Let's break down the questions one by one using the provided data set:

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 numbers and then dividing by the total count of numbers.

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

Sum of all values: $13 + 15 + 20 + 20 + 20 + 20 + 20 + 20 + 20 + 23 + 27 + 31 = 249$

Number of values: $12$

$\text{Mean} = \frac{249}{12} \approx 20.75$

2. Mode:

The mode is the value that appears most frequently in the data set.

Here, the number 20 appears 7 times, more than any other number.

$\text{Mode} = 20$

3. Median:

The median is the middle value when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers.

For this data set (which has 12 values), the two middle values are the 6th and 7th numbers, both of which are 20.

$\text{Median} = 20$

Summary of Answers:

• Mean: 20.75
• Mode: 20
• Median: 20

Unfinished Question:

It seems you were about to ask which of the following statements are true, but the statements are missing. Please provide them, and I'll be happy to help further!

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Do you want more details or have any other questions? Here are 5 related questions you might find interesting:

1. What is the range of the data set?
2. How does the mode compare to the mean in this data set?
3. What would happen to the mean, mode, and median if you added a new value of 25 to the data set?
4. Can there be more than one mode in a data set? If so, what would that be called?
5. How does the distribution of the data (e.g., skewness) affect the relationship between mean, median, and mode?

Tip: When analyzing a data set, it's important to understand how each measure of central tendency (mean, median, and mode) provides different insights into the distribution of the data.