6.5 Box-and-Whisker Plot

Learning Objectives

Order a set of data to find the median, quartiles, and extremes.
Draw a box-and-whisker plot to represent given data.
Identify the median, quartiles, and extremes given a box-and-whisker plot.
Compare and Interpret double box-and-whisker plots of real-world data.

Introduction

Homes for Sale

On Thursday, a real estate agent came to visit the construction site in Roseville. She spent a long time talking with Uncle Rashad while Travis was helping Mr. Whitebird arrange some tile for a bathroom floor. Travis was very curious about what they were discussing. The realtor handed Uncle Rashad a sheet of paper to look at. After the realtor left, Travis decided to ask Uncle Rashad about the meeting.

“What was that all about?” Travis asked.

“Well, the man who owns this house has decided to sell it,” Uncle Rashad 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.

$\#$ 3 - 30 days $\#$ 25 - 32 days

$\#$ 1 - 35 days $\#$ 14 - 40 days

$\#$ 28 - 45 days $\#$ 77 - 60 days

$\#$ 32 - 65 days $\#$ 19 - 90 days

$\#$ 21 - 100 days $\#$ 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.

To organize this data, Travis can build a box-and-whisker plot. In this lesson you will help Travis with this task. Pay close attention and you will be able to use the data and build a box-and-whisker plot at the end of the lesson.

Guided Learning

Order a Set of Data to Find the Median, Quartiles, and Extremes

Today’s lesson focuses on data once again. This time, we will be building box-and-whisker plots. To understand a box-and-whisker plot, there is some vocabulary to learn. Our first keyword 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 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 eight and eight. The lower quartile is eight.

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, six is the lower extreme and 20 is the upper extreme.

Guided Practice

Check your understanding by answering these questions.

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

What is the median of this data set?
What is the lower quartile?
What is the upper extreme?

Take a minute to check your answers with your neighbor.

Solutions:

Median = 7
Lower quartile = 4.5
Upper quartile = 12

Draw a Box-and-Whisker Plot to Represent Given Data

Now that we have identified all of the key parts of a box-and-whisker plot, we can move on to drawing one. Here are the key things that we need to do BEFORE drawing a box-and-whisker plot.

Write the step for creating a box-and-whisker plot in your journal.

We have this information for the data set that we looked at in the last section. Here is the data set again.

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

Here are the steps to drawing a box-and-whisker plot.

Draw a number line labeled to show the range of data from least to greatest.
Mark the median, the upper quartile, the lower quartile, the lower extreme, and the upper extreme on the number line.
Draw in a box around the quartiles. The median is the middle line of the two boxes.
Then draw in the whiskers. These are lines that extend from each quartile to the upper and lower extremes.

Here is a picture of a number line with a completed box-and-whisker plot on it.

Now let’s examine this plot. The first box goes from the lower quartile eight to the median 11. The second box goes from the median 11 to the upper quartile 15.5. The whiskers extend out from the lower quartile to the lower extreme of six, and from the upper quartile to the upper extreme of 20.

Identify the Median, Quartiles, and Extremes Given a Box-and-Whisker Plot

Now that you know how to draw a box-and-whisker plot and find the median, quartiles, and extremes of a set of data, we can work the other way around. We can look at a box-and-whisker plot to identify the median, quartiles, and extremes.

We can use this chart to examine the data. The median divides the two boxes. The median here is 200. The lower quartile is 100 and the upper quartile is 300. The lower extreme is 50 and the upper extreme is 400.

We can use a box-and-whisker plot to analyze data, to show data in a visual way, and to compare two sets of data.

Compare and Interpret Double Box-and-Whisker Plots of Real-World Data

What happens when we have a two box-and-whisker plots? What does this mean?

When we have two box-and-whisker plots on the same set of data we are comparing the similar data. The data probably has close to the same range, but we can get a good idea about the data from looking at the box-and-whisker plot. We can see how much two sets of similar data vary by looking at the plot.

Let’s look at an example.

This box-and-whisker plot looks at the length of the American alligator vs. the Crocodile.

American Alligators range in length from 8.2 to 11.2 ft, with the longest being 17.5 ft long.

Crocodiles range in length from 3.3 to 7.9 ft, with the longest being 15.9 ft long.

The top box-and-whisker plot represents the length of the American Alligator.

The bottom box-and-whisker plot represents the length of the crocodile.

The key thing to notice is that the range of the Crocodile varies more than the American Alligator.

The American alligator ranges from 8.2 to 18 ft, while the crocodile ranges from 3.3 to 16 ft. That is a range of 10 (American) compared to a range of about 13 ft (Crocodile).

Real Life Example Completed

Homes for Sale

Now it is time to draw a box-and-whisker plot for the given data. Here is the problem once again.

“What was that all about?” Travis asked.

“What is a selling season?”

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

$\#$ 3 - 30 days $\#$ 25 - 32 days

$\#$ 1 - 35 days $\#$ 14 - 40 days

$\#$ 28 - 45 days $\#$ 77 - 60 days

$\#$ 32 - 65 days $\#$ 19 - 90 days

$\#$ 21 - 100 days $\#$ 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.

To organize this data, Travis can build a box-and-whisker plot.

First, let’s go back and underline the important information.

Here is the data for us to analyze. Let’s find the median first of all.

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.

To get a visual of when the real estate agent can expect to sell the house, we can look at the boxes of the box-and-whisker plot. Let’s draw it.

First, we can take the number of days that it took to sell a home last year and use this for our data range. Selling days ranged from 30 to 120 days. That is a big range. We can organize the data in tens.

30, 40, 50, 60 70, 80, 90 100, 120

Travis looks at the chart. There is a large time range where the house will probably sell. It could sell in 35 days or in 90 days, but the average time was 60 days.

Travis is excited to show his work to his Uncle Rashad.

Review

Box-and-Whisker Plots are used to evaluate a set of data.

Median

Median is the middle score of a set of data.

Quartile

A quartile is one-fourth of a data set.

Upper Quartile

The median of a quartile on the higher end of the range is called the upper quartile.

Lower quartile

The median of a quartile on the lower range is called the lower quartile.

Extremes

The highest and lowest scores possible in a range of data are called the extremes.

Video Resources

Box-and-Whisker Plot Video

Data Analysis: Box-and-Whisker Plot Video