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Box-and-Whisker Plots

Plotting the five-number summary for ascending data.

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Box-and-Whisker Plots
Credit: CK-12 Foundation
Source: CK-12 Foundation
License: CC BY-NC 3.0

Mr. Hernandez gave his students a science quiz. He has organized the quiz scores in the table below. 

Science Quiz Results
Student  Test Score
AS 89
AS 91
BC 87
CS 77
FR 72
JW 59
ML 76
MY 68
SR 83
ST 91
VS 81
ZS 73

Mr. Hernandez wants to create a graph to display the distribution of the scores. How can he do this? 

In this concept, you will learn how to create and read box-and-whisker plots. 

Creating and Reading Box-and-Whisker Plots

Data is a set of numerical or non-numerical information. Data can be analyzed in many different ways. In this concept you will analyze numerical data using box-and-whisker plots.

Median is a numerical value that represents the middle term in a data set. When ordered sequentially, the value that is in the middle of the data set is the median

The range of a set of data is the difference between the largest and smallest values. The range identifies how far apart the values are in the data set. 

The minimum is the smallest value in the data set.

The maximum is the largest value in the data set. 

Quartiles divide data sets into four equal groups. The first quartile is the median of the lower half of the data set. The second quartile is the median of the data set. The third quartile is the median of the upper half of the data set.

One way to display data is in a box-and-whisker plot. A box plot illustrates the distribution of the data set in quartiles. A box is made around the median, with the sides of the box being the first and third quartiles. The whiskers are the lines extending outwards from either side of the box indicating the range of the data set. Box and whisker plots can be oriented vertically or horizontally. 

Let's look at an example. 

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

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

First, arrange the data in order from smallest to largest. 

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

Next, find the median, which is the middle number. There are nine terms, so the fifth term is the median, 49. The median will not be used in calculating the first or third quartiles

Next, find the first quartile, which is the middle number of the lower half of the data set. Since 49 is not included, there are four terms in the lower half of the set. The median of this set is between the second and third terms, 42 and 45. To find the median we add the two terms together and divide by 2.

\begin{align*}42+45=87\end{align*}42+45=87

\begin{align*}87\div 2=43.5\end{align*}87÷2=43.5

So, the first quartile is 43.5. This will be the left side of the box. 

Next, find the third quartile, which is the middle number of the upper half of the data set. Since 49 is not included, there are four terms in the upper half of the set. The median of this set is between the second and third terms, 51 and 52. To find the median we add the two terms together and divide by 2.

\begin{align*}51+52=103\end{align*}51+52=103

\begin{align*}103÷2=51.5\end{align*}103÷2=51.5

So, the third quartile is 51.5. This will be the right side of the box. 

Next, find the minimum and maximum of the data set. These will be the ends of the whiskers. The minimum is the smallest term in the set. In this case, it's 34. The maximum is the largest term in the set. In this case, it's 58. 

Next, draw the box-and-whisker plot. Begin by drawing a number line that begins below the minimum, 34, and extends beyond the maximum, 58. In this case the number line should be drawn from 30 to 60. Then draw the box and whisker plot above the number line. 

The box-and-whisker plot should look like the one below. 

Examples

Example 1

Earlier, you were given a problem about Mr. Hernandez and his science quiz.

Mr. Hernandez gave his class a science quiz and wants to create a graph to show the distribution of the scores. He organized his data into the table below. 

Science Quiz Results
Student  Test Score
AS 89
AS 91
BC 87
CS 77
FR 72
JW 59
ML 76
MY 68
SR 83
ST 91
VS 81
ZS 73

How can you represent this data in a graph? 

First, decide what kind of graph to make. Since you want to show the distribution of the quiz scores, a box-and-whisker plot is best. 

Next, organize the data from smallest to largest. 

Science Quiz Results (smallest to largest)
Student Test Score
JW 59
MY 68
FR 72
ZS 73
ML 76
CS 77
VS 81
SR 83
BC 87
AS 89
AS 91
ST 91


Next, find the median, which is the middle number. There are twelve terms, so the median is between the sixth and seventh terms, 77 and 81. To find the median, add the two values together and divide by 2.

\begin{align*}77+81=158\end{align*}77+81=158

\begin{align*}158\div 2=79\end{align*}158÷2=79

The median is 79. 

Next, find the first quartile, which is the middle number of the lower half of the data set. There are six terms in the lower half of the set, 59, 58, 72, 73, 76, and 77. The median of this set is between the third and fourth terms, 72 and 73. The number that is halfway between 72 and 73 is 72.5. The first quartile is 72.5, which will be the left side of the box.

Next, find the third quartile, which is the middle number of the upper half of the data set. There are six terms in the upper half of the set, 81, 83, 87, 89, 91, and 91. The median of this set is between the third and fourth terms, 87 and 89. The number that is halfway between 87 and 89 is 88. The third quartile is 88, which will be the right side of the box.

Next, find the minimum and maximum of the data set. These will be the ends of the whiskers. The minimum is the smallest term in the set. In this case, it's 59. The maximum is the largest term in the set. In this case, it's 91.

Next, draw the box-and-whisker plot. Begin by drawing a number line that begins below the minimum, 59, and extends beyond the maximum, 91. In this case, the number line should begin at 55 and go to 95. Then draw the box-and-whisker plot above the number line. 

Lastly, title the graph. The title should be short and clear. In this case, title the graph "Quiz Scores."

The box-and-whisker plot should look like the one below.

Credit: CK-12 Foundation
Source: CK-12 Foundation
License: CC BY-NC 3.0

Example 2

The data values in the table below illustrate the number of TVs 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

First, arrange the data in order from smallest to largest. 

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

Next, find the median, which is the middle number. There are nine terms, so the fifth term is the median, 102. The median will not be used in calculating the first or third quartiles

Next, find the first quartile, which is the middle number of the lower half of the data set. Since 102 is not included, there are four terms in the lower half of the set, 89, 91, 95, and 98. The median of this set is between the second and third terms, 91 and 95. To find the median we add the two terms together and divide by 2.

\begin{align*}91+95=186\end{align*}91+95=186

\begin{align*}186÷2=93\end{align*}186÷2=93

 So, the first quartile is 93. This will be the left side of the box. 

Next, find the third quartile, which is the middle number of the upper half of the data set. Since 102 is not included, there are four terms in the upper half of the set, 108, 110, 118, and 152. The median of this set is between the second and third terms, 110 and 118. To find the median we add the two terms together and divide by 2. 

\begin{align*}110+118=228\end{align*}110+118=228

\begin{align*}228÷2=114\end{align*}228÷2=114

So, the third quartile is 114. This will be the right side of the box. 

Next, find the minimum and maximum of the data set. These will be the ends of the whiskers. The minimum is the smallest term in the set. In this case, it's 89. The maximum is the largest term in the set. In this case, it's 152. 

Next, draw the box-and-whisker plot. Begin by drawing a number line that begins below the minimum, 89, and extends beyond the maximum, 152. In this case, the number line should begin at 80 and go to 160. Then draw the box-and-whisker plot above the number line. 

Lastly, title the graph. The title should be short and clear. In this case, title the graph "Television Sales."

The box-and-whisker plot should look like the one below. 

Example 3

The box plot below shows the weight (in pounds) of black bears. Identify the minimum, maximum, median, first quartile, and third quartile of the data set. 

First, find the minimum value of the data set. To do this look at the end of the left whisker. The left whisker ends at 127, so this is the minimum. 

Next, find the maximum value of the data set. To do this look at the end of the right whisker. The right whisker ends at 201, so this is the maximum. 

Next, find the first quartile. To do this look at the left side of the box. The left side of the box is at 129, so this is the first quartile. 

Next, find the median. To do this look at the vertical line inside the box. This is at 163, so this is the median. 

Next, find the third quartile. To do this look at the right side of the box. The right side of the box is at 196, so this is the third quartile. 

The answer is the minimum is 127 pounds, the maximum is 201 pounds, the median is 163 pounds, the first quartile is 129 pounds, and the third quartile is 196 pounds.

Example 4

The values in the data set below represent the number of dogs that returned from a sled race. Create a box-and-whisker plot of the data. 

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

First, order the data from least to greatest.

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

Next, find the median, which is the middle number. There are ten terms, so the median is between the fifth and sixth terms, 10 and 10. Since these terms are the same, the median is 10. The median will not be used in calculating the first or third quartiles. 

Next, find the first quartile, which is the middle number of the lower half of the data set. Since 10 is not included, there are four terms in the lower half of the set, 7, 7, 9, and 9. The median of this set is between the second and third terms, 7 and 9. To find the median we add the two terms together and divide by 2.

\begin{align*}7+9=16\end{align*}7+9=16

\begin{align*}16÷2=8\end{align*}16÷2=8

So, the first quartile is 8. This will be the left side of the box. 

Next, find the third quartile, which is the middle number of the upper half of the data set. Since 10 is not included, there are four terms in the upper half of the set, 11, 11, 12, and 13. The median of this set is between the second and third terms, 11 and 12. To find the median we add the two terms together and divide by 2. 

\begin{align*}11+12=23\end{align*}11+12=23

\begin{align*}23÷2=11.5\end{align*}23÷2=11.5

So, the third quartile is 11.5. This will be the right side of the box. 

Next, find the minimum and maximum of the data set. These will be the ends of the whiskers. The minimum is the smallest term in the set. In this case, it's 7. The maximum is the largest term in the set. In this case, it's 13. 

Next, draw the box-and-whisker plot. Begin by drawing a number line that begins at the minimum, 7, and extends to the maximum, 13. In this case, the number line can begin at 7 and go to 13. Then draw the box-and-whisker plot above the number line. 

Lastly, title the graph. The title should be short and clear. In this case, title the graph "Sled Dogs."

The box-and-whisker plot should look like the one below. 

Example 5

The box-and-whisker plot below was shows 5-K race times for some runners. Identify the minimum, maximum, median, first quartile, and third quartile of the data set. 

First, find the minimum value of the data set. To do this look at the end of the left whisker. The left whisker ends at 12, so this is the minimum. 

Next, find the maximum value of the data set. To do this look at the end of the right whisker. The right whisker ends at 26, so this is the maximum. 

Next, find the first quartile. To do this look at the left side of the box. The left side of the box is at 14, so this is the first quartile. 

Next, find the median. To do this look at the vertical line inside the box. This is at 17, so this is the median. 

Next, find the third quartile. To do this look at the right side of the box. The right side of the box is at 21, so this is the third quartile. 

The answer is the minimum is 12 minutes, the maximum is 26 minutes, the median is 17 minutes, the first quartile is 14 minutes, and the third quartile is 21 minutes.

Review

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

  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.

100, 105, 107, 109.110, 120

  1. What is the median of the 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?

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.10.

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Vocabulary

Extremes

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

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

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

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

observed data

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

Outlier

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

Quartile

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

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

trends

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

uniform

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

Image Attributions

  1. [1]^ Credit: CK-12 Foundation; Source: CK-12 Foundation; License: CC BY-NC 3.0
  2. [2]^ Credit: CK-12 Foundation; Source: CK-12 Foundation; License: CC BY-NC 3.0

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