1. **Problem Statement:** Calculate the mean for grouped data given class intervals and their frequencies. 2. **Formula:** The mean for grouped data is calculated using the formula: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is the frequency of the $i^{th}$ class and $x_i$ is the midpoint of the $i^{th}$ class interval. 3. **Steps:** - Find the midpoint $x_i$ of each class interval by averaging the lower and upper boundaries. - Multiply each midpoint $x_i$ by its corresponding frequency $f_i$ to get $f_i x_i$. - Sum all $f_i x_i$ values. - Sum all frequencies $f_i$. - Divide the sum of $f_i x_i$ by the sum of $f_i$ to get the mean. 4. **Explanation:** The midpoint represents a typical value for each class interval. Multiplying by frequency weights these values by how often they occur. Dividing by total frequency gives the average value across all data. 5. **Example:** Suppose class intervals are 10-20, 20-30, 30-40 with frequencies 5, 8, 7. - Midpoints: 15, 25, 35 - Products: $5\times15=75$, $8\times25=200$, $7\times35=245$ - Sum of products: $75+200+245=520$ - Sum of frequencies: $5+8+7=20$ - Mean: $\frac{520}{20}=26$ This is the mean of the grouped data.