1. **Problem Statement:** Find the Mean (Direct Method), Median, and Mode of the delivery time from the given frequency distribution. 2. **Given Data:** | Time (mins) | Frequency | |-------------|-----------| | 20–29 | 2 | | 30–39 | 4 | | 40–49 | 7 | | 50–59 | 10 | | 60–69 | 15 | | 70–79 | 9 | | 80–89 | 6 | | 90–99 | 4 | | 100–109 | 3 | 3. **Step 1: Calculate Midpoints ($x_i$) for each class:** - $24.5, 34.5, 44.5, 54.5, 64.5, 74.5, 84.5, 94.5, 104.5$ 4. **Step 2: Calculate $f_i x_i$ (Frequency times Midpoint):** - $2\times24.5=49$ - $4\times34.5=138$ - $7\times44.5=311.5$ - $10\times54.5=545$ - $15\times64.5=967.5$ - $9\times74.5=670.5$ - $6\times84.5=507$ - $4\times94.5=378$ - $3\times104.5=313.5$ 5. **Step 3: Calculate total frequency $N$ and sum of $f_i x_i$:** - $N=2+4+7+10+15+9+6+4+3=60$ - $\sum f_i x_i=49+138+311.5+545+967.5+670.5+507+378+313.5=3880$ 6. **Step 4: Calculate Mean using formula:** $$\text{Mean} = \frac{\sum f_i x_i}{N} = \frac{3880}{60} = 64.67$$ minutes 7. **Step 5: Find Median Class:** - Median position = $\frac{N}{2} = 30$ - Cumulative frequencies: 2, 6, 13, 23, 38, 47, 53, 57, 60 - Median class is where cumulative frequency first exceeds 30, which is 60–69 (cumulative 38) 8. **Step 6: Use Median formula:** $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ Where: - $L=59.5$ (lower boundary of median class 60–69) - $F=23$ (cumulative frequency before median class) - $f_m=15$ (frequency of median class) - $h=10$ (class width) Calculate: $$\text{Median} = 59.5 + \left(\frac{30 - 23}{15}\right) \times 10 = 59.5 + \frac{7}{15} \times 10 = 59.5 + 4.67 = 64.17$$ minutes 9. **Step 7: Find Mode Class:** - Mode class is the class with highest frequency, which is 60–69 with frequency 15 10. **Step 8: Use Mode formula:** $$\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$ Where: - $L=59.5$ (lower boundary of mode class) - $f_1=15$ (frequency of mode class) - $f_0=10$ (frequency before mode class) - $f_2=9$ (frequency after mode class) - $h=10$ Calculate: $$\text{Mode} = 59.5 + \left(\frac{15 - 10}{2 \times 15 - 10 - 9}\right) \times 10 = 59.5 + \left(\frac{5}{30 - 19}\right) \times 10 = 59.5 + \frac{5}{11} \times 10 = 59.5 + 4.55 = 64.05$$ minutes **Final answers:** - Mean = 64.67 minutes - Median = 64.17 minutes - Mode = 64.05 minutes