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Box-and-Whisker Plots

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Practice Box-and-Whisker Plots
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Box-and-Whisker Plots

Remember Travis and the real estate dilemma in the Order a Set of Data to Find Measures Concept? Well, after Travis had worked to find all of the measures, he decided that he wanted to create a visual display of the data. What does this mean? Let's look at the problem from the last Concept and go from there.

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.

$\#$ 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.”

To create a visual display of the data, Travis will need to create a box - and - whisker plot. You will learn exactly how to do that in this Concept.

Guidance

Previously we worked on analyzing all of the measures. Remember these are the quartiles, the median and the extremes. These are the key parts to a box-and-whisker plot. Now that you 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.

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.

1. Draw a number line labeled to show the range of data from least to greatest.
2. Mark the median, the upper quartile, the lower quartile, the lower extreme and the upper extreme on the number line.
3. Draw in a box around the quartiles. The median is the middle line of the two boxes.
4. 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 8 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 6, and from the upper quartile to the upper extreme of 20.

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.

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.

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, with the longest being 17.5 ft long.

Crocodiles range in length from 3.3 to 7.9, with the longest being 15.9 feet 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 feet. That is a range of 10 (American) compared to a range of about 13 feet (Crocodile).

Now it's time for you to try a few. Use this box-and-whisker plot.

Example A

What is the smallest value on the plot?

Solution: $34$

Example B

What is the greatest value on the plot?

Solution: $58$

Example C

What is the median of the whole data set?

Solution: $45$

Now let's go back and see what kind of visual display Travis can make given his data.

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 Larry.

Vocabulary

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.

The weight of bears varies between species. Weight also varies within species as a result of habitat and diet. The box-and-whisker plot was created after recording the weight (in pounds) of several black bears across the country.

Use the box-and-whisker plot to answer the questions below.

How is the number line organized?

The number line is labeled by tens. Notice that each section on the number line has been divided into fifths. Therefore, each mark on the number line represents two.

What are the highest and lowest weights represented on the box-and-whisker plot?

The lowest value or weight is 127 pounds. The highest value or weight is 201 pounds.

What is the median weight for a black bear?

The median weight is 163 pounds.

These are our answers.

Video Review

This video presents box-and-whisker plots.

Practice

Directions: Use the following box-and-whisker plot to answer the questions.

1. What is the median score in this box-and-whisker plot?

2. What is the lower quartile?

3. What is the upper quartile?

4. What is the range of the data?

5. What is the lower extreme?

6. What is the upper extreme?

7. What is the range of the data?

Directions: Use the data to build a box-and-whisker plot. Then answer the questions.

25, 26, 30, 18, 24, 26, 19, 21, 22

8. Box-and-whisker plot

9. Write the data in order from least to greatest.

10. What is the median score?

11. What is the lower quartile?

12. What is the upper quartile?

13. What is the lower extreme?

14. What is the upper extreme?

15. What is the range of the data?

Vocabulary Language: English

arithmetic mean

arithmetic mean

The arithmetic mean is also called the average.
back-to-back stem plots

back-to-back stem plots

A Back-to-Back stem plot is a modified stem-and-leaf plot with the stem in the center and the leaves on the sides, it is used to compare two different related sets of data (bivariate data).
bell shaped

bell shaped

A bell shaped histogram is a histogram with a prominent ‘mound’ in the center and similar tapering to the left and right.
bins

bins

Bins are groups of data plotted on the x-axis.
bivariate data

bivariate data

Bivariate data consists of two paired sets of data.
box- and- whisker plot

box- and- whisker plot

A box- and- whisker plot is a graphic display of quantitative data that demonstrates the five number summary.
calculated data

calculated data

Calculated data has values that are the result of computations performed on the input variable.
dependent variable

dependent variable

The dependent variable is the output variable in an equation or function, commonly represented by $y$ or $f(x)$.
explanatory variables

explanatory variables

Explanatory variables are another name for independent variables.
extreme outliers

extreme outliers

Extreme outliers include points more than 3 times the middle half of your data.      .
Extremes

Extremes

The extremes are the maximum and minimum values in a data set.
first quartile

first quartile

The first quartile, also known as $Q_1$, is the median of the lower half of the data.
five point summary

five point summary

The numbers needed to construct a box-and-whisker plot are called the five-point-summary. The five points are the minimum, the lower median (Q1), the median, the upper median (Q3), and the maximum.
independent variable

independent variable

The independent variable is the input variable in an equation or function, commonly represented by $x$.
input variables

input variables

Input variables are another name for independent variables.
Interquartile range

Interquartile range

The interquartile range is the difference between the third quartile and the first quartile (Q3-Q1).
Leaf

Leaf

The leaves of a stem-and-leaf plot are the rightmost digits of each of the original data values.
line of best fit

line of best fit

A line of best fit is a straight line drawn on a scatter plot such that the sums of the distances to the points on either side of the line are approximately equal and such that there are an equal number of points above and below the line.
line of fit

line of fit

A line of fit is a straight or continuously curved line representing the trend of changes in the comparison of two data sets (or one set of bivariate data).
linear regression

linear regression

In statistics, linear regression is a process that attempts to model the relationship between two variables by fitting a linear equation to the data.
lower median

lower median

The lower median is the first quartile (Q1) in the box-and-whisker plot.
Lower quartile

Lower quartile

The lower quartile, also known as $Q_1$, is the median of the lower half of the data.
Median

Median

The median of a data set is the middle value of an organized data set.
mild outliers

mild outliers

Mild outliers include data points that are more than 1.5 times the middle half of your data above the upper, or below the lower, quartiles.
modified box-plot

modified box-plot

A modified box plot has whiskers that extend to the highest and lowest non-outlier value.
normal distributed

normal distributed

If data is normally distributed, the data set creates a symmetric histogram that looks like a bell.
observed data

observed data

Observed data are the values that result from computations performed on the input variable.
Outlier

Outlier

In statistics, an outlier is a data value that is far from other data values.
output variables

output variables

Output variables are another name for dependent variables.
Quartile

Quartile

A quartile is each of four equal groups that a data set can be divided into.
range

range

The range of a set of data is the difference in value between the least and greatest values in the set.
response variables

response variables

Response variables are another name for dependent variables.
second quartile

second quartile

The second quartile, also known as $Q_2$, is the median of the data.
skewed

skewed

As with the horizontal skewing of a histogram, stem plots with a obvious skew toward one end or the other tend to indicate an increased number of outliers either lesser than or greater than the mode.
statistical correlation

statistical correlation

Statistical correlation is a representation of possible related changes in values between the two sets of data.
stem

stem

A stem  in a stem plot is a values or column of values that represent the greatest place value(s) in a set of data.
Stem-and-leaf plot

Stem-and-leaf plot

A stem-and-leaf plot is a way of organizing data values from least to greatest using place value. Usually, the last digit of each data value becomes the "leaf" and the other digits become the "stem".
third quartile

third quartile

The third quartile, also known as $Q_3$, is the median of the upper half of the data.
trends

trends

Trends in data sets or samples are indicators found by reviewing the data from a general or overall standpoint
uniform

uniform

A uniform shaped histogram indicates data that is very consistent; the frequency of each class is very similar to that of the others.
upper median

upper median

The upper median is the third quartile (Q3) in the box-and-whisker plot.
Upper Quartile

Upper Quartile

The upper quartile, also known as $Q_3$, is the median of the upper half of the data.