To approximate the mean for the given data set, we can use the midpoint of each class interval and the corresponding frequency. Here's how we can calculate it step by step:

Step 1: Identify the Midpoint for Each Class Interval

The midpoint for a class interval is calculated by averaging the lower and upper bounds of the interval.

$\text{Midpoint} = \frac{\text{Lower bound} + \text{Upper bound}}{2}$

Let's calculate the midpoints for each interval:

1. $\text{Class: } 0 - 11$
   Midpoint = $\frac{0 + 11}{2} = 5.5$

2. $\text{Class: } 12 - 23$
   Midpoint = $\frac{12 + 23}{2} = 17.5$

3. $\text{Class: } 24 - 35$
   Midpoint = $\frac{24 + 35}{2} = 29.5$

4. $\text{Class: } 36 - 47$
   Midpoint = $\frac{36 + 47}{2} = 41.5$

5. $\text{Class: } 48 - 59$
   Midpoint = $\frac{48 + 59}{2} = 53.5$

Step 2: Multiply Each Midpoint by the Corresponding Frequency

Now, multiply each midpoint by its corresponding frequency:

1. $5.5 \times 6 = 33$
2. $17.5 \times 12 = 210$
3. $29.5 \times 4 = 118$
4. $41.5 \times 15 = 622.5$
5. $53.5 \times 11 = 588.5$

Step 3: Calculate the Sum of Midpoint × Frequency

Add up the results from Step 2:

$33 + 210 + 118 + 622.5 + 588.5 = 1572$

Step 4: Calculate the Total Frequency

Add up all the frequencies:

$6 + 12 + 4 + 15 + 11 = 48$

Step 5: Calculate the Approximate Mean

Finally, divide the sum from Step 3 by the total frequency:

$\text{Mean} = \frac{1572}{48} \approx 32.8$

Conclusion

The approximate mean for the data set is 32.8.

Would you like more details on any part of the calculation or have any other questions?

Here are five related questions you might find interesting:

1. How do you calculate the median of a grouped frequency distribution?
2. What is the difference between the mean and the mode?
3. How can you estimate the mean using cumulative frequency?
4. How do you find the variance of a grouped data set?
5. What are the advantages of using the mean as a measure of central tendency?

Tip: When calculating the mean of grouped data, always remember that the mean is an estimate based on the midpoints of the intervals.