One simple graph, the stem-and-leaf graph or stemplot, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a final significant digit. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem.
Example \(\PageIndex{1}\)
For Susan Dean's spring pre-calculus class, scores for the first exam were as follows (smallest to largest):
The stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26% \(\left(\frac{8}{31}\right)\) were in the 90s or 100, a fairly high number of As.
Exercise \(\PageIndex{2}\)
For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest):
The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down 50 instead of 500) while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later.
Example \(\PageIndex{3}\)
The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data:
Do the data seem to have any concentration of values?
HINT: The leaves are to the right of the decimal.
Answer
The value 12.3 may be an outlier. Values appear to concentrate at three and four kilometers.
Stem
Leaf
1
1 5
2
3 5 7
3
2 3 3 5 8
4
0 2 5 5 7 8
5
5 6
6
5 7
7
8
9
10
11
12
3
Query \(\PageIndex{1}\)
Exercise \(\PageIndex{4}\)
The following data show the distances (in miles) from the homes of off-campus statistics students to the college. Create a stem plot using the data and identify any outliers:
The value 8.0 may be an outlier. Values appear to concentrate at one and two miles.
Example \(\PageIndex{5}\): side-by-side stem-and-leaf plot
A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) show the ages of presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data.
Table \(\PageIndex{1}\): Presidential Ages at Inauguration
President
Ageat Inauguration
President
Age
President
Age
Washington
57
Lincoln
52
Hoover
54
J. Adams
61
A. Johnson
56
F. Roosevelt
51
Jefferson
57
Grant
46
Truman
60
Madison
57
Hayes
54
Eisenhower
62
Monroe
58
Garfield
49
Kennedy
43
J. Q. Adams
57
Arthur
51
L. Johnson
55
Jackson
61
Cleveland
47
Nixon
56
Van Buren
54
B. Harrison
55
Ford
61
W. H. Harrison
68
Cleveland
55
Carter
52
Tyler
51
McKinley
54
Reagan
69
Polk
49
T. Roosevelt
42
G.H.W. Bush
64
Taylor
64
Taft
51
Clinton
47
Fillmore
50
Wilson
56
G. W. Bush
54
Pierce
48
Harding
55
Obama
47
Buchanan
65
Coolidge
51
Trump
70
\(\PageIndex{2}\) Presidential Age at Death
President
Age
President
Age
President
Age
Washington
67
Lincoln
56
Hoover
90
J. Adams
90
A. Johnson
66
F. Roosevelt
63
Jefferson
83
Grant
63
Truman
88
Madison
85
Hayes
70
Eisenhower
78
Monroe
73
Garfield
49
Kennedy
46
J. Q. Adams
80
Arthur
56
L. Johnson
64
Jackson
78
Cleveland
71
Nixon
81
Van Buren
79
B. Harrison
67
Ford
93
W. H. Harrison
68
Cleveland
71
Reagan
93
Tyler
71
McKinley
58
Polk
53
T. Roosevelt
60
Taylor
65
Taft
72
Fillmore
74
Wilson
67
Pierce
64
Harding
57
Buchanan
77
Coolidge
60
Answer
Ages at Inauguration
Ages at Death
9 9 8 7 7 7 6 3 2
4
6 9
8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 4 2 1 1 1 1 1 0
5
3 6 6 7 7 8
9 5 4 4 2 1 1 1 0
6
0 0 3 3 4 4 5 6 7 7 7 8
7
0 0 1 1 1 4 7 8 8 9
8
0 1 3 5 8
9
0 0 3 3
Exercise \(\PageIndex{6}\)
The table shows the number of wins and losses the Atlanta Hawks have had in 42 seasons. Create a side-by-side stem-and-leaf plot of these wins and losses.
Losses
Wins
Year
Losses
Wins
Year
34
48
1968–1969
41
41
1989–1990
34
48
1969–1970
39
43
1990–1991
46
36
1970–1971
44
38
1991–1992
46
36
1971–1972
39
43
1992–1993
36
46
1972–1973
25
57
1993–1994
47
35
1973–1974
40
42
1994–1995
51
31
1974–1975
36
46
1995–1996
53
29
1975–1976
26
56
1996–1997
51
31
1976–1977
32
50
1997–1998
41
41
1977–1978
19
31
1998–1999
36
46
1978–1979
54
28
1999–2000
32
50
1979–1980
57
25
2000–2001
51
31
1980–1981
49
33
2001–2002
40
42
1981–1982
47
35
2002–2003
39
43
1982–1983
54
28
2003–2004
42
40
1983–1984
69
13
2004–2005
48
34
1984–1985
56
26
2005–2006
32
50
1985–1986
52
30
2006–2007
25
57
1986–1987
45
37
2007–2008
32
50
1987–1988
35
47
2008–2009
30
52
1988–1989
29
53
2009–2010
Answer
\(\PageIndex{1}\) Atlanta Hawks Wins and Losses
Number of Wins
Number of Losses
3
1
9
9 8 8 6 5
2
5 5 9
8 7 6 6 5 5 4 3 1 1 1 1 0
3
0 2 2 2 2 4 4 5 6 6 6 9 9 9
8 8 7 6 6 6 3 3 3 2 2 1 1 0
4
0 0 1 1 2 4 5 6 6 7 7 8 9
7 7 6 3 2 0 0 0 0
5
1 1 1 2 3 4 4 6 7
6
9
Another type of graph that is useful for specific data values is a line graph. In the particular line graph shown in Example, the x-axis (horizontal axis) consists of data values and the y-axis (vertical axis) consists of frequency points. The frequency points are connected using line segments.
Example \(\PageIndex{7}\)
In a survey, 40 mothers were asked how many times per week a teenager must be reminded to do his or her chores. The results are shown in Table and in Figure.
Number of times teenager is reminded
Frequency
0
2
1
5
2
8
3
14
4
7
5
4
Exercise \(\PageIndex{8}\)
In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in Table. Construct a line graph.
Number of times in shop
Frequency
0
7
1
10
2
14
3
9
Answer
Bar graphs consist of bars that are separated from each other. The bars can be rectangles or they can be rectangular boxes (used in three-dimensional plots), and they can be vertical or horizontal. The bar graph shown in Example \(\PageIndex{9}\) has age groups represented on the x-axis and proportions on the y-axis.
Example \(\PageIndex{9}\)
By the end of 2011, Facebook had over 146 million users in the United States. Table shows three age groups, the number of users in each age group, and the proportion (%) of users in each age group. Construct a bar graph using this data.
Age groups
Number of Facebook users
Proportion (%) of Facebook users
13–25
65,082,280
45%
26–44
53,300,200
36%
45–64
27,885,100
19%
Answer
Exercise \(\PageIndex{10}\)
The population in Park City is made up of children, working-age adults, and retirees. Table shows the three age groups, the number of people in the town from each age group, and the proportion (%) of people in each age group. Construct a bar graph showing the proportions.
Age groups
Number of people
Proportion of population
Children
67,059
19%
Working-age adults
152,198
43%
Retirees
131,662
38%
Answer
Example \(\PageIndex{11}\)
The columns in Table contain: the race or ethnicity of students in U.S. Public Schools for the class of 2011, percentages for the Advanced Placement examine population for that class, and percentages for the overall student population. Create a bar graph with the student race or ethnicity (qualitative data) on the x-axis, and the Advanced Placement examinee population percentages on the y-axis.
Race/Ethnicity
AP Examinee Population
Overall Student Population
1 = Asian, Asian American or Pacific Islander
10.3%
5.7%
2 = Black or African American
9.0%
14.7%
3 = Hispanic or Latino
17.0%
17.6%
4 = American Indian or Alaska Native
0.6%
1.1%
5 = White
57.1%
59.2%
6 = Not reported/other
6.0%
1.7%
Solution
Exercise \(\PageIndex{12}\)
Park city is broken down into six voting districts. The table shows the percent of the total registered voter population that lives in each district as well as the percent total of the entire population that lives in each district. Construct a bar graph that shows the registered voter population by district.
District
Registered voter population
Overall city population
1
15.5%
19.4%
2
12.2%
15.6%
3
9.8%
9.0%
4
17.4%
18.5%
5
22.8%
20.7%
6
22.3%
16.8%
Answer
Summary
A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends. That is, finding a general pattern in data sets including temperature, sales, employment, company profit or cost over a period of time. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Some bar graphs present bars clustered in groups of more than one (grouped bar graphs), and others show the bars divided into subparts to show cumulative effect (stacked bar graphs). Bar graphs are especially useful when categorical data is being used.
“Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).
Contributors
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114.