Descriptive Statistics

7 Display Data

Stem-and-Leaf Graphs (Stemplots), 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.

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

Stem-and-Leaf Graph
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% \left(\frac{8}{31}\right) were in the 90s or 100, a fairly high number of As.

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.

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.

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?

NOTE

The leaves are to the right of the decimal.

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
Try It

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

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.

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. (Figure) and (Figure) show the ages of presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data.

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 2 1 1 1 1 1 0 5 3 6 6 7 7 8
9 8 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
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

Another type of graph that is useful for specific data values is a line graph. In the particular line graph shown in (Figure), 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.

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 (Figure) and in (Figure).

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.
Try It

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 (Figure). Construct a line graph.

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

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 (Figure) has age groups represented on the x-axis and proportions on the y-axis.

By the end of 2011, Facebook had over 146 million users in the United States. (Figure) 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%
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.
Try It

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

The columns in (Figure) 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%
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.
Try It

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%
This is a bar graph that matches the supplied data. The x-axis shows Park City voting districts, and the y-axis shows the percentages of the registered voter population.

Below is a two-way table showing the types of pets owned by men and women:

Dogs Cats Fish Total
Men 4 2 2 8
Women 4 6 2 12
Total 8 8 4 20

Given these data, calculate the conditional distributions for the subpopulation of men who own each pet type.

Men who own dogs = 4/8 = 0.5

Men who own cats = 2/8 = 0.25

Men who own fish = 2/8 = 0.25

Note: The sum of all of the conditional distributions must equal one. In this case, 0.5 + 0.25 + 0.25 = 1; therefore, the solution “checks”.

Histograms, Frequency Polygons, and Time Series Graphs

For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.

A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.

The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample.(Remember, frequency is defined as the number of times an answer occurs.) If:

  • f = frequency
  • n = total number of data values (or the sum of the individual frequencies), and
  • RF = relative frequency,

then:

\text{RF}=\frac{f}{n}

For example, if three students in Mr. Ahab’s English class of 40 students received from 90% to 100%, then, <!–<newline count=”1″/>–>f = 3, n = 40, and RF = \frac{f}{n} = \frac{3}{40} = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures.

To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is 6.05 (6.1 – 0.05 = 6.05). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (1.5 – 0.005 = 1.495). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is 0.9995 (1.0 – 0.0005 = 0.9995). If all the data happen to be integers and the smallest value is two, then a convenient starting point is 1.5 (2 – 0.5 = 1.5). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.

The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data, since height is measured.
60; 60.5; 61; 61; 61.5
63.5; 63.5; 63.5
64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5
66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5
68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5
70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71
72; 72; 72; 72.5; 72.5; 73; 73.5
74

The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.

60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.

The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.

Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.

\frac{74.05-59.95}{8}=1.76
NOTE

We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.

The boundaries are:

  • 59.95
  • 59.95 + 2 = 61.95
  • 61.95 + 2 = 63.95
  • 63.95 + 2 = 65.95
  • 65.95 + 2 = 67.95
  • 67.95 + 2 = 69.95
  • 69.95 + 2 = 71.95
  • 71.95 + 2 = 73.95
  • 73.95 + 2 = 75.95

The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.

The following histogram displays the heights on the x-axis and relative frequency on the y-axis.

Histogram consists of 8 bars with the y-axis in increments of 0.05 from 0-0.4 and the x-axis in intervals of 2 from 59.95-75.95.
Try It

The following data are the shoe sizes of 50 male students. The sizes are continuous data since shoe size is measured. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars.
9; 9; 9.5; 9.5; 10; 10; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5
11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5
12; 12; 12; 12; 12; 12; 12; 12.5; 12.5; 12.5; 12.5; 14

Smallest value: 9

Largest value: 14

Convenient starting value: 9 – 0.05 = 8.95

Convenient ending value: 14 + 0.05 = 14.05

\frac{14.05-8.95}{6}=0.85

The calculations suggests using 0.85 as the width of each bar or class interval. You can also use an interval with a width equal to one.

Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data, since books are counted.
1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1
2; 2; 2; 2; 2; 2; 2; 2; 2; 2
3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3
4; 4; 4; 4; 4; 4
5; 5; 5; 5; 5
6; 6

Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.

Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.

Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .

  • 3.5 to 4.5
  • 4.5 to 5.5
  • 6
  • 5.5 to 6.5

Calculate the number of bars as follows:

\frac{6.5-0.5}{\mathrm{number of bars}}=1

where 1 is the width of a bar. Therefore, bars = 6.

The following histogram displays the number of books on the x-axis and the frequency on the y-axis.

Histogram consists of 6 bars with the y-axis in increments of 2 from 0-16 and the x-axis in intervals of 1 from 0.5-6.5.

Using this data set, construct a histogram.

Number of hours my classmates spent playing video games on weekends
9.95 10 2.25 16.75 0
19.5 22.5 7.5 15 12.75
5.5 11 10 20.75 17.5
23 21.9 24 23.75 18
20 15 22.9 18.8 20.5
This is a histogram that matches the supplied data. The x-axis consists of 5 bars in intervals of 5 from 0 to 25. The y-axis is marked in increments of 1 from 0 to 10. The x-axis shows the number of hours spent playing video games on the weekends, and the y-axis shows the number of students.

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x-axis and y-axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

A frequency polygon was constructed from the frequency table below.

Frequency distribution for calculus final test scores
Lower bound Upper bound Frequency Cumulative frequency
49.5 59.5 5 5
59.5 69.5 10 15
69.5 79.5 30 45
79.5 89.5 40 85
89.5 99.5 15 100
A frequency polygon was constructed from the frequency table below.

The first label on the x-axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x-axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Try It

Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in (Figure).

Age at inauguration Frequency
41.5–46.5 4
46.5–51.5 11
51.5–56.5 14
56.5–61.5 9
61.5–66.5 4
66.5–71.5 2

The first label on the x-axis is 39. This represents an interval extending from 36.5 to 41.5. Since there are no ages less than 41.5, this interval is used only to allow the graph to touch the x-axis. The point labeled 44 represents the next interval, or the first “real” interval from the table, and contains four scores. This reasoning is followed for each of the remaining intervals with the point 74 representing the interval from 71.5 to 76.5. Again, this interval contains no data and is only used so that the graph will touch the x-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

This figure shows a graph entitled, 'President's Age at Inauguration.' The x-axis is labeled 'Ages' and is marked off at 39, 44, 49, 54, 59, 64, 69 and 74. The y-axis is labeled, 'Frequency,' and is marked off in intervals of 1 from 0 to 15. The following points are plotted and a line connects one to the other to create the frequency polygon: (39, 0), (44, 4), (49, 11), (54, 14), (59, 9), (64, 4), (69, 2), (74, 0).

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

We will construct an overlay frequency polygon comparing the scores from (Figure) with the students’ final numeric grade.

Frequency distribution for calculus final test scores
Lower bound Upper bound Frequency Cumulative frequency
49.5 59.5 5 5
59.5 69.5 10 15
69.5 79.5 30 45
79.5 89.5 40 85
89.5 99.5 15 100
Frequency distribution for calculus final grades
Lower bound Upper bound Frequency Cumulative frequency
49.5 59.5 10 10
59.5 69.5 10 20
69.5 79.5 30 50
79.5 89.5 45 95
89.5 99.5 5 100
This is an overlay frequency polygon that matches the supplied data. The x-axis shows the grades, and the y-axis shows the frequency.

Constructing a Time Series Graph

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with these data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.

Year Jan Feb Mar Apr May Jun Jul
2003 181.7 183.1 184.2 183.8 183.5 183.7 183.9
2004 185.2 186.2 187.4 188.0 189.1 189.7 189.4
2005 190.7 191.8 193.3 194.6 194.4 194.5 195.4
2006 198.3 198.7 199.8 201.5 202.5 202.9 203.5
2007 202.416 203.499 205.352 206.686 207.949 208.352 208.299
2008 211.080 211.693 213.528 214.823 216.632 218.815 219.964
2009 211.143 212.193 212.709 213.240 213.856 215.693 215.351
2010 216.687 216.741 217.631 218.009 218.178 217.965 218.011
2011 220.223 221.309 223.467 224.906 225.964 225.722 225.922
2012 226.665 227.663 229.392 230.085 229.815 229.478 229.104
Year Aug Sep Oct Nov Dec Annual
2003 184.6 185.2 185.0 184.5 184.3 184.0
2004 189.5 189.9 190.9 191.0 190.3 188.9
2005 196.4 198.8 199.2 197.6 196.8 195.3
2006 203.9 202.9 201.8 201.5 201.8 201.6
2007 207.917 208.490 208.936 210.177 210.036 207.342
2008 219.086 218.783 216.573 212.425 210.228 215.303
2009 215.834 215.969 216.177 216.330 215.949 214.537
2010 218.312 218.439 218.711 218.803 219.179 218.056
2011 226.545 226.889 226.421 226.230 225.672 224.939
2012 230.379 231.407 231.317 230.221 229.601 229.594
This is a times series graph that matches the supplied data. The x-axis shows years from 2003 to 2012, and the y-axis shows the annual CPI.
Try It

The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO2 emissions for the United States.

CO2 emissions
Year Ukraine United Kingdom United States
2003 352,259 540,640 5,681,664
2004 343,121 540,409 5,790,761
2005 339,029 541,990 5,826,394
2006 327,797 542,045 5,737,615
2007 328,357 528,631 5,828,697
2008 323,657 522,247 5,656,839
2009 272,176 474,579 5,299,563
This is a times series graph that matches the supplied data. The x-axis shows years from 2003 to 2012, and the y-axis shows the annual CPI.

Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

How NOT to Lie with Statistics

It is important to remember that the very reason we develop a variety of methods to present data is to develop insights into the subject of what the observations represent. We want to get a “sense” of the data. Are the observations all very much alike or are they spread across a wide range of values, are they bunched at one end of the spectrum or are they distributed evenly and so on. We are trying to get a visual picture of the numerical data. Shortly we will develop formal mathematical measures of the data, but our visual graphical presentation can say much. It can, unfortunately, also say much that is distracting, confusing and simply wrong in terms of the impression the visual leaves. Many years ago Darrell Huff wrote the book How to Lie with Statistics. It has been through 25 plus printings and sold more than one and one-half million copies. His perspective was a harsh one and used many actual examples that were designed to mislead. He wanted to make people aware of such deception, but perhaps more importantly to educate so that others do not make the same errors inadvertently.

Again, the goal is to enlighten with visuals that tell the story of the data. Pie charts have a number of common problems when used to convey the message of the data. Too many pieces of the pie overwhelm the reader. More than perhaps five or six categories ought to give an idea of the relative importance of each piece. This is after all the goal of a pie chart, what subset matters most relative to the others. If there are more components than this then perhaps an alternative approach would be better or perhaps some can be consolidated into an “other” category. Pie charts cannot show changes over time, although we see this attempted all too often. In federal, state, and city finance documents pie charts are often presented to show the components of revenue available to the governing body for appropriation: income tax, sales tax motor vehicle taxes and so on. In and of itself this is interesting information and can be nicely done with a pie chart. The error occurs when two years are set side-by-side. Because the total revenues change year to year, but the size of the pie is fixed, no real information is provided and the relative size of each piece of the pie cannot be meaningfully compared.

Histograms can be very helpful in understanding the data. Properly presented, they can be a quick visual way to present probabilities of different categories by the simple visual of comparing relative areas in each category. Here the error, purposeful or not, is to vary the width of the categories. This of course makes comparison to the other categories impossible. It does embellish the importance of the category with the expanded width because it has a greater area, inappropriately, and thus visually “says” that that category has a higher probability of occurrence.

Time series graphs perhaps are the most abused. A plot of some variable across time should never be presented on axes that change part way across the page either in the vertical or horizontal dimension. Perhaps the time frame is changed from years to months. Perhaps this is to save space or because monthly data was not available for early years. In either case this confounds the presentation and destroys any value of the graph. If this is not done to purposefully confuse the reader, then it certainly is either lazy or sloppy work.

Changing the units of measurement of the axis can smooth out a drop or accentuate one. If you want to show large changes, then measure the variable in small units, penny rather than thousands of dollars. And of course to continue the fraud, be sure that the axis does not begin at zero, zero. If it begins at zero, zero, then it becomes apparent that the axis has been manipulated.

Perhaps you have a client that is concerned with the volatility of the portfolio you manage. An easy way to present the data is to use long time periods on the time series graph. Use months or better, quarters rather than daily or weekly data. If that doesn’t get the volatility down then spread the time axis relative to the rate of return or portfolio valuation axis. If you want to show “quick” dramatic growth, then shrink the time axis. Any positive growth will show visually “high” growth rates. Do note that if the growth is negative then this trick will show the portfolio is collapsing at a dramatic rate.

Again, the goal of descriptive statistics is to convey meaningful visuals that tell the story of the data. Purposeful manipulation is fraud and unethical at the worst, but even at its best, making these type of errors will lead to confusion on the part of the analysis.

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).

Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker

“Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/article/timeline-guide-us-presidents (accessed April 3, 2013).

“Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).

“Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).

“Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).

“CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).

“Births Time Series Data.” General Register Office For Scotland, 2013. Available online at http://www.gro-scotland.gov.uk/statistics/theme/vital-events/births/time-series.html (accessed April 3, 2013).

“Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).

Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach. CRC Press: 1980.

“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).

Chapter 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.

A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y-axis with the frequency being graphed on the x-axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.


For the next three exercises, use the data to construct a line graph.In a survey, 40 people were asked how many times they visited a store before making a major purchase. The results are shown in (Figure).Number of times in storeFrequency142103164654

This is a line graph that matches the supplied data. The x-axis shows the number of times people reported visiting a store before making a major purchase, and the y-axis shows the frequency.

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

Years since last purchase Frequency
0 2
1 8
2 13
3 22
4 16
5 9

Several children were asked how many TV shows they watch each day. The results of the survey are shown in (Figure).

Number of TV shows Frequency
0 12
1 18
2 36
3 7
4 2
This is 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.

The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. (Figure) 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.

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 supplied in (Figure), construct a bar graph showing the percentages.

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.

David County has six high schools. Each school sent students to participate in a county-wide science competition. (Figure) 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.

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 supplied in (Figure). Construct a bar graph that shows the county-wide population percentage of students at each school.

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.

Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Complete the table.

Data value (# cars) Frequency Relative frequency Cumulative relative frequency

What does the frequency column in (Figure) sum to? Why?

65

What does the relative frequency column in (Figure) sum to? Why?

What is the difference between relative frequency and frequency for each data value in (Figure)?

The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value.

What is the difference between cumulative relative frequency and relative frequency for each data value?

To construct the histogram for the data in (Figure), determine appropriate minimum and maximum x and y values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling.

An empty graph template for use with this question.

Answers will vary. One possible histogram is shown:

Construct a frequency polygon for the following:

  1. Pulse rates for women Frequency
    60–69 12
    70–79 14
    80–89 11
    90–99 1
    100–109 1
    110–119 0
    120–129 1
  2. Actual speed in a 30 MPH zone Frequency
    42–45 25
    46–49 14
    50–53 7
    54–57 3
    58–61 1
  3. Tar (mg) in nonfiltered cigarettes Frequency
    10–13 1
    14–17 0
    18–21 15
    22–25 7
    26–29 2

Construct a frequency polygon from the frequency distribution for the 50 highest ranked countries for depth of hunger.

Depth of hunger Frequency
230–259 21
260–289 13
290–319 5
320–349 7
350–379 1
380–409 1
410–439 1

Find the midpoint for each class. These will be graphed on the x-axis. The frequency values will be graphed on the y-axis values.

This is a frequency polygon that matches the supplied data. The x-axis shows the depth of hunger, and the y-axis shows the frequency.

Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlayed frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of women compared to men?

Life expectancy at birth – women Frequency
49–55 3
56–62 3
63–69 1
70–76 3
77–83 8
84–90 2
Life expectancy at birth – men Frequency
49–55 3
56–62 3
63–69 1
70–76 1
77–83 7
84–90 5

Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births.

Sex/Year 1855 1856 1857 1858 1859 1860 1861
Female 45,545 49,582 50,257 50,324 51,915 51,220 52,403
Male 47,804 52,239 53,158 53,694 54,628 54,409 54,606
Total 93,349 101,821 103,415 104,018 106,543 105,629 107,009
Sex/Year 1862 1863 1864 1865 1866 1867 1868 1869
Female 51,812 53,115 54,959 54,850 55,307 55,527 56,292 55,033
Male 55,257 56,226 57,374 58,220 58,360 58,517 59,222 58,321
Total 107,069 109,341 112,333 113,070 113,667 114,044 115,514 113,354
Sex/Year 1870 1871 1872 1873 1874 1875
Female 56,431 56,099 57,472 58,233 60,109 60,146
Male 58,959 60,029 61,293 61,467 63,602 63,432
Total 115,390 116,128 118,765 119,700 123,711 123,578

The following data sets list full time police per 100,000 citizens along with homicides per 100,000 citizens for the city of Detroit, Michigan during the period from 1961 to 1973.

Year 1961 1962 1963 1964 1965 1966 1967
Police 260.35 269.8 272.04 272.96 272.51 261.34 268.89
Homicides 8.6 8.9 8.52 8.89 13.07 14.57 21.36
Year 1968 1969 1970 1971 1972 1973
Police 295.99 319.87 341.43 356.59 376.69 390.19
Homicides 28.03 31.49 37.39 46.26 47.24 52.33
  1. Construct a double time series graph using a common x-axis for both sets of data.
  2. Which variable increased the fastest? Explain.
  3. Did Detroit’s increase in police officers have an impact on the murder rate? Explain.

Homework

(Figure) contains the 2010 obesity rates in U.S. states and Washington, DC.

State Percent (%) State Percent (%) State Percent (%)
Alabama 32.2 Kentucky 31.3 North Dakota 27.2
Alaska 24.5 Louisiana 31.0 Ohio 29.2
Arizona 24.3 Maine 26.8 Oklahoma 30.4
Arkansas 30.1 Maryland 27.1 Oregon 26.8
California 24.0 Massachusetts 23.0 Pennsylvania 28.6
Colorado 21.0 Michigan 30.9 Rhode Island 25.5
Connecticut 22.5 Minnesota 24.8 South Carolina 31.5
Delaware 28.0 Mississippi 34.0 South Dakota 27.3
Washington, DC 22.2 Missouri 30.5 Tennessee 30.8
Florida 26.6 Montana 23.0 Texas 31.0
Georgia 29.6 Nebraska 26.9 Utah 22.5
Hawaii 22.7 Nevada 22.4 Vermont 23.2
Idaho 26.5 New Hampshire 25.0 Virginia 26.0
Illinois 28.2 New Jersey 23.8 Washington 25.5
Indiana 29.6 New Mexico 25.1 West Virginia 32.5
Iowa 28.4 New York 23.9 Wisconsin 26.3
Kansas 29.4 North Carolina 27.8 Wyoming 25.1
  1. Use a random number generator to randomly pick eight states. Construct a bar graph of the obesity rates of those eight states.
  2. Construct a bar graph for all the states beginning with the letter “A.”
  3. Construct a bar graph for all the states beginning with the letter “M.”
  1. Example solution for using the random number generator for the TI-84+ to generate a simple random sample of 8 states. Instructions are as follows.
    • Number the entries in the table 1–51 (Includes Washington, DC; Numbered vertically)
    • Press MATH
    • Arrow over to PRB
    • Press 5:randInt(
    • Enter 51,1,8)

    Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}.

    Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}.

    A bar graph showing 8 states on the x-axis and corresponding obesity rates on the y-axis.

  2. This is a bar graph that matches the supplied data. The x-axis shows states, and the y-axis shows percentages.
  3. This is a bar graph that matches the supplied data. The x-axis shows states, and the y-axis shows percentages.

Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows:

Publisher A
# of books Freq. Rel. freq.
0 10
1 12
2 16
3 12
4 8
5 6
6 2
8 2
Publisher B
# of books Freq. Rel. freq.
0 18
1 24
2 24
3 22
4 15
5 10
7 5
9 1
Publisher C
# of books Freq. Rel. freq.
0–1 20
2–3 35
4–5 12
6–7 2
8–9 1
  1. Find the relative frequencies for each survey. Write them in the charts.
  2. Use the frequency column to construct a histogram for each publisher’s survey. For Publishers A and B, make bar widths of one. For Publisher C, make bar widths of two.
  3. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical.
  4. Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not?
  5. Make new histograms for Publisher A and Publisher B. This time, make bar widths of two.
  6. Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer.

Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group.

Singles
Amount(?) Frequency Rel. frequency
51–100 5
101–150 10
151–200 15
201–250 15
251–300 10
301–350 5
Couples
Amount(?) Frequency Rel. frequency
100–150 5
201–250 5
251–300 5
301–350 5
351–400 10
401–450 10
451–500 10
501–550 10
551–600 5
601–650 5
  1. Fill in the relative frequency for each group.
  2. Construct a histogram for the singles group. Scale the x-axis by ?50 widths. Use relative frequency on the y-axis.
  3. Construct a histogram for the couples group. Scale the x-axis by ?50 widths. Use relative frequency on the y-axis.
  4. Compare the two graphs:
    1. List two similarities between the graphs.
    2. List two differences between the graphs.
    3. Overall, are the graphs more similar or different?
  5. Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x-axis by ?50, scale it by ?100. Use relative frequency on the y-axis.
  6. Compare the graph for the singles with the new graph for the couples:
    1. List two similarities between the graphs.
    2. Overall, are the graphs more similar or different?
  7. How did scaling the couples graph differently change the way you compared it to the singles graph?
  8. Based on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person as a couple? Explain why in one or two complete sentences.
Singles
Amount(?) Frequency Relative frequency
51–100 5 0.08
101–150 10 0.17
151–200 15 0.25
201–250 15 0.25
251–300 10 0.17
301–350 5 0.08
Couples
Amount(?) Frequency Relative frequency
100–150 5 0.07
201–250 5 0.07
251–300 5 0.07
301–350 5 0.07
351–400 10 0.14
401–450 10 0.14
451–500 10 0.14
501–550 10 0.14
551–600 5 0.07
601–650 5 0.07
  1. See (Figure) and (Figure).
  2. In the following histogram data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where both boundary values are included).
    This is a histogram that matches the supplied data supplied for singles. The x-axis shows the total charges in intervals of 50 from 50 to 350, and the y-axis shows the relative frequency in increments of 0.05 from 0 to 0.3.
  3. In the following histogram, the data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where values on both boundaries are included).
    This is a histogram that matches the supplied data for couples. The x-axis shows the total charges in intervals of 50 from 100 to 650, and the y-axis shows the relative frequency in increments of 0.02 from 0 to 0.16.
  4. Compare the two graphs:
    1. Answers may vary. Possible answers include:
      • Both graphs have a single peak.
      • Both graphs use class intervals with width equal to ?50.
    2. Answers may vary. Possible answers include:
      • The couples graph has a class interval with no values.
      • It takes almost twice as many class intervals to display the data for couples.
    3. Answers may vary. Possible answers include: The graphs are more similar than different because the overall patterns for the graphs are the same.
  5. Check student’s solution.
  6. Compare the graph for the Singles with the new graph for the Couples:
      • Both graphs have a single peak.
      • Both graphs display 6 class intervals.
      • Both graphs show the same general pattern.
    1. Answers may vary. Possible answers include: Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different.
  7. Answers may vary. Possible answers include: You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the Couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison.
  8. Answers may vary. Possible answers include: Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals.

Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows.

# of movies Frequency Relative frequency Cumulative relative frequency
0 5
1 9
2 6
3 4
4 1
  1. Construct a histogram of the data.
  2. Complete the columns of the chart.

Use the following information to answer the next two exercises: Suppose one hundred eleven people who shopped in a special t-shirt store were asked the number of t-shirts they own costing more than ?19 each.

A histogram showing the results of a survey. Of 111 respondents, 5 own 1 t-shirt costing more than 💲19, 17 own 2, 23 own 3, 39 own 4, 25 own 5, 2 own 6, and no respondents own 7.

The percentage of people who own at most three t-shirts costing more than ?19 each is approximately:

  1. 21
  2. 59
  3. 41
  4. Cannot be determined

c

If the data were collected by asking the first 111 people who entered the store, then the type of sampling is:

  1. cluster
  2. simple random
  3. stratified
  4. convenience

Following are the 2010 obesity rates by U.S. states and Washington, DC.

State Percent (%) State Percent (%) State Percent (%)
Alabama 32.2 Kentucky 31.3 North Dakota 27.2
Alaska 24.5 Louisiana 31.0 Ohio 29.2
Arizona 24.3 Maine 26.8 Oklahoma 30.4
Arkansas 30.1 Maryland 27.1 Oregon 26.8
California 24.0 Massachusetts 23.0 Pennsylvania 28.6
Colorado 21.0 Michigan 30.9 Rhode Island 25.5
Connecticut 22.5 Minnesota 24.8 South Carolina 31.5
Delaware 28.0 Mississippi 34.0 South Dakota 27.3
Washington, DC 22.2 Missouri 30.5 Tennessee 30.8
Florida 26.6 Montana 23.0 Texas 31.0
Georgia 29.6 Nebraska 26.9 Utah 22.5
Hawaii 22.7 Nevada 22.4 Vermont 23.2
Idaho 26.5 New Hampshire 25.0 Virginia 26.0
Illinois 28.2 New Jersey 23.8 Washington 25.5
Indiana 29.6 New Mexico 25.1 West Virginia 32.5
Iowa 28.4 New York 23.9 Wisconsin 26.3
Kansas 29.4 North Carolina 27.8 Wyoming 25.1

Construct a bar graph of obesity rates of your state and the four states closest to your state. Hint: Label the x-axis with the states.

Answers will vary.

Key Terms

Frequency
the number of times a value of the data occurs
Histogram
a graphical representation in xy form of the distribution of data in a data set; x represents the data and y represents the frequency, or relative frequency. The graph consists of contiguous rectangles.
Relative Frequency
the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes

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