"

Chapter 2: Descriptive Statistics

2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

Learning Objectives

By the end of this section, the student should be able to:

  • Display data graphically and interpret data with stemplots.
  • Construct other types of graphs and interpret information displayed in: line graphs and bar 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

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

Table 1: Stem-and-Leaf Graph of Spring Pre-Calculus Class Scores
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 stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26% [latex]\left(\frac{8}{31}\right)[/latex] were in the 90s or 100, a fairly high number of As.

Your Turn!

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.

Solution
Table 2: Stem-and-Leaf Graph of Park City Basketball Team Scores
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. 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

The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data:

1.1; 1.5; 2.3; 2.5; 2.7; 3.2; 3.3; 3.3; 3.5; 3.8; 4.0; 4.2; 4.5; 4.5; 4.7; 4.8; 5.5; 5.6; 6.5; 6.7; 12.3

Do the data seem to have any concentration of values? (Hint: The leaves are to the right of the decimal.)

Solution

The value 12.3 may be an outlier. Values appear to concentrate at three and four kilometers.

Table 3: Stem-and-Leaf Graph of Distances from Home to Local Supermarket
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

Your Turn!

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:

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

Solution
Table 4: Stem-and-Leaf Graph of Distances from Homes of Off-Campus Students to the College
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.

Example

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 tables below 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 5: 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
Table 6: 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
Solution
Table 7: Side-by-Side Stem-and-Leaf of Ages at Inauguration and Ages at Death
Leaves: Ages at Inauguration Stem Leaves: 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

Your Turn!

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.

Table 8: Atlanta Hawks 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
Solution
Table 9: Side-by-Side Stem-and-Leaf of Number of Wins and Number of Losses
Leaves: Number of Wins Stem Leaves: 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 the next 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

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 below in a table and line graph.

Table 10: Frequency of Number of Times a Teenager is Reminded To Do Chores
Number of times teenager is reminded Frequency
0 2
1 5
2 8
3 14
4 7
5 4
A line graph showing the number of times a teenager needs to be reminded to do chores on the x-axis and frequency on the y-axis.
Figure 1. Line Graph of Number of Times a Teenager is Reminded to Do Chores

Your Turn!

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

Table 11: Frequency of Number of Times People had their Car in the Shop
Number of times in shop Frequency
0 7
1 10
2 14
3 9
Solution
A line graph showing the number of times a car was put in the shop on the x-axis and frequency on the y-axis.
Figure 2. Line Graph of Number of Times a Car is in the Shop for Repairs

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 the next example has age groups represented on the x-axis and proportions on the y-axis.

Examples

By the end of 2011, Facebook had over 146 million users in the United States. The table below 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.

Table 12: Agre Groups, Number of Users, and Proportion of Facebook Users
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%
Solution
This is a bar graph that matches the supplied data. The x-axis shows age groups, and the y-axis shows the percentages of Facebook users.
Figure 3. Bar Graph of Number of Ages of Facebook Users

Your Turn!

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

Table 13: Agre Groups, Number of People, and Proportion of Population
Age groups Number of people Proportion of population
Children 67,059 19%
Working-age adults 152,198 43%
Retirees 131,662 38%
This is a bar graph that matches the supplied data. The x-axis shows age groups, and the y-axis shows the percentages of Park City's population.
Figure 4. Bar Graph of Age Groups of Park City's Population

Example

The columns in the table below 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.
Table 14: Race/Ethnicity, Percent of AP Exam Population, and 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%
Solution
This is a bar graph that matches the supplied data. The x-axis shows race and ethnicity, and the y-axis shows the percentages of AP examinees.
Figure 5. Bar Graph of Race/Ethnicity of AP Examinees

 

Your Turn!

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.

Table 15: District Number, Registered Voter Population and Overall Citiy Population
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%
Solution
A bar graph that matches the supplied data. The x-axis shows Park City voting districts, and the y-axis shows percentages of the registered voter population.
Figure 6. Bar Chart of Districts and Voter Proportions

Videos

Below are helpful videos for the content covered in this Section. Videos are provided from YouTube.

Section 2.1 Review

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.

Section 2.1 Practice

For each of the following data sets, create a stem plot and identify any outliers.

  1. The miles per gallon rating for 30 cars are shown below (lowest to highest): 19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43
  2. The data are the prices of different laptops at an electronics store. Round each value to the nearest ten: 249, 249, 260, 265, 265, 280, 299, 299, 309, 319, 325, 326, 350, 350, 350, 365, 369, 389, 409, 459, 489, 559, 569, 570, 610
Solution

For question 1:

Table 16: Stem-and-Leaf of Miles per Gallon for 30 cars
Stem Leaf
1 9 9 9
2 0 1 1 5 5 5 6 6 8 9
3 1 1 2 2 3 4 5 6 7 7 8 8 8 8
4 1 3 3

For question 2:

Table 17: Stem-and-Leaf of Prices of Different Laptops
Stem Leaf
2 5 5 6 7 7 8
3 0 0 1 2 3 3 5 5 5 7 7 9
4 1 6 9
5 6 7 7
6 1

For each of the following data sets, create a stem plot and identify any outliers.

  1. The height in feet of 25 trees is shown below (lowest to highest): 25, 27, 33, 34, 34, 34, 35, 37, 37, 38, 39, 39, 39, 40, 41, 45, 46, 47, 49, 50, 50, 53, 53, 54, 54
  2. The daily high temperatures in a town for one month: 61, 61, 62, 64, 66, 67, 67, 67, 68, 69, 70, 70, 70, 71, 71, 72, 74, 74, 74, 75, 75, 75, 76, 76, 77, 78, 78, 79, 79, 95

For the next two exercises, use the data to construct a line graph.

  1. In a survey, 40 people were asked how many times they visited a store before making a major purchase. The results are shown in the table below.
  2. Several children were asked how many TV shows they watch each day. The results of the survey are shown in the table below.
Table 18: Frequency Table of How Many Times People Visited a Store Before a Purchase
Number of times in store Frequency
1 4
2 10
3 16
4 6
5 4
Table 19: Frequency Table of How Many TV Shows Watched Each Day
Number of TV Shows Frequency
0 12
1 18
2 36
3 7
4 2
Solution
For question 1:
A line graph that matches the data. The x-axis is the number of times people visited a store before making a purchase, and the y-axis is the frequency.
Figure 7. Line Graph of Number of Times in a Store
For question 2:
A line graph that matches the supplied data. The x-axis shows the number of TV shows a kid watches each day, and the y-axis shows the frequency.
Figure 8. Line Graph of TV Shows Watched Per Day

Use the data to construct a line graph.

In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in the table below.

Table 20: Frequency Table of How Many Years Since Purchasing a Mattress
Years since last purchase Frequency
0 2
1 8
2 13
3 22
4 16
5 9

The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. The table below shows the four seasons, the number of students who have birthdays in each season, and the percentage (%) of students in each group. Construct a bar graph showing the number of students.

Table 21: Seasons, Number of Students Who Have Birthdays, and Proportion
Seasons Number of students Proportion of population
Spring 8 24%
Summer 9 26%
Autumn 11 32%
Winter 6 18%

Using the data from Mrs. Ramirez’s math class, construct a bar graph showing the percentages.

Solution
This is a bar graph that matches the supplied data. The x-axis shows the seasons of the year, and the y-axis shows the proportion of birthdays.
Figure 9. Bar Chart of Proportion of Birthdays Each Season

David County has six high schools. Each school sent students to participate in a county-wide science competition. The following table shows the percentage breakdown of competitors from each school, and the percentage of the entire student population of the county that goes to each school. Construct a bar graph that shows the population percentage of competitors from each school.

Table 22: High School, Science Competition Population, and Overall Student Population
High School Science competition population Overall student population
Alabaster 28.9% 8.6%
Concordia 7.6% 23.2%
Genoa 12.1% 15.0%
Mocksville 18.5% 14.3%
Tynneson 24.2% 10.1%
West End 8.7% 28.8%

Use the data from the David County science competition. Construct a bar graph that shows the county-wide population percentage of students at each school.

Solution
This is a bar graph that matches the supplied data. The x-axis shows the county high schools, and the y-axis shows the proportion of county students.
Figure 10. Bar Chart of Proportion of Students in Science Competition

Student grades on a chemistry exam were: 77, 78, 76, 81, 86, 51, 79, 82, 84, 99

  1. Construct a stem-and-leaf plot of the data.
  2. Are there any potential outliers? If so, which scores are they? Why do you consider them outliers?

References

Burbary, Ken. Facebook Demographics Revisited – 2001 Statistics, 2011. Available online at http://www.kenburbary.com/2011/03/facebook-demographics-revisited-2011-statistics-2/ (accessed August 21, 2013).

“9th Annual AP Report to the Nation.” CollegeBoard, 2013. Available online at http://apreport.collegeboard.org/goals-and-findings/promoting-equity (accessed September 13, 2013).

“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).

License

Icon for the Creative Commons Attribution-ShareAlike 4.0 International License

Introductory Statistics Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book