Suppose that in the first 20 screenings of a movie, the number of paying customers at a theater were as follows: 16, 34, 22, 19, 59, 33, 60, 45, 50, 27, 75, 38, 49, 52, 20, 40, 13, 15, 26, 21. Could you analyze this data with a boxandwhisker plot? If so, what would you need to do first? In this Concept, you'll learn how to use boxandwhisker plots to analyze data sets like this one.
Guidance
A boxandwhisker plot is another type of graph used to display data. It shows how the data are dispersed around a median, but it does not show specific values in the data. It does not show a distribution in as much detail as does a stemandleaf plot or a histogram.
A boxandwhisker plot is a graph based upon medians. It shows the minimum value, the lower median, the median, the upper median, and the maximum value of a data set. It is also known as a box plot.
This type of graph is often used when the number of data values is large or when two or more data sets are being compared.
Example A
You have a summer job working at Paddy’s Pond. Your job is to measure as many salmon as possible and record the results. Here are the lengths (in inches) of the first 15 fish you found: 13, 14, 6, 9, 10, 21, 17, 15, 15, 7, 10, 13, 13, 8, 11
Create a boxandwhisker plot.
Solution:
Since a boxandwhisker plot is based on medians, the first step is to organize the data in order from smallest to largest.
\begin{align*}6, \ 7, \ 8, \ 9, \ 10, \ 10, \ 11, \ \boxed{13}, \ 13, \ 13, \ 14, \ 15, \ 15, \ 17, \ 21\end{align*}
Step 1: Find the median: @$\begin{align*}median=13\end{align*}@$.
Step 2: Find the lower median.
The lower median is the median of the lower half of the data. It is also called the lower quartile or @$\begin{align*}Q_1\end{align*}@$.
@$$\begin{align*}6, \ 7, \ 8, \ & \boxed{9}, \ 10, \ 10, \ 11\\ Q_1& =9\end{align*}@$$
Step 3: Find the upper median.
The upper median is the median of the upper half of the data. It is also called the upper quartile or @$\begin{align*}Q_2\end{align*}@$.
@$$\begin{align*}13, \ 13, \ 14, \ \boxed{15}, \ 15, \ 17, \ 21\end{align*}@$$
@$$\begin{align*}Q_3=15\end{align*}@$$
Step 4: Draw the box plot. The numbers needed to construct a boxandwhisker plot are called the fivenumber summary.
The fivenumber summary are: the minimum value, @$\begin{align*}Q_1\end{align*}@$, the median, @$\begin{align*}Q_2\end{align*}@$, and the maximum value.
@$$\begin{align*}Minimum=6; \ Q_1=9;\ median=13; \ Q_3=15; \ maximum=21\end{align*}@$$
The three medians divide the data into four equal parts. In other words:
 Onequarter of the data values are located between 6 and 9.
 Onequarter of the data values are located between 9 and 13.
 Onequarter of the data values are located between 13 and 15.
 Onequarter of the data values are located between 15 and 21.
From its whiskers, any outliers (unusual data values that can be either low or high) can be easily seen on a boxandwhisker plot. An outlier would create a whisker that would be very long.
Each whisker contains 25% of the data and the remaining 50% of the data is contained within the box. It is easy to see the range of the values as well as how these values are distributed around the middle value. The smaller the box, the more consistent the data values are with the median of the data.
Example B
After one month of growing, the heights of 24 parsley seed plants were measured and recorded. The measurements (in inches) are given here: 6, 22, 11, 25, 16, 26, 28, 37, 37, 38, 33, 40, 34, 39, 23, 11, 48, 49, 8, 26, 18, 17, 27, 14.
Construct a boxandwhisker plot to represent the data.
Solution:
To begin, organize your data in ascending order. There is an even number of data values so the median will be the mean of the two middle values. @$\begin{align*}Med=\frac{26+26}{2}=26\end{align*}@$. The median of the lower quartile is the number between the 6th and 7th positions, which is the average of 16 and 17, or 16.5. The median of the upper quartile is also the number between the 6th and 7th positions, which is the average of 37 and 37, or 37. The smallest number is 6, and the largest number is 49.
Creating BoxandWhisker Plots Using a Graphing Calculator
The TI83 can also be used to create a boxandwhisker plot. The fivenumber summary values can be determined by using the trace function of the calculator.
Example C
'Make a histogram of the data from the previous example on your calculator.
Solution:
Enter the data into @$\begin{align*}[L_1]\end{align*}@$.
Change the [STATPLOT] to a box plot instead of a histogram.
Boxandwhisker plots are useful when comparing multiple sets of data. The graphs are plotted, one above the other, to visualize the median comparisons.
Video Review
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Guided Practice
Using the data from the previous Concept, determine whether the additive improved the gas mileage.
540  550  555  570  570 

580  585  587  588  590 
591  610  615  640  660 
500  589  618  619  629 

633  635  637  638  639 
659  664  689  694  709 
Solution:
Regular Gasoline  Premium Gasoline  

Smallest #  540  500 
@$\begin{align*}Q_1\end{align*}@$  570  619 
Median  587  637 
@$\begin{align*}Q_3\end{align*}@$  610  664 
Largest #  660  709 
From the above boxandwhisker plots, where the blue one represents the regular gasoline and the yellow one the premium gasoline, it is safe to say that the additive in the premium gasoline definitely increases the mileage. However, the value of 500 seems to be an outlier.
Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: BoxandWhisker Plots (13:14)
 Describe a fivenumber summary.
 What is the purpose of a boxandwhisker plot? When it is useful?
 What are some disadvantages to representing data with a boxandwhisker plot?
 The following is the data that represents the amount of money that males spent on prom night. @$$\begin{align*}&25 && 60 && 120 && 64 && 65 && 28 && 110 && 60\\ &70 && 34 && 35 && 70 && 58 && 100 && 55 && 95\\ &55 && 95 && 93 && 50 && 75 && 35 && 40 && 75\\ &90 && 40 && 50 && 80 && 85 && 50 && 80 && 47\\ &50 && 80 && 90 && 42 && 49 && 84 && 35 && 70\end{align*}@$$ Construct a boxandwhisker graph to represent the data.
 Forty students took a college algebra entrance test and the results are summarized in the boxandwhisker plot below. How many students would be allowed to enroll in the class if the pass mark were set at:
 65 %
 60 %
 Harika is rolling three dice and adding the scores together. She records the total score for 50 rolls, and the scores she gets are shown below. Display the data in a boxandwhisker plot, and find both the range and the interquartile range. 9, 10, 12, 13, 10, 14, 8, 10, 12, 6, 8, 11, 12, 12, 9, 11, 10, 15, 10, 8, 8, 12, 10, 14, 10, 9, 7, 5, 11, 15, 8, 9, 17, 12, 12, 13, 7, 14, 6, 17, 11, 15, 10, 13, 9, 7, 12, 13, 10, 12
 The boxandwhisker plots below represent the times taken by a school class to complete a 150yard obstacle course. The times have been separated into boys and girls. The boys and the girls both think that they did best. Determine the fivenumber summary for both the boys and the girls and give a convincing argument for each of them.
 Draw a boxandwhisker plot for the following unordered data. 49, 57, 53, 54, 49, 67, 51, 57, 56, 59, 57, 50, 49, 52, 53, 50, 58
 A simulation of a large number of runs of rolling three dice and adding the numbers results in the following fivenumber summary: 3, 8, 10.5, 13, 18. Make a boxandwhisker plot for the data.
 The boxandwhisker plots below represent the percentage of people living below the poverty line by county in both Texas and California. Determine the fivenumber summary for each state, and comment on the spread of each distribution.
 The fivenumber summary for the average daily temperature in Atlantic City, NJ (given in Fahrenheit) is 31, 39, 52, 68, 76. Draw the boxandwhisker plot for this data and use it to determine which of the following would be considered an outlier if it were included in the data.
 January’s recordhigh temperature of @$\begin{align*}78^\circ\end{align*}@$
 January’s recordlow temperature of @$\begin{align*}8^\circ\end{align*}@$
 April’s recordhigh temperature of @$\begin{align*}94^\circ\end{align*}@$
 The alltime record high of @$\begin{align*}106^\circ\end{align*}@$
 In 1887, Albert Michelson and Edward Morley conducted an experiment to determine the speed of light. The data for the first ten runs (five results in each run) is given below. Each value represents how many kilometers per second over 299,000 km/sec were measured. Create a boxandwhisker plot of the data. Be sure to identify outliers and plot them as such. 900, 840, 880, 880, 800, 860, 720, 720, 620, 860, 970, 950, 890, 810, 810, 820, 800, 770, 850, 740, 900, 1070, 930, 850, 950, 980, 980, 880, 960, 940, 960, 940, 880, 800, 850, 880, 760, 740, 750, 760, 890, 840, 780, 810, 760, 810, 790, 810, 820, 850
 Using the following boxandwhisker plot, list three pieces of information you can determine from the graph.
 In a recent survey done at a high school cafeteria, a random selection of males and females were asked how much money they spent each month on school lunches. The following boxandwhisker plots compare the responses of males to those of females. The lower one is the response by males.
 How much money did the middle 50% of each gender spend on school lunches each month?
 What is the significance of the value of $42 for females and $46 for males?
 What conclusions can be drawn from the above plots? Explain.

Multiple Choice. The following boxandwhisker plot shows final grades last semester. How would you best describe a typical grade in that course?
 Students typically got between 82 and 88.
 Students typically got between 41 and 82.
 Students typically got around 62.
 Students typically got between 58 and 82.
Mixed Review
 Find the mean, median, mode, and range for the following salaries in an office building: 63,450; 45,502; 63,450; 51,769; 63,450; 35,120; 45,502; 63,450; 31,100; 42,216; 49,108; 63,450; 37,904
 Graph @$\begin{align*}g(x)=2\sqrt{x1}3\end{align*}@$.
 Translate into an algebraic sentence: The square root of a number plus six is less than 18.
 Solve for @$\begin{align*}y\end{align*}@$: @$\begin{align*}6(y11)+9=\frac{1}{3} (27+3y)16\end{align*}@$.
 A fundraiser is selling two types of items: pizzas and cookie dough. The club earns $5 for each pizza sold and $4 for each container of cookie dough. They want to earn more than $550.
 Write this situation as an inequality.
 Give four combinations that will make this sentence true.
 Find the equation for a line parallel to @$\begin{align*}x+2y=10\end{align*}@$ containing the point (2, 1).