kg) | Number of Students (Frequency) | |---------------------|-------------------------------| | 1 – 3 | 2 | | 4 – 6 | 3 | | 7 – 9 | 5 | | 10 – 12 | 4 | | 13 – 15 | 6 | | 16 – 18 | 2 | | 19 – 21 | 1 | 3. **Step 1: Calculate the Mean** - The mean for grouped data is given by: $$\text{Mean} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is the frequency and $x_i$ is the midpoint of each class interval. - Calculate midpoints $x_i$ for each class: - 1 – 3: $\frac{1+3}{2} = 2$ - 4 – 6: $\frac{4+6}{2} = 5$ - 7 – 9: $\frac{7+9}{2} = 8$ - 10 – 12: $\frac{10+12}{2} = 11$ - 13 – 15: $\frac{13+15}{2} = 14$ - 16 – 18: $\frac{16+18}{2} = 17$ - 19 – 21: $\frac{19+21}{2} = 20$ - Multiply each midpoint by its frequency: - $2 \times 2 = 4$ - $5 \times 3 = 15$ - $8 \times 5 = 40$ - $11 \times 4 = 44$ - $14 \times 6 = 84$ - $17 \times 2 = 34$ - $20 \times 1 = 20$ - Sum of frequencies $\sum f_i = 2 + 3 + 5 + 4 + 6 + 2 + 1 = 23$ - Sum of $f_i x_i = 4 + 15 + 40 + 44 + 84 + 34 + 20 = 241$ - Calculate mean: $$\text{Mean} = \frac{241}{23} \approx 10.48$$ 4. **Step 2: Calculate the Median** - Median class is the class where the cumulative frequency reaches or exceeds $\frac{N}{2}$, where $N$ is total frequency. - Total frequency $N = 23$, so $\frac{N}{2} = 11.5$ - Calculate cumulative frequencies: - 1 – 3: 2 - 4 – 6: 2 + 3 = 5 - 7 – 9: 5 + 5 = 10 - 10 – 12: 10 + 4 = 14 - 13 – 15: 14 + 6 = 20 - 16 – 18: 20 + 2 = 22 - 19 – 21: 22 + 1 = 23 - Median class is 10 – 12 (since cumulative frequency 14 > 11.5) - Use median formula for grouped data: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times h$$ where: - $L$ = lower boundary of median class = 9.5 (assuming class intervals are continuous) - $F$ = cumulative frequency before median class = 10 - $f$ = frequency of median class = 4 - $h$ = class width = 3 - Calculate median: $$\text{Median} = 9.5 + \left(\frac{11.5 - 10}{4}\right) \times 3 = 9.5 + \left(\frac{1.5}{4}\right) \times 3 = 9.5 + 1.125 = 10.625$$ **Final Answers:** - Mean weight $\approx 10.48$ kg - Median weight $\approx 10.63$ kg