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.
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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 (xi) for each class interval. The midpoint is calculated by averaging the lower and upper limits of the class interval.
Step 2: Calculate fixi
Next, we multiply the frequency (fi) of each class interval by its corresponding midpoint (xi). This gives us the product fixi for each class, which represents the total value contributed by that class.
Step 3: Calculate Sum of Frequencies and Sum of fixi
We then find the sum of all frequencies (∑fi) and the sum of all the fixi values (∑fixi). 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 (xˉ) is equal to the sum of fixi divided by the sum of fi. 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.