Use the following data set to answer the questions.
13    15    20    20    20    20    20    20    20    23    27    31
What is the mean?

What is the mode?

What is the median?

Which of the following statements are correct?
The median is the middle of a data set. Half of the data points are either less than or equal to the median.
The median is the middle of a data set. At least half of the data points are always equal to the median.
The median is not the middle of a data set. You cannot predict the distribution of the numbers in relationship to the median.
The median is the middle of a data set. Half of the data points are always less than the median, and half are always greater than the median.

Let's go through each question step by step.

### Given Data Set:
13, 15, 20, 20, 20, 20, 20, 20, 20, 23, 27, 31

#### 1. **Mean**:
The mean (average) is calculated by summing all the data points and dividing by the number of data points.

\[
\text{Mean} = \frac{13 + 15 + 20 + 20 + 20 + 20 + 20 + 20 + 20 + 23 + 27 + 31}{12} = \frac{249}{12} = 20.75
\]

So, the mean is **20.75**.

#### 2. **Mode**:
The mode is the value that appears most frequently in the data set.

In this data set, **20** appears **7 times**, which is more frequent than any other number.

So, the mode is **20**.

#### 3. **Median**:
The median is the middle value when the data set is ordered. If the data set has an even number of values, the median is the average of the two middle values.

The data set has 12 values, so the median will be the average of the 6th and 7th values:
- Ordered data set: 13, 15, 20, 20, 20, **20**, **20**, 20, 20, 23, 27, 31
- Both the 6th and 7th values are 20.

\[
\text{Median} = \frac{20 + 20}{2} = 20
\]

So, the median is **20**.

### 4. **Correct Statements about the Median**:
Let's evaluate the provided statements:

1. **The median is the middle of a data set. Half of the data points are either less than or equal to the median.**
   - **True**: This statement correctly describes the median. By definition, half the data points will be less than or equal to the median, and the other half will be greater than or equal to it.

2. **The median is the middle of a data set. At least half of the data points are always equal to the median.**
   - **False**: This is incorrect. While it's possible for some data points to equal the median, it's not a requirement that half or more of the data points are exactly equal to it.

3. **The median is not the middle of a data set. You cannot predict the distribution of the numbers in relationship to the median.**
   - **False**: This is incorrect because the median, by definition, is the middle value in an ordered data set.

4. **The median is the middle of a data set. Half of the data points are always less than the median, and half are always greater than the median.**
   - **False**: This is incorrect. Half the data points are less than or equal to the median, and half are greater than or equal to the median, but not necessarily strictly less or greater.

### Summary of Correct Statement:
- The first statement is correct: "The median is the middle of a data set. Half of the data points are either less than or equal to the median."

---

Would you like more details on any of these concepts or have any questions?

### Related Questions:
1. How is the range of a data set calculated?
2. What is the significance of the standard deviation in a data set?
3. How do you determine if a data set is skewed based on its mean, median, and mode?
4. How does the presence of outliers affect the mean and median of a data set?
5. What are quartiles, and how are they related to the median?

### Tip:
The median is less sensitive to extreme values (outliers) compared to the mean, making it a more robust measure of central tendency in skewed distributions.

Learn how to calculate the mean, mode, and median for the data set 13, 15, 20, 23, 27, 31. This problem covers key statistical concepts, including the correct interpretation of median in data sets. Suitable for students in Grades 6-8.

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