# 8.5: Double Box-and-Whisker Plots

**At Grade**Created by: CK-12

**Double box-and-whisker plots** give you a quick visual comparison of 2 sets of data, as was also found with other double graph forms you learned about earlier in this chapter. The difference with double box-and-whisker plots is that you are also able to quickly visually compare the means, the medians, the maximums (upper range), and the minimums (lower range) of the data.

*Example 10*

Emma and Daniel are surveying the times it takes students to arrive at school from home. There are 2 main groups of commuters who were in the survey. There were those who drove their own cars to school, and there were those who took the school bus. Emma and Daniel collected the following data:

\begin{align*}&\text{Bus times (min)} \qquad 14 \quad 18 \quad 16 \quad 22 \quad 25 \quad 12 \quad 32 \quad 16 \quad 15 \quad 18\\ &\text{Car times (min)} \qquad 12 \quad 10 \quad 13 \quad 14 \quad 9 \quad \ \ 17 \quad 11 \quad 10 \quad 8 \quad \ 11\end{align*}

Draw a box-and-whisker plot for both sets of data on the same number line. Use the double box-and-whisker plots to compare the times it takes for students to arrive at school either by car or by bus.

*Solution:*

When plotted, the box-and-whisker plots look like the following:

Using the medians, 50% of the cars arrive at school in 11 minutes or less, whereas 50% of the students arrive by bus in 17 minutes or less. The range for the car times is \begin{align*}17 - 8 = 9\end{align*} minutes. For the bus times, the range is \begin{align*}32 - 12 = 20\end{align*} minutes. Since the range for the driving times is smaller, it means the times to arrive by car are less spread out. This would, therefore, mean that the times are more predictable and reliable.

*Example 11*

A new drug study was conducted by a drug company in Medical Town. In the study, 15 people were chosen at random to take Vitamin X for 2 months and then have their cholesterol levels checked. In addition, 15 different people were randomly chosen to take Vitamin Y for 2 months and then have their cholesterol levels checked. All 30 people had cholesterol levels between 8 and 10 before taking one of the vitamins. The drug company wanted to see which of the 2 vitamins had the greatest impact on lowering people’s cholesterol. The following data was collected:

\begin{align*}& \text{Vitamin X} \qquad 7.2 \quad 7.5 \quad 5.2 \quad 6.5 \quad 7.7 \quad 10 \quad 6.4 \quad 7.6 \quad 7.7 \quad 7.8 \quad 8.1 \quad 8.3 \quad 7.2 \quad 7.1 \quad 6.5\\ & \text{Vitamin Y} \qquad 4.8 \quad 4.4 \quad 4.5 \quad 5.1 \quad 6.5 \quad \ 8 \quad 3.1 \ \quad 4.6 \quad 5.2 \quad 6.1 \quad 5.5 \quad 4.2 \quad 4.5 \quad 5.9 \quad 5.2\end{align*}

Draw a box-and-whisker plot for both sets of data on the same number line. Use the double box-and-whisker plots to compare the 2 vitamins and provide a conclusion for the drug company.

*Solution:*

When plotted, the box-and-whisker plots look like the following:

Using the medians, 50% of the people in the study had cholesterol levels of 7.5 or lower after being on Vitamin X for 2 months. Also 50% of the people in the study had cholesterol levels of 5.1 or lower after being on Vitamin Y for 2 months. Knowing that the participants of the survey had cholesterol levels between 8 and 10 before beginning the study, it appears that Vitamin Y had a bigger impact on lowering the cholesterol levels. The range for the cholesterol levels for people taking Vitamin X was \begin{align*}10 - 5.2 = 4.8\end{align*} points, while the range for the cholesterol levels for people taking Vitamin Y was \begin{align*}8 - 3.1 = 4.9\end{align*} points. Therefore, the range is not useful in making any conclusions.

**Drawing Double Box-and-Whisker Plots Using TI Technology**

The above double box-and-whisker plots were drawn using a program called Autograph. You can also draw double box-and-whisker plots by hand using pencil and paper or by using your TI-84 calculator. Follow the key sequence below to draw double box-and-whisker plots.

After entering the data into L1 and L2, the next step is to graph the data by using STAT PLOT.

The resulting graph looks like the following:

You can then press \begin{align*}\boxed{\text{TRACE}}\end{align*} and find the five-number summary. The five-number summary is shown below for Vitamin X. By pressing the \begin{align*}\boxed{\blacktriangledown}\end{align*} button, you can get to the second box-and-whisker plot (for Vitamin Y) and collect the five-number summary for this box-and-whisker plot.

*Example 12*

2 campus bookstores are having a price war on the prices of their first-year math books. James, a first-year math major, is going into each store to try to find the cheapest books he can find. He looks at 5 randomly chosen first-year books for first-year math courses in each store to determine where he should buy the 5 textbooks he needs for his courses this coming year. He collects the following data:

\begin{align*}& \text{Bookstore A prices} (\$) \qquad 95 \quad \ 75 \quad 110 \quad 100 \quad 80\\ & \text{Bookstore B prices} (\$) \qquad 120 \quad 60 \quad 89 \quad \ 84 \quad 100\end{align*}

Draw a box-and-whisker plot for both sets of data on the same number line. Use the double box-and-whisker plots to compare the 2 bookstores' prices, and provide a conclusion for James as to where to buy his books for his first-year math courses.

*Solution:*

The box-and-whisker plots are plotted and look like the following:

Using the medians, 50% of the books at Bookstore A are likely to be in the price range of $95 or less, whereas at Bookstore B, 50% of the books are likely to be around $89 or less. At first glance, you would probably recommend to James that he go to Bookstore B. Let’s look at the ranges for the 2 bookstores to see the spread of data. For Bookstore A, the range is \begin{align*}\$110 - \$75 = \$35\end{align*}, and for Bookstore B, the range is \begin{align*}\$120 - \$60 = \$60\end{align*}. With the spread of the data much greater at Bookstore B than at Bookstore A, (i.e., the range for Bookstore B is greater than that for Bookstore A), to say that it would be cheaper to buy James’s books at Bookstore A would be more predictable and reliable. You would, therefore, suggest to James that he is probably better off going to Bookstore A.

**Points to Consider**

- What is the difference between categorical and numerical data, and how does this relate to qualitative and quantitative data?
- How is comparing double graphs (pie charts, broken-line graphs, box-and-whisker plots, etc.) useful when doing statistics?

**Vocabulary**

- Categorical data
- Data that are in categories and describe characteristics, or qualities, of a category.

- Double bar graphs
- 2 bar graphs that are graphed side-by-side.

- Double box-and-whisker plots
- 2 box-and-whisker plots that are plotted on the same number line.

- Double line graphs
- 2 line graphs that are graphed on the same coordinate grid. Double line graphs are often called parallel graphs.

- Qualitative data
- Descriptive data, or data that describes categories.

- Quantitative data
- Numerical data, or data that is in the form of numbers.

- Numerical data
- Data that involves measuring or counting a numerical value.

- Two-sided stem-and-leaf plots
- 2 stem-and-leaf plots that are plotted side-by-side. Two-sided stem-and-leaf plots are also called back-to-back stem-and-leaf plots.

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