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CHAPTER
7
Statistics
In grades K–4, students begin to explore basic ideas of statistics by gathering data appropriate
to their grade level, organizing them in charts or graphs, and reading information from displays
of data. These concepts should be expanded in the middle grades. . . . Instruction in statistics
should focus on the active involvement of students in the entire process: formulating key questions; collecting and organizing data; representing the data using graphs, tables, frequency distributions, and summary statistics; analyzing the data; making conjectures; and communicating
information in a convincing way.†
Activity Set 7.1
COLLECTING AND GRAPHING DATA
PURPOSE
To collect and display data and to look for relationships or trends between two sets of data.
MATERIALS
A protractor and a tape measure from Material Card 20 or a ruler and string.
INTRODUCTION
Graphs provide a visual means of looking at sets of data and examining relationships and
trends in the data. There are many ways to display data and a decision on which type of display
to use often depends on the type of data you wish to display and on which display best illustrates the results.
Pie Graphs
Pie graphs, like this one displaying percentage of fruit juices in a fruit drink, show the relative
sizes of certain quantities by dividing a circle into sectors with the same proportions.
Fruit Juice Percentage Breakdown in Fruit Drink
11%
Pears
22%
Apricots
34%
Oranges
33%
Apples
†Curriculum
and Evaluation Standards for School Mathematics, p. 105.
201
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Statistics
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203
Collecting and Graphing Data
Bar Graphs
Bar graphs, such as the triple bar graph showing votes cast in presidential elections,† show data
classified into distinct categories with equal width bars representing frequencies.
Votes Cast for President by Major Political Party: 2000 to 2008
70
Millions of votes
60
50
40
30
20
10
0
2000
2004
Democratic
2008
Republican
Other major candidates
Line Plots
Line plots, such as the plot shown below displaying the number of books read by a class of
students, are often used to display numerical data that naturally falls into distinct numerical
categories.
Number of Books Read by Mrs. Jones’s Students in Last Year
Number of students
10
5 X
X
X
X
X
1
X
X
X
X
X
X
X
2
X
X
X
3
X
X X X
X
X X X X X
X X
X X X X X
X X
X X X X X
X X
X X X X X
X X
X X X X X
X X
X X X X X X
X X X
X X X X X X
X X X X
X X X X X X X X X X X
4 5 6 7 8 9 10 11 12 13 14 15+
Number of books read in last year
Stem-and-Leaf Plots and Histograms
The stem-and-leaf plot and the histogram on the next page both display a graphical representation
of the same set of homework scores. The histogram shows the total number of scores within the
noted interval (for example, there are 4 scores in the 20 to 29 range). The stem-and-leaf plot also
shows scores in the same intervals but retains the actual data (for example, in the interval 20 to
29 the four scores, with stem 2 for 20 and leaves 2, 7, 8, and 8 for the unit digits, are 22, 27, 28,
and 28).
†Statistical
Abstract of the United States (Washington, DC: Board of Census, 2009): Tables 385 and 386.
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Chapter 7
Statistics
Homework Scores
Stem
Leaf
1
9
2
2788
3
022447779
4
01335567889
5
00
Number of students
Homework Scores
12
11
10
9
8
7
6
5
4
3
2
1
0
10–19
20–29
30–39
40–49
Scores
50–59
Scatter Plots
Scatter plots provide a visual means of looking for relationships between two sets of data. In this
scatter plot, a middle school student collected the heights of the mothers and fathers of her classmates. By plotting a point that paired the heights of each set of parents she can look for a trend:
Do taller mothers tend to be married to taller fathers?
73
72
71
Heights of mothers
204
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70
69
68
67
66
65
64
63
62
61
60
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
Heights of fathers
Bar and Pie Graphs
1. As a class, gather data on the eye color of the students in the class. Organize the data in the
provided table.
Eye Color
Brown
Blue
Green
Hazel
Gray
Number of Students
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Collecting and Graphing Data
Number of students
*a. Sketch a single bar graph corresponding to this data set by first marking a scale on the
vertical axis, then marking the heights of the bars, and then sketching single bars of those
heights centered over the eye colors. The bar edges in each data category should not
touch, but each bar should be the same width.
Brown
Blue
Green
Eye colors
Hazel
Gray
*b. Sketch a pie graph corresponding to the class data set of eye color by using a protractor
to divide the circle into a set of pie slices of the appropriate size. To determine the appropriate size you must compute the central angle of each pie slice by multiplying the percentage of students with a particular eye color by 360°. Label each slice with the correct
percentage and the correct eye color.
Eye color comparison
Eye
Color
Number
of Students
Percentage
of Students
Central
Angle
Brown
Blue
Green
Hazel
Gray
Total
c. Which is the most common eye color? The least common?
d. In your opinion, does the bar graph or the pie graph display the eye color data the best?
Explain your thinking.
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Chapter 7
Statistics
e. List at least three observations each about these two data displays; what features does
each display emphasize?
f. List several types of data sets that would be good to display with a bar graph. Explain
your thinking.
g. List several types of data sets that would be good to display with a pie graph. Explain
your thinking.
Line Plots
2. As a class, gather data on the number of individual writing tools (pencils, pens, highlighters,
markers, etc.) that each student in the class has with them today. Organize the data in the
provided table.
Number of
Writing Tools
Number of
Students
Number of
Writing Tools
1
9
2
10
3
11
4
12
5
13
6
14
7
15+
Number of
Students
8
*a. Complete the line plot corresponding to this data set by marking an X for each student
with the indicated number of writing tools above the number.
Number of Writing Tools in Math Class
10
Number of students
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5
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15+
Number of writing tools
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Collecting and Graphing Data
b. What is the most popular number of writing tools? The least popular?
c. List at least three observations about this data display; what feature does this display
emphasize?
d. List several types of data sets that would be good to display with a line plot. Explain your
thinking.
Stem-and-Leaf Plots and Histograms
3. The following stem-and-leaf plot displays one randomly chosen temperature reading for
each of the 50 U.S. states on February 14, 2010, at 12 noon PST (Pacific standard time).
Temperature
Stem
Leaf
2
0678
3
025567777899
4
11246689
5
00555567789
6
0123344456
7
22344
a. How many states had temperatures 50° or less?
b. What is the highest temperature? The lowest?
*c. Another way to organize the temperature data would be to divide the data range into
several intervals and note the frequency at which the temperatures in these intervals
occurs. Organize your temperatures by frequency in the provided table.
Interval
Frequency
15–24
25–34
35–44
45–54
55–64
65–74
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Statistics
d. Sketch a histogram displaying the temperature data by sketching bars of the correct
height above the intervals on the horizontal axis. Consecutive bars should touch.
Current Temperature, February 14, 12 noon
Frequency
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20
18
16
14
12
10
8
6
4
2
15–24
25–34
35–44
45–54
55–64
65–74
Degrees Fahrenheit
e. What is the most common range of temperatures?
f. In your opinion; which graph, stem-and-leaf plot or histogram, displays the temperature
data the best? Explain your thinking.
g. List at least three observations each about these two data displays; what features does
each display emphasize?
h. List several types of data sets that would be good to display with a stem-and-leaf plot.
Explain your thinking.
i. List several types of data sets that would be good to display with a histogram. Explain
your thinking.
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Activity Set 7.1
209
Collecting and Graphing Data
Scatter Plots
4. a. As a class, gather data on the height of your mothers and fathers (in inches); give your
best approximations if you don’t know their exact heights. Organize the class data in
the provided table.
Mother’s
Height
Father’s
Height
Mother’s
Height
Father’s
Height
Mother’s
Height
Father’s
Height
Mother’s
Height
b. Form a scatter plot of these data on the following grid:
74
73
72
71
Heights of mothers in inches
Father’s
Height
70
69
68
67
66
65
64
63
62
61
60
62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
Heights of fathers in inches
Use the scatter plot you formed in part b to answer the following questions.
c. What is the height of the shortest mother? The tallest mother?
d. What are the heights of the shortest and tallest fathers?
e. What is the difference in height between the shortest and tallest mother? The shortest
and tallest father?
f. For how many couples was the woman taller than the man?
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Chapter 7
Statistics
5. One method for determining a relationship or trend on a scatter plot is to draw a single
straight line or curve that approximates the location of the points.
a. On the scatter plot in activity 4, draw a single straight line through the set of points that
you believe best represents the trend of the plotted points. This is called a trend line. (One
strategy: Look for a directional pattern of the points from left to right and then draw a
line in a similar direction so that approximately half the points are above the line and half
of the points are below the line.)
b. Do you think there is a trend or relationship between the two sets of data? That is, from
the data collected, do taller people seem to marry taller people? Justify your conjecture
in a sentence or two.
*6. a. Suppose that the middle school student’s scatter plot in the introduction had looked
as follows. Draw a trend line that you believe best represents this data set. Write a sentence or two summarizing the relationship between heights of fathers and mothers for
this data.
71
Summary:
70
Heights of mothers
69
68
67
66
65
64
63
62
61
60
62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
Heights of fathers
*b. Now suppose that the student’s data had looked like the scatter plot below. Draw a trend
line for these points. Would you say there is a trend or relationship between the heights
of the fathers and mothers for this data? Explain your reasoning.
71
Summary:
70
69
Heights of mothers
210
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68
67
66
65
64
63
62
61
60
62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
Heights of fathers
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Collecting and Graphing Data
c. The relationship between two sets of data can be described as positive, negative, or no
relationship, depending on the slope of the trend line. Classify the relationships for the
scatter plots in activities 4 and 6 on the preceding pages as positive, negative, or no
relationship.
4b.
6a.
6b.
d. For a negative relationship, one set of data increases while the other corresponding set of
data decreases; and for a positive relationship, as one set of data increases the corresponding data also increases. Describe two sets of data that will have a negative relationship and two that will have a positive relationship.
7. Form a scatter plot for each of the following sets of data and in each case sketch a trend line.
Determine if there is a positive or negative relationship, or no relationship.
a. This table contains data on one aspect of child development—the time required to hop a
given distance. The age of each child is rounded to the nearest half year. Use your trend
line to predict the hopping time for an average 7.5-year-old and record it in the table.
Age (years)
5 5 5.5 5.5 6 6 6.5 7 7 7.5 8 8 8.5 8.5 9 9 9.5 10 11
Time (seconds) 10.8 10.8 10.5 9.0 8.4 7.5 9.0 7.1 6.7
7.5 6.3 7.5 6.8 6.7 6.3 6.3 4.8 4.4
to hop 50 feet
11
10
Time in seconds
9
8
7
6
5
5
6
7
8
Age in years
9
10
11
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Chapter 7
Statistics
b. The following table was recorded by a forester who did a sample cutting of oak trees and
recorded their diameters (in inches) and age (by counting rings). Use your trend line to
predict the age of a tree that is 5.5 inches in diameter and the diameter of a tree that is
40 years old.
Age (years)
Diameter (inches)
10
2.0
8
22
30
18
13
38
38
25
8
16
28
34
29
20
4
33
23
14
35
30
12
8
5
10
1.0
5.8
6.0
4.6
3.5
7.0
5.0
6.5
3.0
4.5
6.0
6.5
4.5
5.5
0.8
8.0
4.7
2.5
7.0
7.0
4.9
2.0
0.8
3.5
Age of a tree that is 5.5
inches in diameter:
Diameter of a tree that
is 40 years old:
10
9
8
Diameter in inches
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7
6
5
4
3
2
1
5
10
15
20
25
Age in years
30
35
40
45
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Collecting and Graphing Data
JUST FOR FUN
M&M’S®
Bag Number
Red
Yellow
Blue
Orange
Green
Brown
Totals
1
9
16
12
5
6
8
56
2 3 4 5
12 5 10 9
13 16 17 17
7 11 11 11
12 9 9 9
5 5 6 9
6 9 5 1
55 55 58 56
6
11
8
8
5
10
12
54
7 8 9 10 11
11 11 11 14 10
14 19 13 15 19
11 8 7 11 11
8 5 9 6 9
6 7 8 2 2
5 5 8 8 3
55 55 56 56 54
12
14
12
7
11
4
10
58
13
12
20
7
6
10
5
60
14
8
15
12
9
7
5
56
The table above lists the color distributions in 24 bags of
M&M’s milk chocolate candies (net wt. 1.69 oz.).
16
11
18
11
11
2
5
58
17
12
18
11
5
8
2
56
18
5
16
9
8
5
12
55
19
10
24
5
8
4
10
61
20
18
13
9
7
5
5
57
21
14
10
15
7
6
5
57
22
13
17
8
10
4
3
55
23
11
17
12
5
9
3
57
24
9
9
4
6
16
11
55
20
15
Brown
*1. The total number of pieces of candy varies little from
bag to bag, but some color combinations vary greatly.
As the number of one color changes in a bag, does it
affect the number of any other color? For example, if the
number of brown increases in a bag, does it affect the
number of any other particular color? At the right is a
grid for a scatter plot comparison for the two colors,
blue and brown. Each bag determines a point on the
grid. For example, on the grid point (12, 8) represents
12 blue and 8 brown from bag 1 and (7, 6) represents
7 blue and 6 brown from bag 2. Plot all blue/brown pairs
on the grid. Draw a trend line to see if there is any relationship. (Note: If the number of brown increases as the
number of blue increases, there is a positive relationship. If the number of brown decreases as the number of
blue increases, there is a negative relationship. Or, there
may be no relationship.)
*2. Assume that you purchase the same size bag of candy as
those in the table. If your bag had 14 blue pieces, about
how many brown pieces would you predict using your
15
11
14
8
8
9
7
57
10
5
5
10
Blue
15
20
trend line? If there where 14 brown, how many blue
would you predict?
*3. Try comparing yellow and green. Make a scatter plot
and draw the trend line.
4. Pick other pairs of colors to compare.
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Exercises and Activities 7.1
1. School Classroom: Students studying a scatter plot comparing Monthly Income to Frequency
of Dining Out are having trouble answering interpretive questions that don’t correspond to
plotted points on the scatter plot. For example, they don’t know how to answer, “How often
do families with a monthly income of $4000 eat out?” The students say their trend line does
not go through a point for $4000. How can you help these students?
2. School Classroom: Read through the Mini-Lab on the Elementary School Text page at the
beginning of this section.
a. Explain what prerequisite knowledge the students would have to know before using this
technique to make this circle graph.
b. Compare the construction of the pictured circle graph with the construction of the circle
(pie) graph you made at the beginning of this activity set. What additional skills did you
need to construct the latter?
3. School Classroom: The students in a middle school class have built identical paper bridges
and placed centimeter cubes on the bridges to determine bridge load strengths. In nine trials
the students found the bridges broke under the following weights (in cm cubes):
Number of cubes
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 7
Trial 8
Trial 9
688
851
787
832
1000
187
120
481
730
a. Explain how you would help the students find effective graphical ways to display their
data without just telling them which graphical displays to use.
b. Which graphical display type do you think best displays the data? Why do you think it is
best?
4. Math Concepts: Explore scatter plots and trend lines further by answering the questions
about the distribution of M&M’s in the Just for Fun Activity in this section.
5. Math Concepts: Review current periodicals (newspapers or magazines) and find at least two
different types of data graphs explored in the activities in this activity set. For each graph,
write three observations about the data that you can read from the graph. Include a reference
for each periodical with a copy of the page on which you found the data display.
6. Math Concepts: List at least two similarities and at least two differences between each pair
of graph types:
a. Bar graph and stem-and-leaf graph
b. Bar graph and histogram
c. Stem-and-leaf graph and histogram
7. NCTM Standards: Read the Data Analysis and Probability Standards in the back pages of
this book for each grade level: Pre-K–2, 3–5, and 6–8. For each grade level, list at least one
Expectation that describes the types of graphs that should be studied at that level.
www.mhhe.com/bbn
bennett-burton-nelson website
Virtual Manipulatives
Interactive Chapter Applets
Puzzlers
Grid and Dot Paper
Color Transparencies
Extended Bibliography
Links and Readings
Geometer Sketchpad Modules