11.7: Appropriate Data Displays
Introduction
A Famous Woman
Kelly wants to learn about the famous women who have raced in the Iditarod.
“There is a lot of talk about the men, but there must have been some famous women,” she said in class one day.
“Actually, that is a good point Kelly,” Mr. Hawkins said. “There is one famous woman in particular. Would you be willing to do some research and present some information on her?”
“I would,” Kelly said smiling.
“I’ll help,” Tabitha said from across the room.
“Alright, good, and the rest of you can continue on your projects.”
Kelly and Tabitha got right to work. After looking up some information on the computer and in a few books on the Iditarod, they discovered that this famous woman was Susan Butcher. She was an amazing woman who won the Iditarod in four out of five years.
“Look, she even broke her own record,” Tabitha said.
“Yes, but she died in 2006 from leukemia. That is pretty sad.”
“It is,” Kelly said with a pause. “But she was very committed to her dogs and her racing. She even started a training facility. Her winning times seem slow compared with today, but they were very impressive then.”
“We can include that in our report. Let’s write down her finish times and make a graph to show them,” Tabitha suggested.
“That’s a great idea,” Kelly said smiling.
On a piece of paper the girls wrote down the following statistics.
1986 – 11 days 15 hours
1987 – 11 days 2 hours
1988 – 11 days 11 hours
1990 – 11 days 1 hour
“How are we going to display this? Which graph makes the most sense?” Kelly asked looking at Tabitha.
The girls are a bit stumped as to which visual display to create. To answer this question, you will have to think back over all of the types of data displays that you have learned about in this chapter.
Pay attention and by the end of the lesson you will know how to choose one that will work.
What You Will Learn
By the end of this lesson you will be able to complete the following:
- Associate given conclusions about a set of data with given bar graphs, line graphs, stem-and-leaf plots, box-and-whisker plots and histograms
- Select among bar graphs, line graphs, stem-and-leaf plots, box-and-whisker plots and histograms for appropriate displays of given data.
- Collect, organize and analyze real-world data using selected displays.
Teaching Time
I. Associate Given Conclusions about a Set of Data with Different Displays of Data
When we look at different graphs, we can make conclusions based on the data that we see. Sometimes, you will be given a graph and a set of conclusions and you will need to analyze the data to choose the correct conclusion. This can be a bit tricky.
Keep the following points in mind:
- What information does the data represent?
- Is the graph misleading or is the information accurate?
- What is being recorded or compared?
Let’s look at selecting the correct conclusion based on different data displays.
Example
Use the information on the graph to answer the questions below.
The graph shows that between 1950 and 2000, the population increased approximately:
a. 5 billion
b. 3 billion
c. 3.5 billion
d. 4 billion
The answer is “c.” In the year 2000, the population was approximately 6.1 billion. In the year 1950, the population was approximately 2.5 billion. The difference between 6.1 and 2.5 billion is 3.6 billion.
The smallest increase in population was between the years:
a. 1750 – 1800
b. 1800 – 1850
c. 1850 – 1900
d. 1900 – 1050
The answer is “c.” During 1850 and 1900 the population only increased from 1.5 billion to 1.6 billion.
Example
The box-and-whisker plot depicts the number of minutes ten people exercise daily. Use the information on the box-and-whisker plot to answer the questions below.
The median amount of time spent exercising is:
a. 10 minutes
b. 30 minutes
c. 70 minutes
d. 74 minutes
The answer is “b.” 30 is the median of the set of data. 10 is the median of the lower quartile and 70 is the median of the upper quartile.
What conclusion can be made about the number of minutes exercised each day?
a. The majority of people exercise between ten and thirty minutes each day.
b. About \begin{align*}\frac{2}{3}\end{align*} of the group exercise between ten and thirty minutes each day.
c. The majority of the group exercise between thirty and seventy minutes each day.
d. The greatest amount of time spent exercising by a single individual is seventy-five minutes.
e. None of the above.
The answer is “d.” You can see that the farthest point out at the right hand side of the plot is 75 minutes.
Example
The bar graph below depicts the average price of a gallon of gasoline for five states over a three week period. Use the information from the graph to answer the questions below.
Which state had the greatest decrease in price per gallon of gasoline?
a. Missouri
b. Washington
c. Florida
d. California
The answer is “d.” There was a $0.70 decrease in price per gallon over the three week period in California.
In which state did the price per gallon decrease by $0.45 during the first two weeks?
a. Missouri
b. Hawaii
c. Washington
d. Florida
The answer is “a.” The price decreased from $3.00 to $2.55 in Missouri.
What conclusion can be drawn about the price of gasoline in all five states?
a. Hawaii will continue to have highest cost of gasoline.
b. The prices of a gallon of gasoline will start to level off in week four.
c. The price of a gallon of gasoline will continue to decline.
d. Both “a” and “c” are correct.
The answer is “d.” Looking at the trends on the graph, Hawaii has had the highest price of gasoline all three weeks. Therefore, it is likely that Hawaii will have the highest price of gasoline in week four. The price of a gallon of gasoline has decreased all three weeks in every state. Therefore it is likely that the price will continue to decrease in week four.
Example
The histogram was created to depict the hourly wages of one hundred employees at a department store. Use the information depicted on the graph to answer the questions below.
Identify the difference between the number of people who earn the lowest wages and the highest wages.
a. 17
b. 27
c. 20
d. 7
The answer is “b.” Thirty employees earn between $6.50 and $8.50, the lowest wages. Three employees earn between $16.51 and $18.50, the highest wages. The difference between thirty and three is twenty-seven.
What conclusion can be drawn, on average, about the hourly wages at the department store?
a. As the hourly wage increases, the number of employees increases.
b. The majority of the employees make between $16.51 and $18.00 per hour.
c. As the hourly wage increases, the number of employees decreases.
The answer is “c.” The majority of the employees are earning lower hourly wages. Only a few employees are earning the top wages.
Example
The stem-and-leaf plot was created to compare the scores of two golf teams composed of ten players each. Use the information on the stem-and-leaf plot to answer the questions below.
Leaf: team one | Stem | Leaf: team two |
---|---|---|
9 8 7 | 6 | 4 8 |
7 | 7 | 3 7 |
9 5 2 | 8 | 4 4 8 |
7 5 | 9 | 6 9 |
2 | 10 | 0 |
Key: \begin{align*}7 \big | 6 = 67\end{align*} | Key: \begin{align*}10 \big | 0 = 100\end{align*} |
Which team has a greater range in scores?
a. team one
b. team two
The answer is team two. The range in scores for team two is thirty-six. The range in scores for team one is thirty-five.
Which statement about the data on the stem-and-leaf plot is true?
a. The median score for team one is greater than the median score for team two.
b. The median score for team two is greater than the median score for team one.
c. The mode for both team one and team two is eighty-four.
The answer is “b.” The median score for team one is 83.5. The median score for team two is 84.
II. Select the Appropriate Display of Given Data from Different Data Displays
Think back through all of the different data displays that you have learned about. Different data displays are used for different reasons. Let’s take a look at some of those reasons.
- Bar graphs best depict unconnected values, where as a line graph depicts a continuous trend among the data.
- A stem-and-leaf plot organizes the data by showing the data values in order. A stem-and-leaf plot is useful in determining the mean, median, mode, and range for a set of data.
- The distribution of data items is depicted on a box-and-whisker plot in which the data is divided into four quartiles.
- A histogram shows the frequency of data on a graph.
In the following examples, you will learn to choose the most appropriate graph to display a given set of data.
Example
The data table below depicts the amount of money Mary had in her savings account each month over the course of one year. Choose the most appropriate display for the data.
Month | Deposit |
---|---|
January | $40.00 |
February | $75.00 |
March | $145.00 |
April | $175.00 |
May | $178.00 |
June | $237.00 |
July | $240.00 |
August | $250.00 |
September | $260.00 |
October | $270.00 |
November | $275.00 |
December | $280.00 |
Since the data table depicts data values over time, a line graph would most appropriately display the data.
Example
The data listed below depicts the ages of students who participated in the National Spelling Bee. Choose the most appropriate display for the data.
14, 12, 13, 14, 11, 13, 12, 11, 12, 13, 14, 10, 10, 14, 13
Since the data values are similar in range, create a histogram to depict the frequency of the ages of students who participated in the spelling bee. Recall that to create a histogram, first arrange the data on a frequency table.
Ages | Tally | Frequency |
---|---|---|
10 – 11 | I I I I | 4 |
12 – 13 | I I I I I I I | 7 |
14 – 15 | I I I I | 4 |
Example
The data values listed below depict two periods of students’ scores on a history exam (out of 100). The teacher grading the exams would like to create a display that will compare the two sets of data and would allow him to eventually determine the overall mean, median, mode, and range of the test scores. What is the most appropriate display for this set of data?
\begin{align*}& \text{Per.1} \qquad 87 \quad 92 \quad 81 \quad 79 \quad 95 \quad 83 \quad 75 \quad 92 \quad 88 \quad 77 \quad 65 \quad 69 \quad 80 \quad 91 \quad 99\\ & \text{Per.2} \qquad 91 \quad 97 \quad 98 \quad 85 \quad 81 \quad 79 \quad 70 \quad 62 \quad 82 \quad 73 \quad 69 \quad 99 \quad 78 \quad 74 \quad 85\end{align*}
A stem-and-leaf plot will allow for the best comparison of the two sets of data. A stem-and-leaf plot will also arrange the data values for determining the mean, median, mode, and range. Recall that the first step in creating a stem-and-leaf plot is to arrange the data in order.
\begin{align*}& \text{Period 1:} \qquad 65, \quad 69, \quad 75, \quad 77, \quad 79, \quad 80, \quad 81, \quad 83, \quad 87, \quad 88, \quad 91, \quad 92, \quad 92, \quad 95, \quad 99\\ & \text{Period 2:} \qquad 62, \quad 69, \quad 70, \quad 73, \quad 74, \quad 78, \quad 79, \quad 81, \quad 82, \quad 85, \quad 85, \quad 91, \quad 97, \quad 98, \quad 99\end{align*}
Leaf (Per. 1) | Stem | Leaf (Per. 2) |
---|---|---|
9 5 | 6 | 2 9 |
9 7 5 | 7 | 0 3 4 8 9 |
8 7 3 1 0 | 8 | 1 2 5 5 |
9 5 2 2 1 | 9 | 1 7 8 9 |
Key: \begin{align*}1 \big | 9 = 91\end{align*} | Key: \begin{align*}9 \big | 1 = 91\end{align*} |
III. Collect, Organize and Analyze Real-World Data Using Selected Displays
You can use all you have learned about displays to collect, organize, and analyze real-world data. In the following examples, you will be asked to create multiple displays to depict the same set of data.
Example
The table below depicts the mean temperature (measured in Kelvin) of each planet in our solar system. Use what you have learned in the past lessons to display the data from the table on a bar and line graph, a stem-and-leaf plot, a box-and-whisker plot, and a histogram.
Planet | Mean Temperature (Kelvin) |
---|---|
Mercury | 452 |
Venus | 726 |
Earth | 305 |
Mars | 285 |
Jupiter | 120 |
Saturn | 59 |
Uranus | 48 |
Neptune | 37 |
Looking at the decreasing bars and lines on the graphs, you can conclude that a planet’s location affects its temperature. Planets closest to the sun have a higher temperature. Planets furthest from the sun have the lowest temperatures. It is evident that Venus has the highest surface temperature. Neptune has the coolest surface temperature.
Stem-and-Leaf Plot
37, 48, 59, 120, 285, 305, 452, 726
Stem | Leaf |
---|---|
3 | 7 |
4 | 8 |
5 | 5 |
12 | 0 |
28 | 5 |
30 | 5 |
45 | 2 |
72 | 6 |
Looking at the stem-and-leaf plot, it can be determined that none of the data share a stem. Therefore, there is no mode for this set of data.
You can see that the coolest temperature is 37 K and the warmest temperature is 726 K. The difference between the two extremes (the range) is 689 K.
The data values in the middle are 120 and 285. Therefore, the median planet temperature is 202.5 K.
\begin{align*}120 + 285 &= 405\\ 405 \div 8 &= 202.5\end{align*}
The mean planet temperature is 254 K.
\begin{align*}37 + 48 + 59 + 120 + 285 + 305 + 452 + 726 &= 2,032\\ 2,032 \div 8 &= 254\end{align*}
We can create a histogram of the data too. Look at the frequency table and the histogram created below.
Temperature | Tally | Frequency |
---|---|---|
0 – 200 K | I I I I | 4 |
201 – 400 K | I I I | 3 |
401 – 600 K | 0 | |
601 – 800 K | I | 1 |
Now it’s time to draw some conclusions based on our data displays.
Looking at the histogram, it is apparent that most temperatures fall between zero and two hundred Kelvin. Three planets have surface temperatures that fall between two hundred one Kelvin and four hundred Kelvin. None of the planets have a surface temperature between four hundred one and six hundred Kelvin. One planet has a temperature between six hundred one and eight hundred Kelvin.
By looking at the same data in different ways, we become very familiar with the data. One visual way may make more sense to you than another. No matter which ones you create, you can use visual displays of data to answer questions and draw conclusions.
Real Life Example Completed
A Famous Woman
Here is the original problem once again. Reread the problem and underline any important information.
Kelly wants to learn about the famous women who have raced in the Iditarod.
“There is a lot of talk about the men, but there must have been some famous women,” she said in class one day.
“Actually, that is a good point, Kelly,” Mr. Hawkins said. “There is one famous woman in particular. Would you be willing to do some research and present some information on her?”
“I would,” Kelly said smiling.
“I’ll help,” Tabitha said from across the room.
“Alright, good, and the rest of you can continue on your projects.”
Kelly and Tabitha got right to work. After looking up some information on the computer and in a few books on the Iditarod, they discovered that this famous woman was Susan Butcher. She was an amazing woman who won the Iditarod in four out of five years.
“Look, she even broke her own record,” Tabitha said.
“Yes, but she died in 2006 from leukemia. That is pretty sad.”
“It is,” Kelly said with a pause. “But she was very committed to her dogs and her racing. She even started a training facility. Her winning times seem slow compared with today, but they were very impressive then.”
“We can include that in our report. Let’s write down her finish times and make a graph to show them,” Tabitha suggested.
“That’s a great idea,” Kelly said smiling.
On a piece of paper the girls wrote down the following statistics.
1986 – 11 days 15 hours
1987 – 11 days 2 hours
1988 – 11 days 11 hours
1990 – 11 days 1 hour
“How are we going to display this? Which graph makes the most sense?” Kelly asked looking at Tabitha.
“Let’s create a line graph. That way we can show how her times changed over the course of the four races that she won.” Tabitha suggested.
“Great. The \begin{align*}x\end{align*} axis can be the years that she won.”
“Yes, and the \begin{align*}y\end{align*} axis can be the times. We can create 15 lines-each one has 11 days as a starter and then we add the number of hours from 1 – 15 with the days.”
Here is the graph that the girls created.
If you look at the graph, you will see that the second time and the last time that Susan Butcher raced and won the Iditarod were her best times!! She is definitely a hero who never stopped believing or backed down from a challenge!!
You can learn more about Susan Butcher at:
http://en.wikipedia.org/wiki/Susan_Butcher
http://www.achievement.org/autodoc/page/but0bio-1
http://www.galenfrysinger.com/iditarod_alaska.htm
Time to Practice
Directions: The graph below depicts the quarterly sales for two competing computer companies. Use the information on the graph to answer the question below.
1. True or false. Each company has increased its computer sales each quarter.
2. True or false. One company has had a more significant increase in sales than the other.
3. If number two is true, which company has had a more significant increase in sales?
4. What were the average sales for Computer Company 1 in quarter 3?
5. What were the average sales for Computer Company 2 in quarter 4?
6. Which company did not show much of a change in profits from quarter 3 to quarter 4?
7. What were those average earnings for the last two quarters?
The data table below depicts the final score for ten football games played this past season.
\begin{align*}48 \quad 56 \quad 42 \quad 59 \quad 62 \quad 45 \quad 36 \quad 58 \quad 49 \quad 50\end{align*}
8. What is the most appropriate display for the data?
9. Create a display of that data.
The data below depicts the number of hours that 10 students in seventh grade spent working on a science project.
\begin{align*}& \# 1 \quad \# 2 \quad \# 3 \quad \# 4 \quad \# 5 \quad \# 6 \quad \# 7 \quad \# 8 \quad \# 9 \quad \# 10\\ & 12 \quad \ 14 \quad \ 11 \quad \ 15 \quad \ 16 \quad \ 13 \ \quad 10 \quad \ \ 9 \quad \ \ 16 \quad \ \ 8\end{align*}
10. Create a bar graph of the data.
11. Create a line graph of the data.
The data table below depicts the amount of weekly allowance received by fifteen teenagers.
\begin{align*}25 \quad 15 \quad 20 \quad 30 \quad 10 \quad 5 \quad 8 \quad 12 \quad 18 \quad 23 \quad 27 \quad 15 \quad 18 \quad 25 \quad 10\end{align*}
12. Create a frequency table of the data.
13. Create a histogram of the data.
14. Create a box-and-whisker plot of the data.
The data table below depicts the magnitude of the last ten earthquakes that occurred in Sacramento, California.
\begin{align*}3.8 \quad 4.2 \quad 4.3 \quad 5.1 \quad 3.6 \quad 4.6 \quad 3.6 \quad 3.5 \quad 4.1 \quad 4.5\end{align*}
15. Create a stem-and-leaf plot of the data.
16. Create a histogram of the data.