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This lesson submitted by jmdavidson01

**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.

**Introduction**

Throughout this book, you have learned about variables. You have learned about random variables, discrete variables, continuous variables, numerical (or quantitative) variables, and categorical (or qualitative) variables. The various forms of graphical representations you have learned about in the previous lessons can be added to your learning of variables. The graphic below may help to summarize what you have learned.

Broken-line graphs, histograms, pie charts, stem-and-leaf plots, and box-and-whisker plots all represent useful (often very useful) tools in determining trends. Broken-line graphs, for example, allow you to show situations such as the distance traveled in specific time spans. Histograms use continuous grouped data to show the frequency trend in the data. Bar graphs are a little different from histograms in that they use grouped discrete data, as do stem-and-leaf plots. Bar graphs, as you know, have gaps between the columns, while histograms do not. Stem-and-leaf plots are excellent for giving you a quick visual representation of data. Used for only smaller sets of data, stem-and-leaf plots are a good example of representations of grouped discrete data. Box-and-whisker plots are a final visual way of representing grouped data that you have learned about in the previous lessons. In a box-and-whisker plot, you are able to find the five-number summary to describe the spread of the data.

## Review

In previous lessons, you learned about discrete and continuous data and were introduced to categorical and numerical forms of displaying data. In this final lesson, you will learn how to display discrete and continuous data in both categorical and numerical displays, but in a way that allows you to compare sets of data.

Remember that discrete data is represented by exact values that result from counting, as in the number of people in the households in your neighborhood. Continuous data is represented by a range of data that results from measuring. For example, taking the average temperatures for each month during a year is an example of continuous data. Also remember from an earlier lesson how you distinguished between these types of data when you graphed them.

The graph above shows discrete data. Remember that you know this because the data points are not joined. The graph below represents the average temperatures during the months in 2009. This data is continuous. You can easily tell this by looking at the graph and seeing the data points connected together.

2 newer terms used are the categorical and numerical data forms. **Categorical data** forms are just what the term suggests. These are data forms that are in categories and describe characteristics, or qualities, of a category. These data forms are more **qualitative data** and, therefore, are less numerical than they are descriptive. Graphs such as pie charts and bar graphs show descriptive data, or qualitative data. Below are 2 examples of categorical data represented in these types of graphs:

**Numerical data** is **quantitative data**. Numerical data involves measuring or counting a numerical value. Therefore, when you talk about discrete and continuous data, you are talking about numerical data. Line graphs, frequency polygons, histograms, and stem-and-leaf plots all involve numerical data, or quantitative data, as is shown below:

Box-and-whisker plots are also considered numerical displays of data, as they are based on quantitative data (the mean and median), as well as the maximum (upper) and minimum (lower) values found in the data. The figure below is a typical box-and-whisker plot:

You will spend the remainder of the lesson learning about how to compare sets of categorical and numerical data.

## Double Line Graphs

Remember a line graph, by definition, can be the result of a linear function or can simply be a graph of plotted points, where the points are joined together by line segments. Line graphs that are linear functions are normally in the form , where is the slope and is the -intercept. The graph below is an example of a linear equation with a slope of and a -intercept of :

The second type of line graph is known as a broken-line graph. In a broken-line graph, the slope represents the rate of change, and the -intercept is actually the starting point. The graph below is a broken-line graph:

When the measurements began, the number of sales (the -intercept) was 645. The graph shows a significant increase in the number of sales from weeks 5 through 10 and a significant reduction in the number of sales from weeks 16 through 20.

In this lesson, you will be learning about comparing 2 line graphs that each contain data points. In statistics, when line graphs are in the form of broken-line graphs, they are of more use. Linear functions (i.e., ) are more for algebraic reasoning. **Double line graphs**, as with any double graphs, are often called parallel graphs, due to the fact that they allow for the quick comparison of 2 sets of data. In this lesson , you will see them referred to only as double graphs.

*Example 1*

Christopher and Jack are each opening businesses in their neighborhoods for the summer. Christopher is going to sell lemonade for per glass. Jack is going to sell popsicles for $1.00 each. The following graph represents the sales for each boy for the 8 weeks in the summer.

a. Explain the slopes of the line segments for Christopher’s graph.

b. Explain the slopes of the line segments for Jack’s graph.

c. Are there any negative slopes? What does this mean?

d. Where is the highest point on Christopher’s graph? What does this tell you?

e. Where is the highest point on Jack’s graph? What does this tell you?

f. Can you provide some reasons for the shape of Jack’s graph?

g. Can you provide some reasons for the shape of Christopher’s graph?

*Solution:*

a. The slope of the line segments for Christopher’s graph (red) are positive for the first 5 weeks, meaning he was increasing his sales each week. This is also true from weeks 6 to 7. From weeks 5 to 6 and weeks 7 to 8, the slopes were decreasing, meaning there was a decrease in sales.

b. The same trend that is seen for Christopher’s graph (red) is also seen for Jack’s graph (blue). The slope of the line segments for Jack’s graph (blue) are positive for the first 5 weeks, meaning he was increasing his sales each week. This is also true from weeks 6 to 7. From weeks 5 to 6 and weeks 7 to 8, the slopes were decreasing, meaning there was a decrease in sales.

c. Negative sales from weeks 5 to 6 and weeks 7 to 8 (for both boys) mean there was a decrease in sales during these 2-week periods.

d. The highest point on Christopher’s graph occurred in week 7, when he sold 65 glasses of lemonade. This must have been a very good weeknice and hot!

e. The highest point on Jack’s graph occurred in week 5, when he sold 74 popsicles. This must have been a very hot week as well!

f. Popsicles are a great food when you are warm and want a light snack. You can see how as the summer became hotter, the sales increased. Even in the weeks where it looks like Jack had a decrease in sales (maybe a few rainy days occurred, or it was not as hot), his sales still remained at a good level.

g. Lemonade is a very refreshing drink when you are warm. You can see how as the summer became hotter, the sales increased. Even in the weeks where it looks like Christopher had a decrease in sales (maybe a few rainy days occurred, or it was not as hot), his sales still remained at a good level, just as Jack's sales did.

*Example 2*

Thomas and Abby are training for the cross country meet at their school. Both students are in the 100 yard dash. The coach asks them to race 500 yards and time each 100 yard interval. The following graph represents the times for both Thomas and Abby for each of the five 100 yard intervals.

a. Who won the race? How do you know?

b. Between what times did Thomas (blue) appear to slow down? How do you know?

c. Between what times was Abby (red) ahead of Thomas? How do you know?

d. At what time did Thomas pass Abby? How do you know?

*Solution:*

a. Thomas (blue) won the race, because he finished the 500 yards in the least amount of time.

b. Between 20 and 40 seconds, Thomas (blue) seems to slow down, because the slope of the graph is less steep.

c. Between 0 and 57 seconds, Abby (pink) is ahead of Thomas (blue). You can see this, because the pink line is above the blue line.

d. At 57 seconds, Thomas (blue) passes Abby (pink). From this point onward, the blue line is above the pink line, meaning Thomas is running faster 100 yard intervals.

*Example 3*

Brenda and Ervin are each planting corn in a section of garden in their back yard. Brenda says that they need to put fertilizer on the plants 3 to 5 times per week. Ervin contradicts Brenda, saying that they need to fertilize only 1 to 2 times per week. Each gardener plants his or her garden of corn and measures the heights of their plants. The graph for the growth of their corn is found below:

a. Who was right? How do you know?

b. Between what times did Brenda’s (pink) garden appear to grow more? How do you know?

c. Between what times were Ervin's (blue) heights ahead of Brenda’s? How do you know?

*Solution:*

a. Brenda (pink) is correct, because her plants grew more in the same amount of time.

b. Between 4 and 8 weeks, Brenda’s plants seemed to grow faster (taller) than Ervin’s plants. You can tell this, because the pink line is above the blue line after the 4-week mark.

c. From 0 and 4 weeks, Ervin’s plants seemed to grow faster (taller) than Brenda’s plants. You can tell this, because the blue line is above the pink line before the 4-week mark.

*Example 4*

Nicholas and Jordan went on holidays with their families. They decided to monitor the mileage they traveled by keeping track of the time and the distance they were on the road. The boys collected the following data:

\\

& \text{Time (hr)} \qquad \qquad \quad 1 \qquad 2 \qquad 3 \qquad 4 \qquad 5 \qquad 6\\

& \text{Distance (miles)} \quad \quad 60 \quad 110 \quad 175 \quad 235 \quad 280 \quad 320\\

\\

&\text{Jordan}\\

\\

& \text{Time (hr)} \qquad \qquad \quad 1 \quad \ 2 \qquad 3 \qquad 4 \qquad 5 \qquad 6 \\

& \text{Distance (miles)} \qquad 50 \quad 90 \quad 125 \quad 125 \quad 165 \quad 210" />

a. Draw a graph to show the trip for each boy.

b. What conclusions could you draw by looking at the graphs?

*Solution:*

a.

You can also use TI technology to graph this data. First, you need to enter in all of the data Nicholas and Jordan collected.

Now you need to graph the 2 sets of data.

The resulting graph looks like the following:

b. Looking at the speed of Nicholas’s family vehicle and the shape of the graph, it could be concluded that Nicholas’s family was traveling on the highway going toward their family vacation destination. The family did not stop and continued on at a pretty steady speed until they reached where they were going.

Jordan’s trip was more relaxed. The speed indicates they were probably not on a highway, but more on country-type roads, and that they were traveling through a scenic route. In fact, from hours 3 to 4, the family stopped for some reason (maybe lunch), and then they continued on their way.

## Two-Sided Stem-and-Leaf Plots

As you have learned in an earlier lesson, stem-and-leaf plots are an excellent tool for organizing data. Remember that stem-and-leaf plots are a visual representation of grouped discrete data, but they can also be referred to as a modal representation. This is because by looking at a stem-and-leaf plot, we can determine the mode by quick visual inspection. In the last lesson, you learned about single-sided stem-and-leaf plots. In this lesson, you will learn about **two-sided stem-and-leaf plots**, which are also often called back-to-back stem-and-leaf plots.

*Example 5*

The girls and boys in one of BDF High School's AP English classes are having a contest. They want to see which group can read the most number of books. Mrs. Stubbard, their English teacher, says that the class will tally the number of books each group has read, and the highest mode will be the winner. The following data was collected for the first semester of AP English:

& \text{Boys} \qquad 15 \quad 18 \quad 22 \quad 22 \quad 23 \quad 26 \quad 34 \quad 35 \quad 35 \quad 35 \quad 40 \quad 40 \quad 42 \quad 47 \quad 49 \quad 50 \quad 50 \quad 51" />

a. Draw a two-sided stem-and-leaf plot for the data.

b. Determine the mode for each group.

c. Help Mrs. Stubbard decide which group won the contest.

*Solution:*

a.

b. The mode for the girls is 23 books. It is the number in the girls column that appears most often. The mode for the boys is 35 books. It is the number in the boys column that appears most often.

c. Mrs. Stubbard should decide that the boys group has won the contest.

*Example 6*

Mrs. Cameron teaches AP Statistics at GHI High School. She recently wrote down the class marks for her current grade 12 class and compared it to the previous grade 12 class. The data can be found below. Construct a two-sided stem-and-leaf plot for the data and compare the distributions.

& 82 \quad 82 \quad 82 \quad 83 \quad 84 \quad 85 \quad 85 \quad 86 \quad 87 \quad 93 \quad 98 \quad 100\\

\text{2009 class} \qquad & 76 \quad 76 \quad 76 \quad 76 \quad 77 \quad 78 \quad 78 \quad 78 \quad 79 \quad 80 \quad 80 \quad 82 \quad 82 \quad 83 \quad 83 \quad 83 \quad 85 \\

& 85 \quad 88 \quad 91 \quad 95" />

*Solution:*

There is a wide variation in the marks for both years in Mrs. Cameron’s AP Statistics Class. In 2009, her class had marks anywhere from 76 to 95. In 2010, the class marks ranged from 70 to 100. The mode for the 2009 class was 76, but for the 2010 class, it was 74. It would seem that the 2009 class had, indeed, done slightly better than Mrs. Cameron’s current class.

*Example 7*

The following data was collected in a survey done by Connor and Scott for their statistics project. The data represents the ages of people who entered into a new hardware store within its first half hour of opening on its opening weekend. The M's in the data represent males, and the F's represent females.

& 26F \quad \ 29F \quad \ 29F \quad \ 31M \quad 33M \quad 35M \quad 35M \quad 35M\\

& 41F \quad \ 42F \quad \ 42M \quad 45M \quad 46F \quad \ 48F \quad \ 51M \quad 51M\\

& 55F \quad \ 56M \quad 58M \quad 59M \quad 60M \quad 60F \quad \ \ 61F \quad 65M\\

& 65M \quad 66M \quad 70M \quad 70M \quad 71M \quad 71M \quad \ 72M \quad 72F" />

Construct a back-to-back stem-and-leaf plot showing the ages of male customers and the ages of female customers. Compare the distributions.

**Solution:**

For the male customers, the ages ranged from 10 to 72. The ages for the male customers were spread out throughout this range, with the mode being age 35. In other words, for the males found to be at the store in the first half hour of opening day, there was no real age category where a concentration of males could be found.

For the female customers, the ages ranged from 15 to 72, but they were concentrated between 21 and 48. The mode for the ages of the female customers was 29 years of age.

## Double Bar Graphs

In lesson 7, you studied both histograms and bar graphs. Remember that histograms have measurements on the horizontal axis and frequencies on the vertical axis . A bar graph, on the other hand, displays categories on the horizontal axis and frequencies on the vertical axis. This means that bar graphs are more qualitative, and, therefore, display categorical data. The figure below shows 1 bar graph (on the top) and 1 histogram (on the bottom):

Let’s look at an example of **double bar graphs**.

*Example 8*

Kerry-Sue is surveying a random sample of students to determine which sports they would like to have set up at the end-of-year Safe Grad event. She collects the following data:

**Sports**

**Girls**

**Boys**Racquetball 6 3 Basketball 3 6 Volleyball 5 5 Swimming 7 8

Draw a double bar graph and help Kerry-Sue determine which 2 sports would be most equally-liked by both boys and girls at the end-of-year Safe Grad event.

*Solution:*

According to the double bar graph, volleyball and swimming seem to be almost equally liked by both the girls and the boys. Therefore, these 2 sports would be the ones Kerry-Sue should choose to set up at the end-of-year Safe Grad event.

*Example 9*

Mrs. Smith teaches both academic and advanced math. She has been teaching these 2 courses for the past 4 years. She decided she wanted to compare her grades to see how each class was doing over the past few years and see if she has improved her class instruction at all. Her data can be found below:

**Marks**

**Academic Math**

**Advanced Math**2007 61.3 74.7 2008 67.9 80.3 2009 50.9 86.8 2010 63.7 81.5

Draw a double bar graph and help Mrs. Smith determine if her class instruction has improved over the past 4 years.

*Solution:*

Both the academic and advanced math marks went up and down over the past 4 years. If Mrs. Smith looks at the difference between 2007 and 2010, she can see that there is an improvement in the final grades for her students. Although there are many factors that can affect these grades, she can say that the change in her instruction is making some difference in the results for her students. Other factors might have contributed to the huge decline in grades for the academic math students in 2009. You can make this conclusion for Mrs. Smith, as there was a marked improvement in her advanced math course. In 2009, it seemed her instruction methods were working well with the advanced students, but other factors were affecting the academic students.

## Double Box-and-Whisker Plots

**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 lesson. 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:

&\text{Car times (min)} \qquad 12 \quad 10 \quad 13 \quad 14 \quad 9 \quad \ \ 17 \quad 11 \quad 10 \quad 8 \quad \ 11" />

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 minutes. For the bus times, the range is 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:

& \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" />

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 points, while the range for the cholesterol levels for people taking Vitamin Y was 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 and find the five-number summary. The five-number summary is shown below for Vitamin X. By pressing the 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:

& \text{Bookstore B prices} (\$) \qquad 120 \quad 60 \quad 89 \quad \ 84 \quad 100" />

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 , and for Bookstore B, the range is . 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?

## Review Questions

Answer the following questions and show all work (including diagrams) to create a complete answer.

- In the table below, match the following types of graphs with the types of variables used to create the graphs.

**Type of Graph**

**Type of Variable**a. Histogram _____ discrete b. Stem-and-leaf plot _____ discrete c. Broken-line graph _____ discrete d. Bar graph _____ continuous e. Pie chart _____ continuous

- In the table below, match the following types of graphs with the types of variables used to create the graphs.

**Type of Graph**

**Type of Variable**a. Broken-line graph _____ qualitative b. Bar graph _____ numerical c. Pie chart _____ categorical d. Stem-and-leaf plot _____ quantitative e. Histogram _____ numerical

- Jack takes a pot of water at room temperature and puts it on the stove to boil , which takes about 5 minutes. He then takes a cup of this water, adds a package of hot chocolate, and mixes it up. He places the cup on the counter to cool for 10 minutes to before having his first sip. After 30 minutes, the hot chocolate is now at room
temperature.

Thomas is making chocolate chip cookies. He mixes all of the ingredients together at room temperature, which takes him about 5 minutes, and then places the cookies in the oven at for 8 minutes. After cooking, he takes them off the pan and places them on a cooling rack. After 15 minutes, the cookies are still warm (about ), but he samples them for taste. After 30 minutes, the cookies are at room temperature and ready to be served.

Draw a broken-line graph for each set of data. Label the graphs to show what is happening.

- Scott is asked to track his daily video game playing. He gets up at 7 A.M. and plays for 1 hour. He then eats his breakfast and gets ready for school. He runs to catch the bus at 8:25 A.M. On the bus ride (about 35 minutes), he plays his IPOD until arriving for school. He is not allowed games at school, so he waits for the bus ride home at 3:25 P.M. When he gets home, he does homework for 1 hour and plays games for 1 hour until dinner. There are no games in the
evening.

Michael gets up at 7:15 A.M., eats breakfast, and gets ready for school. It takes him 30 minutes to get ready. He then plays games until he goes to meet the bus with Scott. Michael is in Scott’s class, but he has a free period from 11:00 A.M. until 11:45 A.M., when he goes outside to play a game. He goes home and plays his 1 hour of games immediately, and he then works on his homework until dinner. He, like Scott, is not allowed to play games in the evening.

Draw a broken-line graph for each set of data. Label the graphs to show what is happening.

- The following graph shows the gasoline remaining in a car during a family trip east. Also found on the graph is the gasoline remaining in a truck traveling west to deliver goods. Describe what is happening for each graph. What other conclusions may you
draw?

- Mr. Dugas, the senior high physical education teacher, is doing fitness testing this week in gym class. After each test, students are required to take their pulse rate and record it on the chart in the front of the gym. At the end of the week, Mr. Dugas looks at the data in order to analyze it. The data is shown
below:

& \qquad \qquad 82 \quad 78 \quad 60 \quad 64 \quad 64 \quad 65 \quad 81 \quad 84 \quad 84 \quad 79 \quad 78 \quad 70\\

&\text{Boys} \qquad 76 \quad 88 \quad 87 \quad 86 \quad 85 \quad 70 \quad 76 \quad 70 \quad 70 \quad 79 \quad 80 \quad 82 \quad 82 \quad 82 \quad 83 \quad 84 \quad 85\\

& \qquad \qquad 85 \quad 78 \quad 81 \quad 85" />Construct a two-sided stem-and-leaf plot for the data and compare the distributions.

- Starbucks prides itself on its low line-up times in order to be served. A new coffee house in town has also boasted that it will have your order in your hands and have you on your way quicker than the competition. The following data was collected for the line-up times (in minutes) for both coffee
houses:

& \text{Just Us Coffee} \qquad 17 \quad 16 \quad 15 \quad 10 \quad 16 \quad 10 \quad 10 \quad 29 \quad 20 \quad 22 \quad 22 \quad 12 \quad 13 \quad 24 \quad 15" />

Construct a two-sided stem-and-leaf plot for the data. Determine the median and mode using the two-sided stem-and-leaf plot. What can you conclude from the distributions?

- The boys and girls basketball teams had their heights measured at practice. The following data was recorded for their heights
(in centimeters):

& \qquad \qquad 168 \quad 178 \quad 174 \quad 170 \quad 155 \quad 155 \quad 154 \quad 164 \quad 145 \quad 171 \quad 161\\

& \text{Boys} \qquad 168 \quad 170 \quad 162 \quad 153 \quad 176 \quad 167 \quad 158 \quad 180 \quad 181 \quad 176 \quad 172\\

& \qquad \qquad 168 \quad 167 \quad 165 \quad 159 \quad 185 \quad 184 \quad 173 \quad 177 \quad 167 \quad 169 \quad 177" />Construct a two-sided stem-and-leaf plot for the data. Determine the median and mode using the two-sided stem-and-leaf plot. What can you conclude from the distributions?

- The grade 12 biology class did a survey to see what color eyes their classmates had and if there was a connection between eye color and sex. The following data was recorded:

**Eye color**

**Males**

**Females**Blue 5 5 Green 6 8 Brown 3 4 Hazel 4 3

Draw a double bar graph to represent the data, and draw any conclusions that you can from the resulting chart.

- Robbie is in charge of the student organization for new food selections in the cafeteria. He designed a survey to determine if 4 new food options would be good to put on the menu. The results are shown below:

**Food Option**

**Yes votes**

**No votes**Fish burgers 10 5 Vegetarian pizza 7 18 Brown rice 23 9 Carrot soup 20 20

Draw a double bar graph to represent the data, and draw any conclusions that you can from the resulting chart.

- The guidance counselor at USA High School wanted to know what future plans the graduating class had. She took a survey to determine the intended plans for both boys and girls in the school’s graduating class. The following data was recorded:

**Future Plans**

**Boys**

**Girls**University 35 40 College 27 22 Military 23 9 Employment 10 5 Other/unsure 5 10

Draw a double bar graph to represent the data, and draw any conclusions that you can from the resulting chart.

- International Baccalaureate has 2 levels of courses, which are standard level (SL) and higher level (HL). Students say that study times are the same for both the standard level exams and the higher level exams. The following data represents the results of a survey conducted to determine how many hours a random sample of students studied for their final
exams at each level:

& \text{SL Exams} \qquad \ 10 \quad 6 \quad \ 6 \quad \ \ 7 \quad \ 9 \ \quad 12 \quad \ 2 \quad 6 \quad 2 \quad 5 \quad 7 \quad 20 \quad 18 \quad 8 \quad \ 18" />

Draw a box-and-whisker plot for both sets of data on the same number line. Use the double box-and-whisker plots to determine the five-number summary for both sets of data. Compare the times students prepare for each level of exam.

- Students in the AP math class at BCU High School took their SATs for university entrance. The following scores were
obtained for the math and verbal sections:

& \text{Verbal} \qquad \ 499 \quad 509 \quad 524 \quad 530 \quad 550 \quad 499 \quad 545 \quad 560 \quad 579 \quad 524 \quad 478 \quad 487 \quad 482 \quad 570" />

Draw a box-and-whisker plot for both sets of data on the same number line. Use the double box-and-whisker plots to determine the five-number summary for both sets of data. Compare the data for the 2 sections of the SAT using the five-number summary data.

- The following box-and-whisker plots were drawn to analyze the data collected in a survey of scores for the doubles performances in the figure skating competitions at 2 Winter Olympic games. The box-and-whisker plot on the top represents the scores obtained at the 2010 winter games in Whistler, BC. The box-and-whisker plot on the bottom represents the scores obtained at the 2006 winter games in Torino, Italy.

Here is what the double box-and-whisker plots look like when created with a TI-84 calculator:

- Use the double box-and-whisker plots to determine the five-number summary for both sets of data. Compare the scores obtained at each of the Winter Olympic games.

## Lesson feedback

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