Approximate the mean for the following data set. Round your answer to one decimal place.

Class Frequency

-

-

-

-

-

0 11 6

12 23 12

24 35 4

36 47 15

48 59 11

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.

Learn how to approximate the mean for a grouped frequency distribution using midpoints and frequencies. This problem involves calculating the mean from grouped data intervals and provides a detailed step-by-step solution. Suitable for high school students studying statistics.

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