To find the mean of grouped data, we follow these steps:

1. Determine the midpoints (class marks) for each interval.
2. Multiply each midpoint by the corresponding frequency to get the product $f_i \times x_i$.
3. Sum all the products.
4. Divide the sum by the total frequency.

Step 1: Determine the midpoints for each interval

• For $0.5 - 2.5$, the midpoint $x_1 = \frac{0.5 + 2.5}{2} = 1.5$
• For $2.5 - 4.5$, the midpoint $x_2 = \frac{2.5 + 4.5}{2} = 3.5$
• For $4.5 - 6.5$, the midpoint $x_3 = \frac{4.5 + 6.5}{2} = 5.5$
• For $6.5 - 8.5$, the midpoint $x_4 = \frac{6.5 + 8.5}{2} = 7.5$

Step 2: Multiply the midpoints by the corresponding frequencies

Interval	Frequency $f_i$	Midpoint $x_i$	$f_i \times x_i$
0.5-2.5	5	1.5	5 × 1.5 = 7.5
2.5-4.5	2	3.5	2 × 3.5 = 7.0
4.5-6.5	6	5.5	6 × 5.5 = 33.0
6.5-8.5	8	7.5	8 × 7.5 = 60.0

Step 3: Sum the products

$\sum (f_i \times x_i) = 7.5 + 7.0 + 33.0 + 60.0 = 107.5$

Step 4: Divide by the total frequency

The total frequency is: $\sum f_i = 5 + 2 + 6 + 8 = 21$

Thus, the mean is: $\text{Mean} = \frac{\sum (f_i \times x_i)}{\sum f_i} = \frac{107.5}{21} \approx 5.12$

Conclusion:

The mean of the given data is approximately 5.12.

Would you like further clarification or more details?

Here are 5 related questions:

1. How would the mean change if an additional interval was included?
2. What is the difference between the mean and the median in grouped data?
3. How do you calculate the mode in a frequency distribution?
4. What is the impact of outliers on the mean?
5. How do you calculate the variance for grouped data?

Tip: The mean is sensitive to extreme values (outliers), so be mindful when interpreting data with wide variation.