1. **Stating the problem:** We are given a frequency distribution and asked to calculate the mean using the relationship among mode, median, and mean. 2. **Understanding the relationship:** For moderately skewed distributions, the empirical relationship is: $$\text{Mean} = 3 \times \text{Median} - 2 \times \text{Mode}$$ 3. **Extracting mode and median:** - The mode is the value with the highest frequency. - The median is the middle value when data is ordered. 4. **Finding the mode:** From the data, the highest frequency is 15 for the value 8, so: $$\text{Mode} = 8$$ 5. **Finding the median:** - Total frequency $N = 1+1+2+5+5+15+26+45+17+38+18+6+2+1 = 182$ - Median position = $\frac{N+1}{2} = \frac{182+1}{2} = 91.5$th value 6. **Cumulative frequencies:** - Up to 8: $1+1+2+5+5+15 = 29$ - Up to 9: $29 + 26 = 55$ - Up to 10: $55 + 45 = 100$ Since 91.5 lies between 56 and 100, the median class is 10. 7. **Calculate mean using the formula:** $$\text{Mean} = 3 \times 10 - 2 \times 8 = 30 - 16 = 14$$ **Final answer:** $$\boxed{14}$$