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# Box-and-Whisker Plots

Plotting the five-number summary for ascending data.
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Practice Box-and-Whisker Plots
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Box-and-Whisker Plots

Have you ever studied sled dogs?

Kelly loves dogs. She has been researching the sled dogs connected with the Iditarod. She has noticed that the dogs are very well loved by the mushers who care for them and that there seems to be a very unique connection between them.

Kelly was pleased to see that there are veterinary services for the dogs along the trail route. Here the dogs can receive medical attention if they become injured along the way.

Most of the teams begin with between 12 and 16 dogs, but most don’t finish with that many. Some of the dogs can become tired or hurt, and sometimes a dog can die along the journey too.

Kelly did some research about the 2010 dog teams and discovered that the top ten teams had arrived back with somewhere between 7 and 13 dogs.

She wrote these statistics down in her notebook.

11, 11, 12, 10, 9, 10, 13, 7, 9, 7

Kelly wants to create a display of this information. She has decided to create a box-and-whisker plot to show the number of dogs who finished in 2010 in the top 10 teams.

Do you have an idea how to do this? If you do, then draw a box-and-whisker plot now in your notebook using this data. If not, then pay attention to this Concept and you will learn all that you need to know about box-and-whisker plots.

### Guidance

Previously we learned all about different ways to analyze and display data. Now we are going to learn about a new one it is called a box-and-whisker plot.

A box-and-whisker plot depicts the distribution of data items.

Recall that the median is the middle number when the data is arranged in order from the least to greatest. The median separates the data into two equal parts. On a box-and-whisker plot, the median represents half or fifty percent of all of the data points.

Data can then be separated into quartiles. Quartiles divide data into four equal parts. The median is the middle quartile. The lower quartile is the median of the lower half of the data. The lower quartile represents one fourth or twenty-five percent of the smaller data points. The upper quartile is the median of the upper part of the data. The upper quartile represents one fourth or twenty-five percent of the largest data points.

$& \underline{62, \quad 67, \quad 75, \quad 76, \quad 78}, \quad 81, \quad 81, \quad \underline{83, \quad 85, \quad 88, \quad 90, \quad 92}\\& \text{Upper Quartile} \qquad \qquad \qquad \text{Median} \qquad \text{Lower Quartile}\\& \qquad \qquad 81 = \text{Median}\\& \qquad 75 = \text{Median of Upper Quartile}\\& \qquad 88 = \text{Median of Lower Quartile}\\& 62 \ \text{is the smallest value.} \ 92 \ \text{is the largest value.}$

Create a box-and-whisker plot to display the data below.

45, 58, 34, 42, 52, 49, 50, 45, 51

Step 1: To determine the median of the set of data, arrange the data in order from least to greatest. Identify the data value in the middle of the data set. In this case, the median is 49.

34, 42, 45, 45, 49, 50, 51, 52, 58

Step 2: Identify the median for the lower quartile. In this case, two data values share the middle position in the lower quartile. Recall that when two data values share the middle position, find the mean. To find the mean, add the data values and then divide by two. The median of the lower quartile is 43.5.

$& \underline{34, \ 42, \ 45, \ 45,} \ 49, \ 50, \ 51, \ 52, \ 58\\& \qquad \qquad 42 + 45 = 87\\& \qquad \qquad \ 87 \div 2 = 43.5$

Step 3: Identify the median of the upper quartile. Again, two data values share the middle position. Therefore, you must determine the mean of the two numbers. Since the numbers 51 and 52 are only one away from each other, the median is the number in the exact middle of the two. In this case, the median of the upper quartile is 51.5. This method works whenever the two numbers that share the middle position are one away from each other.

$& 34, \ 42, \ 45, \ 45, \ 49, \ \underline{50, \ 51, \ 52, \ 58}\\& \qquad \qquad 51 + 52 = 103\\& \qquad \qquad 103 \div 2 = 51.5$

Step 4: Draw a number line. The first value on the number line should be near the smallest number in the data set. In this case, the smallest number is 34. Therefore, the number line will start at 30. The last value on the number line should be near the largest number in the set of data. The largest number in the data set is 58. Therefore, the number line will end at 60. Because the difference in the data values is not that great, the number line will be labeled by fives.

The smallest value, 34 is marked on the number line as “I.” The largest value, 58 is marked on the number line as “I.”

The median of the first, second, and third quartiles are marked as “+.”

Step 5: Draw a box around the first, second, and third quartiles. Draw whiskers from the box to the smallest and largest values.

Step 6: Give the box-and-whisker plot at title.

Take a few minutes to write these steps down in your notebook. Then continue with the Concept.

The data values on the table below depict the number of televisions sold at a department store each month for nine months. Create a box-and-whisker plot to display the data.

April May June July August September October November December
110 98 91 102 89 95 108 118 152

Step 1: To determine the median of the set of data, arrange the data in order from least to greatest. Identify the data value in the middle of the data set. For this set of data, 102 is the median.

89, 91, 95, 98, 102, 108, 110, 118, 152

Step 2: Identify the median for the lower quartile. Again, since two data values share the middle position, find their mean. The median for the lower quartile is 93.

$& \underline{89, \ 91, \ 95, \ 98}, \ 102, \ 108, \ 110, \ 118, \ 152\\& \qquad \qquad \qquad \quad \ 91 + 95 = 186\\& \qquad \qquad \qquad \quad \ 186 \div 2 = 93$

Step 3: Identify the median of the upper quartile. Remember to find the mean of the two data values that share the middle position. The median of the upper quartile is 114.

$& 89, \ 91, \ 95, \ 98, \ 102, \ \underline{108, \ 110, \ 118, \ 152}\\& \qquad \qquad \qquad \quad \ 110 + 118 = 228\\& \qquad \qquad \qquad \quad \quad \ 228 \div 2 = 114$

Step 4: Draw a number line. The first value on the number line should be near the smallest number in the data set. In this case, the smallest number is 89. Therefore, the number line will start at 80. The last value on the number line should be near the largest number in the set of data. The largest number in the data set is 152. Therefore, the number line will end at 160. In this case, label the number line by tens.

The smallest value, 89 is marked with a “I” at the end of the whisker in the lower quartile. The largest value, 151is marked with a “I” at the end of the whisker in the upper quartile.

The median of the first, second, and third quartiles are marked with a “+.”

Step 5: Draw a box around the first, second, and third quartiles. Draw whiskers from the box to the smallest and largest values.

Step 6: Give the box-and-whisker plot a title.

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.

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. This is important to note prior to answering the questions below.

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.

What is the median weight for the lower quartile? The median weight of the lower quartile is 129 pounds.

What is the median weight for the upper quartile? The median weight of the upper quartile is 196 pounds.

Now it's time for you to answer a few questions about box-and-whisker plots.

#### Example A

True or false. The median is the middle value in a data set.

Solution: True

#### Example B

True or false. Every box-and-whisker plot has an upper quartile.

Solution: True

#### Example C

True or false. An outlier can't be part of either the upper quartile or lower quartile.

Solution: False

Here is the original problem once again. Reread it and then create a box-and-whisker plot using the data provided. After that, check your work with Kelly’s box-and-whisker plot.

Kelly loves dogs. She has been researching the sled dogs connected with the Iditarod. She has noticed that the dogs are very well loved by the mushers who care for them and that there seems to be a very unique connection between them.

Kelly was pleased to see that there are veterinary services for the dogs along the trail route. Here the dogs can receive medical attention if they become injured along the way.

Most of the teams begin with between 12 and 16 dogs, but most don’t finish with that many. Some of the dogs can become tired or hurt, and sometimes a dog can die along the journey too.

Kelly did some research about the 2010 dog teams and discovered that the top ten teams had arrived back with somewhere between 7 and 13 dogs.

She wrote these statistics down in her notebook.

11, 11, 12, 10, 9, 10, 13, 7, 9, 7

Kelly wants to create a display of this information. She has decided to create a box-and-whisker plot to show the number of dogs who finished in 2010 in the top 10 teams.

Now let’s create a box-and-whisker plot to display the data. First, we write the data in order from least to greatest.

7, 7, 9, 9, 10, 10, 11, 11, 12, 13

The median of all the data is 10.

The median of the lower quartile is 8.

The median of the upper quartile is 11.5

Here is the box-and-whisker plot.

### Guided Practice

Here is one for you to try on your own.

The box-and-whisker plot below was created after recording amount of time it took for several runners to finish a 5K race. Use the box-and-whisker plot to answer the questions below.

The number line on the box-and-whisker plot is labeled by twos. Notice that there is only one section in between each labeled value. Therefore, each mark on the number line represents one. This is important to note when answering the questions below.

Identify the first and last finish times for the race. The first finish time or the smallest value identified on the box-and-whisker plot is 12 minutes. The last finish time or largest value on the box-and-whisker plot is 26 minutes.

Identify the median finish time for the race. The median finish time is 17 minutes.

What was the median finishing time in the lower quartile? The median of the lower quartile is 14 minutes.

What was the median finishing time in the upper quartile? The median of the upper quartile is 21 minutes.

### Explore More

Directions: Use each set of data to work with box-and-whisker plots.

12, 13, 15, 17, 21, 22, 24, 26, 28, 30, 31

1. What is the median of the set of data?

2. What is the median of the lower quartile?

3. What is the median of the upper quartile?

4. What is the lowest value whisker?

5. What is the highest value whisker?

6. Use the data to create a box-and-whisker plot.

26, 27, 29, 30, 32, 35, 41, 42, 44

7. What is the median of the set of data?

8. What is the median of the lower quartile?

9. What is the median of the upper quartile?

10. What is the lowest value whisker?

11. What is the highest value whisker?

12. Use the data to create a box-and-whisker plot.

100, 105, 107, 109.110, 120

13. What is the median of the data?

14. What is the median of the lower quartile?

15. What is the median of the upper quartile?

16. What is the lowest value whisker?

17. What is the highest value whisker?

### 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.
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.
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.
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".
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.