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2.1: Stem-and-Leaf Graphs, Line Graphs, Bar Graphs, and Pie Charts

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    Stem-and-leaf Graphs

    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 1

    For Susan Dean’s spring pre-calculus class, scores for the first exam were as follows (smallest to largest):
    33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100
    Form a stem-and-leaf graph.
    [reveal-answer q=”65536″]Show Answer[/reveal-answer]
    [hidden-answer a=”65536″]

    Stem Leaf
    3 3
    4 2 9 9
    5 3 5 5
    6 1 3 7 8 8 9 9
    7 2 3 4 8
    8 0 3 8 8 8
    9 0 2 4 4 4 4 6
    10 0

    The stem-and-leaf plot shows that most scores fell in the 60s, 70s, 80s, and 90s.

    Eight out of the 31 scores or approximately 26% were in the 90s or 100, a fairly high number of As. [/hidden-answer]

    Try It

    For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest):

    32; 32; 33; 34; 38; 40; 42; 42; 43; 44; 46; 47; 47; 48; 48; 48; 49; 50; 50; 51; 52; 52; 52; 53; 54; 56; 57; 57; 60; 61

    Construct a stem plot for the data.[practice-area rows=”3″][/practice-area]
    [reveal-answer q=”265621″]Show Answer[/reveal-answer]
    [hidden-answer a=”265621″]

    Stem Leaf
    3 2 2 3 4 8
    4 0 2 2 3 4 6 7 7 8 8 8 9
    5 0 0 1 2 2 2 3 4 6 7 7
    6 0 1

    The stemplot is a quick way to graph data and gives an exact picture of the data.[/hidden-answer]

    Try It

    The following data show the distances (in miles) from the homes of off-campus statistics students to the college.

    0.5; 0.7; 1.1; 1.2; 1.2; 1.3; 1.3; 1.5; 1.5; 1.7; 1.7; 1.8; 1.9; 2.0; 2.2; 2.5; 2.6; 2.8; 2.8; 2.8; 3.5; 3.8; 4.4; 4.8; 4.9; 5.2; 5.5; 5.7; 5.8; 8.0

    Create a stem plot using the data and identify any outliers:
    [reveal-answer q=”294219″]Show Answer[/reveal-answer]
    [hidden-answer a=”294219″]

    Stem Leaf
    0 5 7
    1 1 2 2 3 3 5 5 7 7 8 9
    2 0 2 5 6 8 8 8
    3 5 8
    4 4 8 9
    5 2 5 7 8
    6
    7
    8 0

    The value 8.0 may be an outlier. Values appear to concentrate at one and two miles.[/hidden-answer]

     

    Watch this video to see an example of how to create a stem plot.

    Thumbnail for the embedded element "Stem and Leaf Plots aka Stemplots"

    A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/esm/?p=42

    Example 2

    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. The two following tables show the ages of presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data.

    Presidential Ages at Inauguration

    President Age 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

    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

    [reveal-answer q=”154052″]Show Answer[/reveal-answer]
    [hidden-answer a=”154052″]

    Ages at Inauguration

    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

    [/hidden-answer]

     


    Example 3

    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

    [reveal-answer q=”863761″]Show Answer[/reveal-answer]
    [hidden-answer a=”863761″]

    Number of losses

    Atalanta Hawks Wins and Leaves  
    Number of Wins   Number of Loses
    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

    [/hidden-answer]

     

    Line Graph

    Another type of graph that is useful for specific data values is a line graph.

    Example 1

    In a survey, 40 mothers were asked how many times per week a teenager must be reminded to do his or her chores.

    Number of times teenager is reminded Frequency
    0 2
    1 5
    2 8
    3 14
    4 7
    5 4

    [reveal-answer q=”510661″]Show Answer[/reveal-answer]
    [hidden-answer a=”510661″]fig-ch_02_03_01.jpg[/hidden-answer]

    Try It

    In a survey, 40 people were asked how many times per year they had their car in the shop for repairs.

    Number of times in shop Frequency
    0 7
    1 10
    2 14
    3 9

    Construct a line graph.
    [reveal-answer q=”727928″]Show Answer[/reveal-answer]
    [hidden-answer a=”727928″]CNX_Stats_C02_M03_tryit003.png[/hidden-answer]

     

    Bar Graphs & Pie Charts

    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.

    Pie Charts is another method to “visualize” the data. Each proportion on a pie chart represents the proportion of the corresponding class in the data set.

    Example 1

    By the end of 2011, Facebook had over 146 million users in the United States. The 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 and a pie chart 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%

    [reveal-answer q=”310877″]Bar Graph[/reveal-answer]
    [hidden-answer a=”310877″]fig-ch02_03_02.jpgThe bar graph has age groups represented on the x-axis and proportions on the y-axis.[/hidden-answer]
    [reveal-answer q=”283830″]Pie Chart[/reveal-answer]
    [hidden-answer a=”283830″]pie-chart-1.jpg[/hidden-answer]

    Try It

    The population in Park City is made up of children, working-age adults, and retirees. The 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.

    Age groups Number of people Proportion of population
    Children 67,059 19%
    Working-age adults 152,198 43%
    Retirees 131,662 38%

    Construct a bar graph and a pie chart showing the proportions.
    [reveal-answer q=”759238″]Bar Graph[/reveal-answer]
    [hidden-answer a=”759238″]CNX_Stats_C02_M03_tryit004.png[/hidden-answer]
    [reveal-answer q=”951719″]Pie Chart[/reveal-answer]
    [hidden-answer a=”951719″]pie-chart-2.jpg[/hidden-answer]

    Example 2

    The columns in the 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.

    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%

    Create a bar graph and a pie chart with the student race or ethnicity (qualitative data) on the x-axis, and the Advanced Placement examinee population percentages on the y-axis.
    [reveal-answer q=”875617″]Show Answer[/reveal-answer]
    [hidden-answer a=”875617″]fig-ch02_03_03.jpg[/hidden-answer]
    [reveal-answer q=”523702″]Pie Chart[/reveal-answer]
    [hidden-answer a=”523702″]pie-chart-3.jpg[/hidden-answer]

     

    Try It

    Park city is broken down into six voting districts. The table shows the percentage 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.

    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%

    Construct a bar graph and a pie chart that shows the registered voter population by district.

    [reveal-answer q=”16013″]Bar Graph[/reveal-answer]
    [hidden-answer a=”16013″]CNX_Stats_C02_M03_tryit005.png[/hidden-answer]
    [reveal-answer q=”533178″]Pie Chart[/reveal-answer]
    [hidden-answer a=”533178″]pie-chart-4.jpg

    [/hidden-answer]

     

     

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