We then find the sum of all frequencies (Σ f_i) and the sum of all the f_i x_i values (Σ f_i x_i). These summations are crucial for calculating the mean using the direct method formula.

Find the mean of the following data by the direct method: class intervals 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 16, 6.

Answer: The mean of the given data is 27.

Step-by-step solution

Step 1: Calculate Midpoints

To apply the direct method for finding the mean of grouped data, we first need to find the midpoint ( $x_i$ ) for each class interval. The midpoint is calculated by averaging the lower and upper limits of the class interval.

Step 2: Calculate $f_i x_i$

Next, we multiply the frequency ( $f_i$ ) of each class interval by its corresponding midpoint ( $x_i$ ). This gives us the product $f_i x_i$ for each class, which represents the total value contributed by that class.

Step 3: Calculate Sum of Frequencies and Sum of $f_i x_i$

We then find the sum of all frequencies ( $\sum f_i$ ) and the sum of all the $f_i x_i$ values ( $\sum f_i x_i$ ). These summations are crucial for calculating the mean using the direct method formula.

Step 4: Apply Direct Method Formula for Mean

Finally, we apply the direct method formula for the mean, which states that the mean ( $\bar{x}$ ) is equal to the sum of $f_i x_i$ divided by the sum of $f_i$ . This formula gives us the weighted average of the midpoints, which is the mean of the grouped data.

Step 5: Substitute Values and Calculate Mean

Substituting the calculated sums into the direct method formula, we perform the division to find the mean of the given data. The result is 27.

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