# 2.3: Other Graphical Representations of Data

There are many other types of graphs. Some of the more common ones are the frequency polygon, the dot plot, the stem plot, scatter plot, and a time-series plot. There are also many different graphs that have emerged lately for qualitative data. Many are found in publications and websites. The following is a description of the stem plot, the scatter plot, and the time-series plot.

## Stem Plots

Stem plots are a quick and easy way to look at small samples of numerical data. You can look for any patterns or any strange data values. It is easy to compare two samples using stem plots.

The first step is to divide each number into 2 parts, the stem (such as the leftmost digit) and the leaf (such as the rightmost digit). There are no set rules, you just have to look at the data and see what makes sense.

Example $$\PageIndex{1}$$ stem plot for grade distribution

The following are the percentage grades of 25 students from a statistics course. Draw a stem plot of the data.

 62 87 81 69 87 62 45 95 76 76 62 71 65 67 72 80 40 77 87 58 84 73 93 64 89

Table 2.3.1: Data of Test Grades

Solution:

Divide each number so that the tens digit is the stem and the ones digit is the leaf. 62 becomes 6|2.

Make a vertical chart with the stems on the left of a vertical bar. Be sure to fill in any missing stems. In other words, the stems should have equal spacing (for example, count by ones or count by tens). The Graph 2.3.1 shows the stems for this example.

Graph 2.3.1: Stem Plot for Test Grades Step 1

Now go through the list of data and add the leaves. Put each leaf next to its corresponding stem. Don’t worry about order yet just get all the leaves down.

When the data value 62 is placed on the plot it looks like the plot in Graph 2.3.2.

Graph 2.3.2: Stem Plot for Test Grades Step 2

When the data value 87 is placed on the plot it looks like the plot in Graph 2.3.3.

Graph 2.3.3: Stem Plot for Test Grades Step 3

Filling in the rest of the leaves to obtain the plot in Graph 2.3.4.

Graph 2.3.4: Stem Plot for Test Grades Step 4

Now you have to add labels and make the graph look pretty. You need to add a label and sort the leaves into increasing order. You also need to tell people what the stems and leaves mean by inserting a legend. Be careful to line the leaves up in columns. You need to be able to compare the lengths of the rows when you interpret the graph. The final stem plot for the test grade data is in Graph 2.3.5.

Graph 2.3.5: Stem Plot for Test Grades

Now you can interpret the stem-and-leaf display. The data is bimodal and somewhat symmetric. There are no gaps in the data. The center of the distribution is around 70.

You can create a stem and leaf plot on R. the command is:

stem(variable) – creates a stem and leaf plot, if you do not get a stem plot that shows all of the stems then use scale = a number. Adjust the number until you see all of the stems. So you would have stem(variable, scale = a number)

For Example 2.3.1, the command would be

grades<-c(62, 87, 81, 69, 87, 62, 45, 95, 76, 76, 62, 71, 65, 67, 72, 80, 40, 77, 87, 58, 84, 73, 93, 64, 89)

Output:

The decimal point is 1 digit(s) to the right of the |

Now just put a title on the stem plot.

## Scatter Plot

Sometimes you have two different variables and you want to see if they are related in any way. A scatter plot helps you to see what the relationship would look like. A scatter plot is just a plotting of the ordered pairs.

Example $$\PageIndex{2}$$ scatter plot

Is there any relationship between elevation and high temperature on a given day? The following data are the high temperatures at various cities on a single day and the elevation of the city.

 Elevation Temperature (°F) (in feet) 7000 4000 6000 3000 7000 4500 5000 50 60 48 70 55 55 60

Table 2.3.2: Data of Temperature versus Elevation

Solution:

Preliminary: State the random variables

Let x = altitude

y = high temperature

Now plot the x values on the horizontal axis, and the y values on the vertical axis. Then set up a scale that fits the data on each axes. Once that is done, then just plot the x and y values as an ordered pair. In R, the command is:

independent variable<-c(type in data with commas in between values)
dependent variable<-c(type in data with commas in between values)
plot(independent variable, dependent variable, main="type in a title you want", xlab="type in a label for the horizontal axis", ylab="type in a label for the vertical axis", ylim=c(0, number above maximum y value)

For this example, that would be:
elevation<-c(7000, 4000, 6000, 3000, 7000, 4500, 5000)
temperature<-c(50, 60, 48, 70, 55, 55, 60)
plot(elevation, temperature, main="Temperature versus Elevation", xlab="Elevation (in feet)", ylab="Temperature (in degrees F)", ylim=c(0, 80))

Graph 2.3.6: Scatter Plot of Temperature versus Elevation

Looking at the graph, it appears that there is a linear relationship between temperature and elevation. It also appears to be a negative relationship, thus as elevation increases, the temperature decreases.

## Time-Series

A time-series plot is a graph showing the data measurements in chronological order, the data being quantitative data. For example, a time-series plot is used to show profits over the last 5 years. To create a time-series plot, the time always goes on the horizontal axis, and the other variable goes on the vertical axis. Then plot the ordered pairs and connect the dots. The purpose of a time-series graph is to look for trends over time. Caution, you must realize that the trend may not continue. Just because you see an increase, doesn’t mean the increase will continue forever. As an example, prior to 2007, many people noticed that housing prices were increasing. The belief at the time was that housing prices would continue to increase. However, the housing bubble burst in 2007, and many houses lost value, and haven’t recovered.

Example $$\PageIndex{3}$$ Time-series plot

The following table tracks the weight of a dieter, where the time in months is measuring how long since the person started the diet

 Time (months) 0 1 2 3 4 5 Weight (pounds) 200 195 192 193 190 187

Table 2.3.3: Data of Weights versus Time

Make a time-series plot of this data

Solution:

In R, the command would be:

variable1<-c(type in data with commas in between values, this should be the time variable)
variable2<-c(type in data with commas in between values)
plot(variable1, variable2, ylim=c(0,number over max), main="type in a title you want", xlab="type in a label for the horizontal axis", ylab="type in a label for the vertical axis")
lines(variable1, variable2) – connects the dots

For this example:
time<-c(0, 1, 2, 3, 4, 5)
weight<-c(200, 195, 192, 193, 190, 187)
plot(time, weight, ylim=c(0,250), main="Weight over Time", xlab="Time (Months) ", ylab="Weight (pounds)")
ines(time, weight)

Graph 2.3.7: Time-Series Graph of Weight versus Time

Notice, that over the 5 months, the weight appears to be decreasing. Though it doesn’t look like there is a large decrease.

Be careful when making a graph. If you don’t start the vertical axis at 0, then the change can look much more dramatic than it really is. As an example, Graph 2.3.8 shows the Graph 2.3.7 with a different scaling on the vertical axis. Notice the decrease in weight looks much larger than it really is.

Graph 2.3.8: Example of a Poor Graph

## Homework

Exercise $$\PageIndex{1}$$

1. Students in a statistics class took their first test. The data in Table 2.3.4 are the scores they earned. Create a stem plot.
 80 79 89 74 73 67 79 93 70 70 76 88 83 73 81 79 80 85 79 80 79 58 93 94 74

Table 2.3.4: Data of Test 1 Grades

2. Students in a statistics class took their first test. The data in Table 2.3.5 are the scores they earned. Create a stem plot. Compare to the graph in question 1.
 67 67 76 47 85 70 87 76 80 72 84 98 84 64 65 82 81 81 88 74 87 83

Table 2.3.5: Data of Test 1 Grades

3. When an anthropologist finds skeletal remains, they need to figure out the height of the person. The height of a person (in cm) and the length of one of their metacarpal bone (in cm) were collected and are in Table 2.4.6 ("Prediction of height," 2013). Create a scatter plot and state if there is a relationship between the height of a person and the length of their metacarpal.
Length of Metacarpal Height of Person
45 171
51 178
39 157
41 163
48 172
49 183
46 173
43 175
47 173

Table 2.3.6: Data of Metacarpal versus Height

4. Table 2.3.7 contains the value of the house and the amount of rental income in a year that the house brings in ("Capital and rental," 2013). Create a scatter plot and state if there is a relationship between the value of the house and the annual rental income.
Value Rental Value Rental Value Rental Value Rental
81000 6656 77000 4576 75000 7280 67500 6864
95000 7904 94000 8736 90000 6240 85000 7072
121000 12064 115000 7904 110000 7072 104000 7904
135000 8320 130000 9776 126000 6240 125000 7904
145000 8320 140000 9568 140000 9152 135000 7488
165000 13312 165000 8528 155000 7488 148000 8320
178000 11856 174000 10400 170000 9568 170000 12688
200000 12272 200000 10608 194000 11232 190000 8320
214000 8528 280000 10400 200000 10400 200000 8320
240000 10192 240000 12064 240000 11648 225000 12480
289000 11648 270000 12896 262000 10192 244500 11232
325000 12480 310000 12480 303000 12272 300000 12480

Table 2.3.7: Data of House Value versus Rental

5. The World Bank collects information on the life expectancy of a person in each country ("Life expectancy at," 2013) and the fertility rate per woman in the country ("Fertility rate," 2013). The data for 24 randomly selected countries for the year 2011 are in Table 2.3.8. Create a scatter plot of the data and state if there appears to be a relationship between life expectancy and the number of births per woman.
Life Expectancy Fertility Rate Life Expectancy Fertility rate
77.2 1.7 72.3 3.9
55.4 5.8 76.0 1.5
69.9 2.2 66.0 4.2
76.4 2.1 5.9 5.2
75.0 1.8 54.4 6.8
78.2 2.0 62.9 4.7
73.0 2.6 78.3 2.1
70.8 2.8 72.1 2.9
82.6 1.4 80.7 1.4
68.9 2.6 74.2 2.5
81.0 1.5 73.3 1.5
54.2 6.9 67.1 2.4

Table 2.3.8: Data of Life Expectancy versus Fertility Rate

6. The World Bank collected data on the percentage of gross domestic product (GDP) that a country spends on health expenditures ("Health expenditure," 2013) and the percentage of woman receiving prenatal care ("Pregnant woman receiving," 2013). The data for the countries where this information is available for the year 2011 is in Table 2.3.9. Create a scatter plot of the data and state if there appears to be a relationship between percentage spent on health expenditure and the percentage of woman receiving prenatal care.
Prenatal Care (%) Health Expenditure (% of GDP)
47.9 9.6
54.6 3.7
93.7 5.2
84.7 5.2
100.0 10.0
42.5 4.7
96.4 4.8
77.1 6.0
58.3 5.4
95.4 4.8
78.0 4.1
93.3 6.0
93.3 9.5
93.7 6.8
89.8 6.1

Table 2.3.9: Data of Prenatal Care versus Health Expenditure

7. The Australian Institute of Criminology gathered data on the number of deaths (per 100,000 people) due to firearms during the period 1983 to 1997 ("Deaths from firearms," 2013). The data is in Table 2.3.10. Create a time-series plot of the data and state any findings you can from the graph.
 Year Rate Year Rate 1983 1984 1985 1986 1987 1988 1989 1990 4.31 4.42 4.52 4.35 4.39 4.21 3.4 3.61 1991 1992 1993 1994 1995 1996 1997 3.67 3.61 2.98 2.95 2.72 2.95 2.3

Table 2.3.10: Data of Year versus Number of Deaths due to Firearms

8. The economic crisis of 2008 affected many countries, though some more than others. Some people in Australia have claimed that Australia wasn’t hurt that badly from the crisis. The bank assets (in billions of Australia dollars (AUD)) of the Reserve Bank of Australia (RBA) for the time period of March 2007 through March 2013 are contained in Table 2.3.11 ("B1 assets of," 2013). Create a time-series plot and interpret any findings.
Date Assets in Billions of AUD
Mar-2006 96.9
Jun-2006 107.4
Sep-2006 107.2
Dec-2006 116.2
Mar-2007 123.7
Jun-2007 134.0
Sep-2007 123.0
Dec-2007 93.2
Mar-2008 93.7
Jun-2008 105.6
Sep-2008 101.5
Dec-2008 158.8
Mar-2009 118.7
Jun-2009 111.9
Sep-2009 87.0
Dec-2009 86.1
Mar-2010 83.4
Jun-2010 85.7
Sep-2010 74.8
Dec-2010 76.0
Mar-2011 75.7
Jun-2011 75.9
Sep-2011 75.2
Dec-2011 87.9
Mar-2012 91.0
Jun-2012 90.1
Sep-2012 83.9
Dec-2012 95.8
Mar-2013 90.5

Table 2.3.11: Data of Date versus RBA Assets

9. The consumer price index (CPI) is a measure used by the U.S. government to describe the cost of living. Table 2.3.12 gives the cost of living for the U.S. from the years 1947 through 2011, with the year 1977 being used as the year that all others are compared (DeNavas-Walt, Proctor & Smith, 2012). Create a time-series plot and interpret.
Year CPI-U-RS1 index (December 1977=100) Year CPI-U-RS1 index (December 1977=100)
1947 37.5 1980 127.1
1948 40.5 1981 139.2
1949 40.0 1982 147.6
1950 40.5 1983 153.9
1951 43.7 1984 160.2
1952 44.5 1985 165.7
1953 44.8 1986 168.7
1954 45.2 1987 174.4
1955 45.0 1988 180.8
1956 45.7 1989 188.6
1957 47.2 1990 198.0
1958 48.5 1991 205.1
1959 48.9 1992 210.3
1960 49.7 1993 215.5
1961 50.2 1994 220.1
1962 50.7 1995 225.4
1963 51.4 1996 231.4
1964 52.1 1997 236.4
1965 52.9 1998 239.7
1966 54.4 1999 244.7
1967 56.1 2000 252.9
1968 58.3 2001 260.0
1969 60.9 2002 264.2
1970 63.9 2003 270.1
1971 66.7 2004 277.4
1972 68.7 2005 286.7
1973 73.0 2006 296.1
1974 80.3 2007 304.5
1975 86.9 2008 316.2
1976 91.9 2009 315.0
1977 97.7 2010 320.2
1978 104.4 2011 330.3
1979 114.4

Table 2.3.12: Data of Time versus CPI

10. The median incomes for all households in the U.S. for the years 1967 to 2011 are given in Table 2.3.13 (DeNavas-Walt, Proctor & Smith, 2012). Create a time-series plot and interpret.
Year Median Income Year Median Income
1967 42,056 1990 49,950
1968 43,868 1991 48,516
1969 45,499 1992 48,117
1970 45,146 1993 47,884
1971 44,707 1994 48,418
1972 46,622 1995 49,935
1973 47,563 1996 50,661
1974 46,057 1997 51,704
1975 44,851 1998 53,582
1976 45,595 1999 54,932
1977 45,884 2000 54,841
1978 47,659 2001 53,646
1979 47,527 2002 53,019
1980 46,024 2003 52,973
1981 45,260 2004 52,788
1982 45,139 2005 53,371
1983 44,823 2006 53,768
1984 46,215 2007 54,489
1985 47,079 2008 52,546
1986 48,746 2009 52,195
1987 49,358 2010 50,831
1988 49,737 2011 50,054
1989 50,624

Table 2.3.13: Data of Time versus Median Income

11. State everything that makes Graph 2.3.9 a misleading or poor graph.

Graph 2.3.9: Example of a Poor Graph
12. State everything that makes Graph 2.3.10 a misleading or poor graph (Benen, 2011).

Graph 2.3.10: Example of a Poor Graph
13. State everything that makes Graph 2.3.11 a misleading or poor graph ("United States unemployment," 2013).

Graph 2.3.11: Example of a Poor Graph
14. State everything that makes Graph 2.3.12 a misleading or poor graph.

Graph 2.3.12: Example of a Poor Graph

See solutions

## Data Sources:

Capital and rental values of Auckland properties. (2013, September 26). Retrieved from http://www.statsci.org/data/oz/rentcap.html

DeNavas-Walt, C., Proctor, B., & Smith, J. U.S. Department of Commerce, U.S. Census Bureau. (2012). Income, poverty, and health insurance coverage in the United States: 2011 (P60-243). Retrieved from website: www.census.gov/prod/2012pubs/p60-243.pdf

Density of people in Africa. (2013, October 9). Retrieved from http://www.prb.org/DataFinder/Topic/...249,250,251,25 2,253,254,34227,255,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,27 2,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,294, 295,296,297,298,299,300,301,302,304,305,306,307,308

Health expenditure. (2013, October 14). Retrieved from http://data.worldbank.org/indicator/SH.XPD.TOTL.ZS Hinatov, M. U.S. Consumer Product Safety Commission, Directorate of Epidemiology. (2012). Incidents, deaths, and in-depth investigations associated with non-fire carbon monoxide from engine-driven generators and other engine-driven tools, 1999-2011. Retrieved from website: http://www.cpsc.gov/PageFiles/129857/cogenerators.pdf