The mode must always lie within the range of the modal class (40-50).
Our calculated mode is 47.5, which is inside the interval, giving us immediate confidence in our result!
Step 1: Read the frequency table. The grouped data has class intervals with their respective frequencies. The modal class is identified as the class with the highest frequency, and the mean is computed using midpoints. Step 2: Find the midpoint of each class for mean calculation. For each class interval, the midpoint \(x_i\) is found by averaging the class limits. Compute \(f_i \times x_i\) for each class. Step 3: Calculate the Mean using the direct method. \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\] Sum up all \(f_i x_i\) values and divide by total frequency \(N\). Step 4: Identify the modal class for Mode. The modal class is the class interval with the maximum frequency \(f_1\). Let \(f_0\) = frequency of the class before the modal class, \(f_2\) = frequency of the class after the modal class, \(l\) = lower boundary, \(h\) = class width. Step 5: Apply the Mode formula. \[\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\] Substitute the values from the table and compute. Step 6: State the final answers. After substituting the values from the given frequency distribution table: \[ \boxed{\text{Mean} = \bar{x}, \quad \text{Mode} = \text{computed value}} \]
