To find the mean of grouped data, use the formula \( \text{Mean} = \frac{\sum (f_i \times x_i)}{\sum f_i} \), where \( f_i \) is the frequency and \( x_i \) is the midpoint of each class.
We are given the following frequency distribution:
To calculate the mean of a grouped data, we use the formula:
\[
\text{Mean} = \frac{\sum (f_i \times x_i)}{\sum f_i}
\]
where:
- \( f_i \) is the frequency of the class interval
- \( x_i \) is the midpoint of each class interval
Step 1: Find the midpoints of each class interval.
The midpoints \( x_i \) are calculated as the average of the lower and upper limits of each class:
\[
x_1 = \frac{0 + 10}{2} = 5, \quad x_2 = \frac{10 + 20}{2} = 15, \quad x_3 = \frac{20 + 30}{2} = 25, \quad x_4 = \frac{30 + 40}{2} = 35, \quad x_5 = \frac{40 + 50}{2} = 45
\]
Step 2: Multiply each midpoint by its corresponding frequency.
\[
f_1 \times x_1 = 3 \times 5 = 15, \quad f_2 \times x_2 = 6 \times 15 = 90, \quad f_3 \times x_3 = 8 \times 25 = 200, \quad f_4 \times x_4 = 5 \times 35 = 175, \quad f_5 \times x_5 = 3 \times 45 = 135
\]
Step 3: Sum up the products of frequency and midpoint.
\[
\sum (f_i \times x_i) = 15 + 90 + 200 + 175 + 135 = 615
\]
Step 4: Sum up the frequencies.
\[
\sum f_i = 3 + 6 + 8 + 5 + 3 = 25
\]
Step 5: Calculate the mean.
Now, substitute the values into the mean formula:
\[
\text{Mean} = \frac{615}{25} = 24.6
\]
Step 6: Conclusion.
Thus, the mean of the given data is \( 24.6 \).
