A box plot (sometimes called a box-and-whisker plot) is a visual representation of a data set created using its quartile information.
Suppose a particular data set has the following features:
| Smallest Value | Lower Quartile | Median | Upper Quartile | Greatest Value |
To draw a box plot, we start by drawing a horizontal axis and a box (rectangle) that runs from the lower quartile to the upper quartile.
Then, we put a vertical bar within the box at the median.
Finally, we put two vertical bars outside the box at the smallest and greatest values and connect them to the box with horizontal segments.
Note the following:
The spaces to the box's right and left (i.e., the "whiskers") represent the top and bottom quarter of the data, respectively.
The box itself represents the middle half of the data.
The vertical bar inside the box gives the center of the distribution.
The box plot below represents the distribution of the daily wages of the workers in a fast-food restaurant. What is the median of the data set?
The median is represented by the vertical line that goes through the box.
In our case, the median line corresponds to , as shown below.
Therefore, the median daily wage of the workers is
The box plot above shows the distribution of students' ages in a particular school. What is the upper quartile of the data set?
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$8$ years old |
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b
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$13$ years old |
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c
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$9$ years old |
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$15$ years old |
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$10$ years old |
The box plot above represents the distribution of some students' scores on a math test. What is the interquartile range of the data set?
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$3$ |
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$2$ |
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$1$ |
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$6$ |
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$4$ |
A certain data set has the features listed in the table below.
| Smallest Value | Lower Quartile | Median | Upper Quartile | Greatest Value |
Draw a box plot for this data.
Using the data from the table, we obtain the following boxplot:
A certain data set has the features listed in the table below.
| Smallest Value | Lower Quartile | Median | Upper Quartile | Greatest Value |
| $5$ | $10$ | $12$ | $18$ | $20$ |
What is the box plot for this data?
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The lower quartile of a data set is $100$ while the greatest value in the data set is $120.$ Which of the following could be a boxplot for this data?
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The data set below shows the number of vanilla ice creams sold by a particular ice cream parlor every day during a certain week. Which of the following gives the box plot for this data?
Notice that our data set is already arranged from smallest to greatest, where we have
Since the number of data points is odd (), the median of the data set is the middle number. Therefore,
Now, let's find the quartiles by considering the lower and upper halves of the distribution:
The lower quartile is the median of the lower half. In this case, it's the middle number:
The upper quartile is the median of the upper half. In this case, it's the middle number:
Finally, we draw the corresponding boxplot, as shown below.
Therefore, the boxplot representing this data is diagram
The data set below shows the ages of the members of a rock band: \[ 31, \: 33, \: 34, \: 35, \: 38, \: 40, \: 43 \] Which of the following gives the box plot for this data?
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The data set below shows the number of comic books that each of Georgia's $8$ friends read last summer: \[ 2, \: 2, \: 5, \: 7, \: 7, \: 8, \: 10, \: 12 \] Which of the following gives the box plot for this data?
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Let's consider the box plot below.
We can identify the tails of the data by finding the distance between the median and the quartiles:
The left tail is the distance between the median and the lower quartile.
The right tail is the distance between the median and the upper quartile.
We can use the tails of a box plot to state the skew of the data set.
If the right tail is longer than the left, we say the data is right-skewed (or positively skewed).
If the left tail is longer than the right, we say the data is left-skewed (or negatively skewed).
A data set is symmetric if it's symmetric about the median.
The box plot in our example is right-skewed because the right tail is longer than the left.
The boxplot below shows the distribution of the weekly waste produced by families in a residential building. Describe the skewness of the distribution.
Notice that the median is closer to the upper quartile than the lower quartile. So, we have a longer tail on the left, as shown below.
Therefore, the distribution is left-skewed.
The boxplot above shows the distribution of the number of electronic devices on employees' desks in a new company. Which of the following best describes the distribution?
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the distribution is right-skewed and symmetric |
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b
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the distribution is symmetric |
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c
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the distribution is right-skewed |
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the distribution is left-skewed and symmetric |
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the distribution is left-skewed |
The boxplot above shows the distribution of the number of siblings for students in a particular school. Which of the following best describes the distribution?
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the distribution is symmetric |
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b
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the distribution is left-skewed |
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c
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the distribution is both symmetric and right-skewed |
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d
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the distribution is right-skewed |
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e
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the distribution is both symmetric and left-skewed |
