Find the mean of the following grouped frequency distribution: Class 0-10 (freq 5), 10-20 (freq 8), 20-30 (freq 12), 30-40 (freq 7), 40-50 (freq 8).

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Step-by-Step Solution

Step 1: Calculate Class Marks (Midpoints)

To find the mean of a grouped frequency distribution, we first need to determine the class mark (midpoint) for each class interval. The class mark $x_i$ is calculated by adding the lower limit and the upper limit of the class and dividing by 2.

Step 2: Calculate $f_i x_i$ for each class

Next, we multiply the frequency ($f_i$) of each class by its corresponding class mark ($x_i$). This product, $f_i x_i$, represents the sum of the values within that class interval.

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

Now, we need to find the sum of all frequencies, denoted as $\Sigma f_i$, and the sum of all the $f_i x_i$ products, denoted as $\Sigma f_i x_i$. These sums are essential for calculating the mean.

Step 4: Calculate the Mean

Finally, we calculate the mean ($\bar{x}$) using the formula: the sum of $f_i x_i$ divided by the sum of $f_i$. This gives us the average value of the grouped data.