1. **Problem Statement:** Calculate the mean, median, and mode of the daily wages given grouped data. 2. **Given Data:** Wage intervals: 100–120, 120–140, 140–160, 160–180, 180–200 Frequencies: 5, 9, 14, 8, 4 3. **Step 1: Calculate the Mean** - Use the formula for mean of grouped data: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is frequency and $x_i$ is the class midpoint. - Find midpoints: - 100–120: $\frac{100+120}{2} = 110$ - 120–140: $\frac{120+140}{2} = 130$ - 140–160: $\frac{140+160}{2} = 150$ - 160–180: $\frac{160+180}{2} = 170$ - 180–200: $\frac{180+200}{2} = 190$ - Multiply midpoints by frequencies: - $110 \times 5 = 550$ - $130 \times 9 = 1170$ - $150 \times 14 = 2100$ - $170 \times 8 = 1360$ - $190 \times 4 = 760$ - Sum of $f_i x_i = 550 + 1170 + 2100 + 1360 + 760 = 5940$ - Sum of frequencies $\sum f_i = 5 + 9 + 14 + 8 + 4 = 40$ - Calculate mean: $$\bar{x} = \frac{5940}{40} = 148.5$$ 4. **Step 2: Calculate the Median** - Median class is the class where cumulative frequency just exceeds $\frac{N}{2} = 20$ - Cumulative frequencies: - 5, 14, 28, 36, 40 - Median class is 140–160 (since cumulative frequency 28 > 20) - Use median formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ where: - $L = 140$ (lower boundary of median class) - $N = 40$ - $F = 14$ (cumulative frequency before median class) - $f_m = 14$ (frequency of median class) - $h = 20$ (class width) - Substitute values: $$\text{Median} = 140 + \left(\frac{20 - 14}{14}\right) \times 20 = 140 + \frac{6}{14} \times 20 = 140 + 8.57 = 148.57$$ 5. **Step 3: Calculate the Mode** - Modal class is the class with highest frequency: 140–160 (frequency 14) - Use mode formula: $$\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$ where: - $L = 140$ (lower boundary of modal class) - $f_1 = 14$ (frequency of modal class) - $f_0 = 9$ (frequency before modal class) - $f_2 = 8$ (frequency after modal class) - $h = 20$ (class width) - Substitute values: $$\text{Mode} = 140 + \left(\frac{14 - 9}{2 \times 14 - 9 - 8}\right) \times 20 = 140 + \left(\frac{5}{28 - 17}\right) \times 20 = 140 + \frac{5}{11} \times 20 = 140 + 9.09 = 149.09$$ **Final answers:** - Mean wage = $148.5$ - Median wage = $148.57$ - Modal wage = $149.09$