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6.16: Quartiles

Difficulty Level: At Grade Created by: CK-12
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On Thursday, a real estate agent came to visit the construction site. She spent a long time talking with Uncle Larry while Travis was helping Mr. Wilson arrange some tile for a bathroom floor. Travis was very curious about what they were discussing. The realtor handed Uncle Larry a sheet of paper to look at. After the realtor left, Travis decided to ask Uncle Larry about the meeting. “What was that all about?” Travis asked.

“Well, the man who owns this house has decided to sell it,” Uncle Larry explained. “The realtor wants to know when it will be finished so that she can be sure that she has enough time in the selling season to sell it.”

“What is a selling season?”

“Certain times of the year are better for buying and selling houses. Spring and summer are the best times in this area. This sheet says about how long it took houses in this area to sell last spring and summer. We want to be sure to be finished in time so that the realtor can sell this house.”

Travis takes a look at the paper. Here is what he sees.

\begin{align*}\#\end{align*}3 - 30 days

\begin{align*}\#\end{align*}25 - 32 days

\begin{align*}\#\end{align*}1 - 35 days

\begin{align*}\#\end{align*}14 - 40 days

\begin{align*}\#\end{align*}28 - 45 days

\begin{align*}\#\end{align*}77 - 60 days

\begin{align*}\#\end{align*}32 - 65 days

\begin{align*}\#\end{align*}19 - 90 days

\begin{align*}\#\end{align*}21 - 100 days

\begin{align*}\#\end{align*}22 - 120 days

“Wow, that’s a big range,” Travis says.

“Yes, now we need to get back to work.”

Travis is puzzled by the data. He wonders what the average selling time was for the houses that sold last summer. He is also curious about the shortest and the longest sale.

When looking at this data, Travis will need to analyze it to figure out certain measures. In this Concept, you will learn the same skills, then you will know how Travis can complete this task.

Guidance

Today’s Concept focuses on data once again. This time, we will be ordering data. Eventually, we will be building box-and-whisker plots.

To understand a box-and-whisker plot, there is some vocabulary to learn. We can practice with these new terms before building a data display.

Our first key word when working with box-and-whisker plots is median.

When working with data, we often have series of numbers that tell us important information. Here is a data set showing the number of hours that the average teenager works in a part time job.

16, 10, 8, 8, 11, 11, 12, 15, 10, 20, 6, 16, 8

To work with this set of data, the first thing that we need to do is to order it. To order it means that we write the data in order from least to greatest including any repeated numbers.

6, 8, 8, 8, 10, 10, 11, 11, 12, 15, 16, 16, 20

Next, we find the median. Remember that the median is the middle number in a set of data. Here there are 13 values. The median is 11. The next key term that we need to understand is a quartile. A quartile divides the data set into four parts. With the median, our data set is divided into two parts. The first part is the first half up to 10 and the second half starts at 11 and goes to 20. Take a look.

6, 8, 8, 8, 10, 10, 11, 11, 12, 15, 16, 16, 20

To use quartiles, we need to divide this data set into four sections, not just two. To do this, we find the median of the first half of the data and the median of the second half of the data. The median of the first half of the data is called the lower quartile. The median of the second half of the data is called the upper quartile.

6, 8, 8, 8, 10, 10, 11, 11, 12, 15, 16, 16, 20

The lower quartile is the average between 8 and 8. The lower quartile is 8.

The upper quartile is the average between 15 and 16. The upper quartile is 15.5.

The next term that we need to know is the extremes.

The term extremes refers to the lowest value in a data set (the lower extreme) and the highest value in a data set (the upper extreme).

In the set we just looked at, 6 is the lower extreme and 20 is the upper extreme.

Check your understanding by answering these questions.

4, 4, 5, 6, 7, 8, 11, 13, 16

Example A

What is the median of this data set?

Solution: 7

Example B

What is the lower quartile?

Solution: 4.5

Example C

What is the upper extreme?

Solution: 16

Now let's go back to Travis and the real estate market.

Travis is puzzled by the data. He wonders what the average selling time was for the houses that sold last summer. He is also curious about the shortest and the longest sale.

To work with this data, Travis will need to begin by analyzing it for the different measures. These are the ones that you just learned about in the Concept.

30, 32, 35, 40, 45, 60, 65, 78, 90, 100, 120

The median is 60 days. That was the median number of days that it took to sell a house. What is the lower quartile number of days? This is the lowest number of days on average.

30, 32, 35, 40, 45, 60, 65, 78, 90, 100, 120

35 days is the average of the lower quartile.

What is the upper quartile number of days? This is the highest number of days on average.

30, 32, 35, 40, 45, 60, 65, 78, 90, 100, 120

90 days is the average of the upper quartile.

Then we have two extremes-the lowest number of days is 30-that is the lower extreme. The highest number of days is 120; that is the upper extreme.

Now Travis wonders if there is a way to show this data. That is what you will learn in the next Concept.

Vocabulary

Here are the vocabulary words in this Concept.

Median
the middle score of a set of data.
Quartile
dividing a data into four sections.
Upper Quartile
the median of a quartile on the higher end of the range.
Lower quartile
the median of a quartile on the lower range
Extremes
the highest and lowest scores possible in a range of data.

Guided Practice

Here is one for you to try on your own.

What is the median of this data set?

4, 5, 12, 11, 9, 8, 7, 4, 3

To figure this out, we must first write the data set in order from least to greatest.

3, 4, 4, 5, 7, 8, 9, 11, 12

The median is the middle score.

There are nine values in this data set.

The median is 7.

This is our answer.

Video Review

Here are videos for review.

This video presents box-and-whisker plots.

Practice

Directions: Use each data set to answer the questions following it.

3, 5, 6, 8, 11, 13, 15, 17, 19

1. How many values are there in this data set?

2. What is the median of the data?

3. What is the range?

4. What is the upper quartile?

5. What is the lower quartile?

6. What are the extremes?

100, 112, 115, 122, 123, 126, 130, 131

7. How many values are there in this data set?

8. What is the median of the data?

9. What is the range?

10. What is the upper quartile?

11. What is the lower quartile?

12. What are the extremes?

113, 120, 131, 142, 150, 155, 157, 161, 167

13. How many values are there in this data set?

14. What is the median of the data?

15. What is the range?

16. What is the upper quartile?

17. What is the lower quartile?

18. What are the extremes?

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Vocabulary Language: English

TermDefinition
Extremes The extremes are the maximum and minimum values in a data set.
first quartile The first quartile, also known as $Q_1$, is the median of the lower half of the data.
Lower quartile The lower quartile, also known as $Q_1$, is the median of the lower half of the data.
Median The median of a data set is the middle value of an organized data set.
Quartile A quartile is each of four equal groups that a data set can be divided into.
second quartile The second quartile, also known as $Q_2$, is the median of the data.
third quartile The third quartile, also known as $Q_3$, is the median of the upper half of the data.
Upper Quartile The upper quartile, also known as $Q_3$, is the median of the upper half of the data.

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