To find the five-number summary of the given data, we need to determine:

1. Minimum: The smallest value in the dataset.
2. First Quartile (Q1): The median of the lower half of the data.
3. Median (Q2): The middle value of the dataset.
4. Third Quartile (Q3): The median of the upper half of the data.
5. Maximum: The largest value in the dataset.

Step 1: Organize the Data

First, we need to expand the table into a list of values by considering their frequencies:

• 12 appears 2 times: 12, 12
• 13 appears 1 time: 13
• 14 appears 1 time: 14
• 15 appears 4 times: 15, 15, 15, 15
• 17 appears 5 times: 17, 17, 17, 17, 17
• 18 appears 2 times: 18, 18
• 19 appears 3 times: 19, 19, 19
• 21 appears 1 time: 21

Thus, the data in ascending order is: $12, 12, 13, 14, 15, 15, 15, 15, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 21$

Step 2: Determine the Five-Number Summary

• Minimum: The smallest value is 12.
• Maximum: The largest value is 21.

Next, we calculate the median (Q2) and the quartiles (Q1 and Q3).

Median (Q2):

The median is the middle value of the dataset. There are 19 data points, so the median is the 10th value.

• The 10th value is 17.

First Quartile (Q1):

Q1 is the median of the lower half of the data (values below the median). The lower half is: $12, 12, 13, 14, 15, 15, 15, 15, 17$ The median of this group is the 5th value, which is 15.

Third Quartile (Q3):

Q3 is the median of the upper half of the data (values above the median). The upper half is: $17, 17, 17, 17, 18, 18, 19, 19, 19, 21$ The median of this group is the 5th value, which is 18.

Step 3: Final Five-Number Summary

• Minimum: 12
• Q1: 15
• Median (Q2): 17
• Q3: 18
• Maximum: 21

Let me know if you need any further explanation or details!

Related Questions:

1. How do you calculate the interquartile range (IQR)?
2. How would you construct a box plot from the five-number summary?
3. How can you identify outliers using the IQR method?
4. What is the significance of the median and quartiles in summarizing data?
5. What is the effect of extreme values on the five-number summary?

Tip:

Remember that the median divides the data into two halves, and quartiles further break down those halves to provide deeper insights into the spread and central tendency of the data.