https://file.aiquickdraw.com/mathbot/file/1725386927145d9e4nzd0.jpg

The problem in the image provides a data set representing the average annual total returns (in percentage points) for 8 mutual funds: 

**24, 35, 40, 16, 18, 40, 25, 16.**

It asks three questions about the data set:

1. **What is the median of this data set?**  
   The median is the middle value of a data set when the numbers are arranged in order. If the number of observations is even, the median is the average of the two middle numbers.

2. **What is the mean of this data set?**  
   The mean is the average of the data points, calculated by summing all the values and dividing by the number of values.

3. **How many modes does the data set have, and what are their values?**  
   The mode is the number(s) that appear most frequently in the data set. A data set can have zero, one, or multiple modes.

Let's go through each step:

### 1. Median Calculation:
First, arrange the data in ascending order:
**16, 16, 18, 24, 25, 35, 40, 40**

Since there are 8 data points (an even number), the median will be the average of the 4th and 5th numbers:
\[
\text{Median} = \frac{24 + 25}{2} = \frac{49}{2} = 24.5
\]

### 2. Mean Calculation:
Sum all the data points:
\[
16 + 16 + 18 + 24 + 25 + 35 + 40 + 40 = 214
\]
Now, divide by the number of data points:
\[
\text{Mean} = \frac{214}{8} = 26.75
\]

### 3. Mode Calculation:
Identify the frequency of each number:
- 16 appears **twice**
- 18 appears **once**
- 24 appears **once**
- 25 appears **once**
- 35 appears **once**
- 40 appears **twice**

The numbers 16 and 40 each appear twice, so there are **two modes: 16 and 40**.

### Final Answers:
1. **Median:** 24.5
2. **Mean:** 26.75
3. **Modes:** 16 and 40 (two modes)

Would you like to explore any of these concepts in more detail or have any other questions?

Here are 5 related questions to deepen your understanding:
1. How would the median change if another data point, 30, was added to the set?
2. What is the range of this data set, and how is it calculated?
3. How would the mean be affected if the largest value, 40, were increased to 50?
4. Can a data set have more than two modes, and what would that imply?
5. How does the concept of a weighted mean differ from a regular mean?

**Tip:** Always double-check the order of data when calculating the median and mode, as errors in ordering can lead to incorrect results.

What is the median, mean, and mode(s) of the following data set: 24, 35, 40, 16, 18, 40, 25, 16?

Learn how to calculate the median, mean, and mode(s) for the data set 24, 35, 40, 16, 18, 40, 25, 16, representing the average annual total returns for 8 mutual funds. This step-by-step solution is ideal for students in Grades 6-8 and covers key statistical concepts in descriptive statistics.

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