Let's go through the image and calculate the measures of dispersion step by step based on the data provided.

Step 1: Sorted Data

The data given is: $0, 2, 4, 5, 8, 9, 10, 12, 12, 16, 19, 20$ After sorting, it remains the same: $0, 2, 4, 5, 8, 9, 10, 12, 12, 16, 19, 20$

Step 2: Minimum and Maximum

• Minimum: The smallest number in the dataset is $0$.
• Maximum: The largest number in the dataset is $20$.

Step 3: Quartiles

To find the quartiles, we first need to divide the data into quarters.

• Median (Q2): This is the middle value of the data. Since there are 12 data points, the median is the average of the 6th and 7th values: $\text{Median} = \frac{9 + 10}{2} = 9.5$

• Quartile 1 (Q1): This is the median of the first half of the data (lower half), which is: $0, 2, 4, 5, 8, 9$ The median here is the average of the 3rd and 4th values: $\text{Q1} = \frac{4 + 5}{2} = 4.5$

• Quartile 3 (Q3): This is the median of the second half of the data (upper half), which is: $10, 12, 12, 16, 19, 20$ The median here is the average of the 3rd and 4th values: $\text{Q3} = \frac{12 + 16}{2} = 14$

Step 4: Range

The range is the difference between the maximum and minimum values: $\text{Range} = 20 - 0 = 20$

Step 5: Interquartile Range (IQR)

The interquartile range (IQR) is the difference between Quartile 3 and Quartile 1: $\text{IQR} = 14 - 4.5 = 9.5$

Summary:

• Minimum: 0
• Quartile 1 (Q1): 4.5
• Median: 9.5
• Quartile 3 (Q3): 14
• Maximum: 20
• Range: 20
• Interquartile Range (IQR): 9.5

Let me know if you need any further details or explanations.

Here are 5 related questions to expand this:

1. How is the variance calculated for this data set?
2. What is the standard deviation of this data?
3. What are outliers and how can we find them using IQR?
4. Can the range be a reliable measure of dispersion in every situation?
5. How do Quartiles help in understanding the data distribution?

Tip: Always use the interquartile range (IQR) to identify and deal with outliers, as it focuses on the middle 50% of the data and is less sensitive to extreme values than the range.