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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.

62,67,75,76,78,81,81,83,85,88,90,92Upper QuartileMedianLower Quartile81=Median75=Median of Upper Quartile88=Median of Lower Quartile62 is the smallest value. 92 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.

34, 42, 45, 45, 49, 50, 51, 52, 5842+45=87 87÷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, 50, 51, 52, 5851+52=103103÷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.

89, 91, 95, 98, 102, 108, 110, 118, 152 91+95=186 186÷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, 108, 110, 118, 152 110+118=228 228÷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.

### Vocabulary

Box-and-Whisker Plot
A visual way of organizing data by using median scores to organize data.
Median
the middle value of a set of data.
Quartile
the two halves of the data around the median.
Lower quartile
the half of the data from the median to the lowest value in the set.
Upper quartile
the half of the data from the median to the upper value in the set.

### 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.

### Practice

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?

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

19. Create a box-and-whisker plot for the given data: 87, 85, 89, 92, 94, 97, 102, 105, 105, 113, and 108.

20. The table below depicts the number of students in each grade at Keys School. Create a box-and-whisker plot to display the data from the table.

K1st2nd3rd 4th5th6th7th8th647576 71 8083 95 9291

21. Use the data below to create a double box-and-whisker plot

Data Set 1: 25, 28, 22, 21, 23, 28, 29, 32, 31, 29

Data Set 2: 35, 38, 28, 31, 26, 30, 32, 25, 26, 34

### Vocabulary Language: English

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.
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).
Median

Median

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

Quartile

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