Learning objectives
Box plots, also called box-and-whisker plots or box-whisker plots, give a good graphical image of the concentration of the data. They also show how far the extreme values are from most of the data. As mentioned previously, a box plot is constructed from five values: the minimum value, the first quartile, the median, the third quartile, and the maximum value. We use these values to compare how close other data values are to them.
To construct a box plot, use a horizontal or vertical number line and a rectangular box. The smallest and largest data values label the endpoints of the axis. The first quartile marks one end of the box, and the third quartile marks the other end of the box. Approximately the middle 50 percent of the data fall inside the box. The whiskers extend from the ends of the box to the smallest and largest data values. A box plot easily shows the range of a data set, which is the difference between the largest and smallest data values (or the difference between the maximum and minimum). Unless the median, first quartile, and third quartile are the same value, the median will lie inside the box or between the first and third quartiles. The box plot gives a good, quick picture of the data.
Consider, again, this data set:
1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5
The first quartile is two, the median is seven, and the third quartile is nine. The smallest value is one, and the largest value is 11.5. The following image shows the constructed box plot.
View original figure (external; image not reproduced in this text edition)
Original figure description: Horizontal boxplot's first whisker extends from the smallest value, 1, to the first quartile, 2, the box begins at the first quartile and extends to the third quartile, 9, a vertical dashed line is drawn at the median, 7, and the second whisker extends from the third quartile to the largest value of 11.5.
The two whiskers extend from the first quartile to the smallest value and from the third quartile to the largest value. The median is shown with a dashed line.
The following data are the heights of 40 students in a statistics class:
59, 60, 61, 62, 62, 63, 63, 64, 64, 64, 65, 65, 65, 65, 65, 65, 65, 65, 65, 66, 66, 67, 67, 68, 68, 69, 70, 70, 70, 70, 70, 71, 71, 72, 72, 73, 74, 74, 75, 77.
Construct a box plot with the following properties. Calculator instructions for finding the five number summary follow this example:
- Minimum value = 59
- Maximum value = 77
- Q1: First quartile = 64.5
- Q2: Second quartile or median = 66
- Q3: Third quartile = 70
View original figure (external; image not reproduced in this text edition)
Original figure description: Horizontal boxplot with first whisker extending from smallest value, 59, to Q1, 64.5, box beginning from Q1 to Q3, 70, median dashed line at Q2, 66, and second whisker extending from Q3 to largest value, 77.
- Each quarter has approximately 25 percent of the data.
- The spreads of the four quarters are 64.5 – 59 = 5.5 (first quarter), 66 – 64.5 = 1.5 (second quarter), 70 – 66 = 4 (third quarter), and 77 – 70 = 7 (fourth quarter). So, the second quarter has the smallest spread, and the fourth quarter has the largest spread.
- Range = maximum value – minimum value = 77 – 59 = 18.
- Interquartile Range: IQR = Q3 – Q1 = 70 – 64.5 = 5.5.
- The interval 59–65 has more than 25 percent of the data, so it has more data in it than the interval 66–70, which has 25 percent of the data.
- The middle 50 percent (middle half) of the data has a range of 5.5 inches.
For some sets of data, some of the largest value, smallest value, first quartile, median, and third quartile may be the same. For instance, you might have a data set in which the median and the third quartile are the same. In this case, the diagram would not have a dotted line inside the box displaying the median. The right side of the box would display both the third quartile and the median. For example, if the smallest value and the first quartile were both one, the median and the third quartile were both five, and the largest value was seven, the box plot would look like the following:
View original figure (external; image not reproduced in this text edition)
Original figure description: Horizontal boxplot box begins at the smallest value and Q1, 1, until the Q3 and median, 5, no median line is designated, and has its lone whisker extending from the Q3 to the largest value, 7.
In this case, at least 25 percent of the values are equal to one. Twenty-five percent of the values are between one and five, inclusive. At least 25 percent of the values are equal to five. The top 25 percent of the values fall between five and seven, inclusive.
Test scores for Mr. Ramirez's class held during the day are as follows:
99, 56, 78, 55.5, 32, 90, 80, 81, 56, 59, 45, 77, 84.5, 84, 70, 72, 68, 32, 79, 90.
Test scores for Ms. Park's class held during the evening are as follows:
98, 78, 68, 83, 81, 89, 88, 76, 65, 45, 98, 90, 80, 84.5, 85, 79, 78, 98, 90, 79, 81, 25.5.
- Find the smallest and largest values, the median, and the first and third quartile for Mr. Ramirez's class.
- Find the smallest and largest values, the median, and the first and third quartile for Ms. Park's class.
- For each data set, what percentage of the data is between the smallest value and the first quartile? the first quartile and the median? the median and the third quartile? the third quartile and the largest value? What percentage of the data is between the first quartile and the largest value?
- Create a box plot for each set of data. Use one number line for both box plots.
- Which box plot has the widest spread for the middle 50 percent of the data,the data between the first and third quartiles? What does this mean for that set of data in comparison to the other set of data?
- Min = 32
- Q1 = 56
- M = 74.5
- Q3 = 82.5
- Max = 99
-
- Min = 25.5
- Q1 = 78
- M = 81
- Q3 = 89
- Max = 98
- Mr. Ramirez's class: There are six data values ranging from 32 to 56: 30 percent. There are six data values ranging from 56 to 74.5: 30 percent. There are five data values ranging from 74.5 to 82.5: 25 percent. There are five data values ranging from 82.5 to 99: 25 percent. There are 16 data values between the first quartile, 56, and the largest value, 99: 80 percent. Ms. Park’s class: There are seven data values ranging from 25.5 to 78: 32 percent. There are seven data values ranging from 78 through 32 percent. There are seven data values ranging 81 to 89: 32 percent. There are six data values ranging from 89 to 98: 27 percent. There are 17 values between the first quartile, 78, and the largest value, 98: 77 percent.
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View original figure (external; image not reproduced in this text edition)
Original figure description: Two box plots over a number line from 0 to 100. The top plot shows a whisker from 32 to 56, a solid line at 56, a dashed line at 74.5, a solid line at 82.5, and a whisker from 82.5 to 99. The lower plot shows a whisker from 25.5 to 78, solid line at 78, dashed line at 81, solid line at 89, and a whisker from 89 to 98.
- The first data set has the wider spread for the middle 50 percent of the data. The IQR for the first data set is greater than the IQR for the second set. This means that there is more variability in the middle 50 percent of the first data set.
Graph a box-and-whisker plot for the following data values shown:
10, 10, 10, 15, 35, 75, 90, 95, 100, 175, 420, 490, 515, 515, 790
The five numbers used to create a box-and-whisker plot are as follows:
- Min: 10
- Q1: 15
- Med: 95
- Q3: 490
- Max: 790
The following graph shows the box-and-whisker plot.
View original figure (external; image not reproduced in this text edition)
Original figure description: This graph and whisker plot shows the numbers 10, 15, 95, 490, and 790 represented on a line plot. The markings are from 10 to 790 with the range being from 15 to 490 and the median being 95.
References
West Magazine. (n.d.). Retrieved from https://westmagazine.net/
Chapter Review
Box plots are a type of graph that can help visually organize data. Before a box plot can be graphed, the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. Once the box plot is graphed, you can display and compare distributions of data.
Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars.
Construct a box plot below. Use a ruler to measure and scale accurately.
Looking at your box plot, does it appear that the data are concentrated together, spread out evenly, or concentrated in some areas but not in others? How can you tell?
More than 25 percent of salespersons sell four cars in a typical week. You can see this concentration in the box plot because the first quartile is equal to the median. The top 25 percent and the bottom 25 percent are spread out evenly; the whiskers have the same length.
Homework
In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results:
View original figure (external; image not reproduced in this text edition)
Original figure description: This shows three boxplots graphed over a number line from 0 to 11. The boxplots match the supplied data, and compare the countries' results. The China boxplot has a single whisker from 0 to 5. The Germany box plot's median is equal to the third quartile, so there is a dashed line at right edge of box. The America boxplot does not have a left whisker.
- In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected.
- Have more Americans or more Germans surveyed been to more than eight foreign countries?
- Compare the three box plots. What do they imply about the foreign travel of 20-year-old residents of the three countries when compared to each other?
Given the following box plot, answer the questions.
View original figure (external; image not reproduced in this text edition)
Original figure description: This is a boxplot graphed over a number line from 0 to 150. There is no first, or left, whisker. The box starts at the first quartile, 0, and ends at the third quartile, 80. A vertical, dashed line marks the median, 20. The second whisker extends the third quartile to the largest value, 150.
- Think of an example (in words) where the data might fit into the above box plot. In two to five sentences, write down the example.
- What does it mean to have the first and second quartiles so close together, while the second to third quartiles are far apart?
- Answers will vary. Possible answer: State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year.
- Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile.
Given the following box plots, answer the questions.
View original figure (external; image not reproduced in this text edition)
Original figure description: This shows two boxplots graphed over number lines from 0 to 7. The first whisker in the data 1 boxplot extends from 0 to 2. The box begins at the firs quartile, 2, and ends at the third quartile, 5. A vertical, dashed line marks the median at 4. The second whisker extends from the third quartile to the largest value, 7. The first whisker in the data 2 box plot extends from 0 to 1.3. The box begins at the first quartile, 1.3, and ends at the third quartile, 2.5. A vertical, dashed line marks the medial at 2. The second whisker extends from the third quartile to the largest value, 7.
- In complete sentences, explain why each statement is false.
- Data 1 has more data values above two than Data 2 has above two.
- The data sets cannot have the same mode.
- For Data 1, there are more data values below four than there are above four.
- For which group, Data 1 or Data 2, is the value of 7 more likely to be an outlier? Explain why in complete sentences.
A survey was conducted of 130 purchasers of new black sports cars, 130 purchasers of new red sports cars, and 130 purchasers of new white sports cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results:
View original figure (external; image not reproduced in this text edition)
Original figure description: This shows three boxplots graphed over a number line from 25 to 80. The first whisker on the BMW 3 plot extends from 25 to 30. The box begins at the firs quartile, 30 and ends at the thir quartile, 41. A verical, dashed line marks the median at 34. The second whisker extends from the third quartile to 66. The first whisker on the BMW 5 plot extends from 31 to 40. The box begins at the firs quartile, 40, and ends at the third quartile, 55. A vertical, dashed line marks the median at 41. The second whisker extends from 55 to 64. The first whisker on the BMW 7 plot extends from 35 to 41. The box begins at the first quartile, 41, and ends at the third quartile, 59. A vertical, dashed line marks the median at 46. The second whisker extends from 59 to 68.
- In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series.
- Which group is most likely to have an outlier? Explain how you determined that.
- Compare the three box plots. What do they imply about the age of purchasing a sports car from the series when compared to each other?
- Look at the red sports cars. Which quarter has the smallest spread of data? What is the spread?
- Look at the red sports cars. Which quarter has the largest spread of data? What is the spread?
- Look at the red sports cars. Estimate the interquartile range (IQR).
- Look at the red sports cars. Are there more data in the interval 31–38 or in the interval 45–55? How do you know this?
- Look at the red sports cars. Which interval has the fewest data in it? How do you know this?
- 31–35
- 38–41
- 41–64
- Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50 percent of buyers are more variable than the ages of the lower 50 percent.
- The black sports car is most likely to have an outlier. It has the longest whisker.
- Comparing the median ages, younger people tend to buy the black sports car, while older people tend to buy the white sports car. However, this is not a rule, because there is so much variability in each data set.
- The second quarter has the smallest spread. There seems to be only a three-year difference between the first quartile and the median.
- The third quarter has the largest spread. There seems to be approximately a 14-year difference between the median and the third quartile.
- IQR ~ 17 years
- There is not enough information to tell. Each interval lies within a quarter, so we cannot tell exactly where the data in that quarter is are concentrated.
- The interval from 31 to 35 years has the fewest data values. Twenty-five percent of the values fall in the interval 38 to 41, and 25 percent fall between 41 and 64. Since 25 percent of values fall between 31 and 38, we know that fewer than 25 percent fall between 31 and 35.
Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:
| Number of Movies | Frequency |
|---|---|
| 0 | 5 |
| 1 | 9 |
| 2 | 6 |
| 3 | 4 |
| 4 | 1 |
Construct a box plot of the data.
Bringing It Together
Santa Clara County, California, has approximately 27,873 Japanese Americans. fs-idm41920944 shows their ages by group and each age-group's percentage of the Japanese American community.
| Age-Group | Percentage of Community |
|---|---|
| 0–17 | 18.9 |
| 18–24 | 8.0 |
| 25–34 | 22.8 |
| 35–44 | 15.0 |
| 45–54 | 13.1 |
| 55–64 | 11.9 |
| 65+ | 10.3 |
- Construct a histogram of the Japanese American community in Santa Clara County. The bars will not be the same width for this example. Why not? What impact does this have on the reliability of the graph?
- What percentage of the community is under age 35?
- Which box plot most resembles the information above?
View original figure (external; image not reproduced in this text edition)
Original figure description: Three box plots with values between 0 and 100. Plot i has Q1 at 24, M at 34, and Q3 at 53; Plot ii has Q1 at 18, M at 34, and Q3 at 45; Plot iii has Q1 at 24, M at 25, and Q3 at 54.
- For graph, check student's solution.
- 49.7 percent of the community is under the age of 35
- Based on the information in the table, graph (a) most closely represents the data.
Glossary
- box plot
- a graph that gives a quick picture of the middle 50 percent of the data
- first quartile
- the value that is the median of the lower half of the ordered data set
- frequency polygon
- a data display that looks like a line graph but uses intervals to display ranges of large amounts of data
- interval
- also called a class interval; an interval represents a range of data and is used when displaying large data sets
- paired data set
- two data sets that have a one-to-one relationship so that
- both data sets are the same size, and
- each data point in one data set is matched with exactly one point from the other set
- skewed
- used to describe data that is not symmetrical; when the right side of a graph looks chopped off compared to the left side, we say it is skewed to the left.
When the left side of the graph looks chopped off compared to the right side, we say the data are skewed to the right. Alternatively, when the lower values of the data are more spread out, we say the data are skewed to the left. When the greater values are more spread out, the data are skewed to the right.