In this Concept, we will investigate the different types of graphs that can be used to represent single numerical variables (univariate data). We will compare the distribution of the data, and look at the effect of outliers.
Watch This
For a description of how to draw a stemandleaf plot, as well as how to derive information from one (14.0) , see APUS07, StemandLeaf Plot (8:08).
Guidance
Dot Plots
A dot plot is one of the simplest ways to represent numerical data. After choosing an appropriate scale on the axes, each data point is plotted as a single dot. Multiple points at the same value are stacked on top of each other using equal spacing to help convey the shape and center.
Example A
The following is a data set representing the percentage of paper packaging manufactured from recycled materials for a select group of countries.
Country  % of Paper Packaging Recycled 

Estonia  34 
New Zealand  40 
Poland  40 
Cyprus  42 
Portugal  56 
United States  59 
Italy  62 
Spain  63 
Australia  66 
Greece  70 
Finland  70 
Ireland  70 
Netherlands  70 
Sweden  70 
France  76 
Germany  83 
Austria  83 
Belgium  83 
Japan  98 
The dot plot for this data would look like this:
Notice that this data set is centered at a manufacturing rate for using recycled materials of between 65 and 70 percent. It is spread from 34% to 98%, and appears very roughly symmetric, perhaps even slightly skewed left. Dot plots have the advantage of showing all the data points and giving a quick and easy snapshot of the shape, center, and spread. Dot plots are not much help when there is little repetition in the data. They can also be very tedious if you are creating them by hand with large data sets, though computer software can make quick and easy work of creating dot plots from such data sets.
StemandLeaf Plots
One of the shortcomings of dot plots is that they do not show the actual values of the data. You have to read or infer them from the graph. From the previous example, you might have been able to guess that the lowest value is 34%, but you would have to look in the data table itself to know for sure. A stemandleaf plot is a similar plot in which it is much easier to read the actual data values. In a stemandleaf plot, each data value is represented by two digits: the stem and the leaf. In this example, it makes sense to use the ten's digits for the stems and the one's digits for the leaves. The stems are on the left of a dividing line as follows:
Once the stems are decided, the leaves representing the one's digits are listed in numerical order from left to right:
It is important to explain the meaning of the data in the plot for someone who is viewing it without seeing the original data. For example, you could place the following sentence at the bottom of the chart:
Note: means 56% and 59% are the two values in the 50's.
If you could rotate this plot on its side, you would see the similarities with the dot plot. The general shape and center of the plot is easily found, and we know exactly what each point represents. This plot also shows the slight skewing to the left that we suspected from the dot plot. Stem plots can be difficult to create, depending on the numerical qualities and the spread of the data. If the data values contain more than two digits, you will need to remove some of the information by rounding. A data set that has large gaps between values can also make the stem plot hard to create and less useful when interpreting the data.
Example B
Consider the following populations of counties in California.
Butte  220,748
Calaveras  45,987
Del Norte  29,547
Fresno  942,298
Humboldt  132,755
Imperial  179,254
San Francisco  845,999
Santa Barbara  431,312
To construct a stem and leaf plot, we need to first make sure each piece of data has the same number of digits. In our data, we will add a 0 at the beginning of our 5 digit data points so that all data points have six digits. Then, we can either round or truncate all data points to two digits.
Value  Value Rounded  Value Truncated 

149  15  14 
657  66  65 
188  19  18 
represents when data has been truncated
represents when data has been rounded.
If we decide to round the above data, we have:
Butte  220,000
Calaveras  050,000
Del Norte  030,000
Fresno  940,000
Humboldt  130,000
Imperial  180,000
San Francisco  850,000
Santa Barbara  430,000
And the stem and leaf will be as follows:
where:
represents .
Source: California State Association of Counties http://74.205.125.191/default.asp?id=399
BacktoBack Stem Plots
Stem plots can also be a useful tool for comparing two distributions when placed next to each other. These are commonly called backtoback stem plots .
Example C
In a previous example, we looked at recycling in paper packaging. Here are the same countries and their percentages of recycled material used to manufacture glass packaging:
Country  % of Glass Packaging Recycled 

Cyprus  4 
United States  21 
Poland  27 
Greece  34 
Portugal  39 
Spain  41 
Australia  44 
Ireland  56 
Italy  56 
Finland  56 
France  59 
Estonia  64 
New Zealand  72 
Netherlands  76 
Germany  81 
Austria  86 
Japan  96 
Belgium  98 
Sweden  100 
In a backtoback stem plot, one of the distributions simply works off the left side of the stems. In this case, the spread of the glass distribution is wider, so we will have to add a few extra stems. Even if there are no data values in a stem, you must include it to preserve the spacing, or you will not get an accurate picture of the shape and spread.
We have already mentioned that the spread was larger in the glass distribution, and it is easy to see this in the comparison plot. You can also see that the glass distribution is more symmetric and is centered lower (around the mid50's), which seems to indicate that overall, these countries manufacture a smaller percentage of glass from recycled material than they do paper. It is interesting to note in this data set that Sweden actually imports glass from other countries for recycling, so its effective percentage is actually more than 100.
Vocabulary
A dot plot is a convenient way to represent univariate numerical data by plotting individual dots along a single number line to represent each value. They are especially useful in giving a quick impression of the shape, center, and spread of the data set, but are tedious to create by hand when dealing with large data sets.
Stemandleaf plots show similar information with the added benefit of showing the actual data values.
Guided Practice
Here are the ages, arranged order, for the CEOs of the 60 topranked small companies in America in 1993 http://lib.stat.cmu.edu/DASL/Datafiles/ceodat.html
32, 33, 36, 37, 38, 40, 41, 43, 43, 44, 44, 45, 45, 45, 45,46, 46, 47, 47, 47, 48, 48, 48, 48, 49, 50, 50, 50, 50, 50, 50, 51, 51, 52, 53, 53, 53, 55, 55, 55, 56, 56, 56, 56, 57, 57, 58, 58, 59, 60, 61, 61, 61, 62, 62, 63, 69, 69, 70, 74
a) Create a stemandleaf plot for these ages.
b) Create a dot plot for these ages.
c) Describe the shape of this dataset.
d) Are there any outliers in this dataset?
Solutions:
1. Here is the stemandleaf plot:
b. Here is the dot plot:
c. The data set is approximately symmetric with most CEOs in their fifties.
d. There do not appear to be any outliers.
Practice
For 14, the following table gives the percentages of municipal waste recycled by state in the United States, including the District of Columbia, in 1998. Data was not available for Idaho or Texas.
State  Percentage 

Alabama  23 
Alaska  7 
Arizona  18 
Arkansas  36 
California  30 
Colorado  18 
Connecticut  23 
Delaware  31 
District of Columbia  8 
Florida  40 
Georgia  33 
Hawaii  25 
Illinois  28 
Indiana  23 
Iowa  32 
Kansas  11 
Kentucky  28 
Louisiana  14 
Maine  41 
Maryland  29 
Massachusetts  33 
Michigan  25 
Minnesota  42 
Mississippi  13 
Missouri  33 
Montana  5 
Nebraska  27 
Nevada  15 
New Hampshire  25 
New Jersey  45 
New Mexico  12 
New York  39 
North Carolina  26 
North Dakota  21 
Ohio  19 
Oklahoma  12 
Oregon  28 
Pennsylvania  26 
Rhode Island  23 
South Carolina  34 
South Dakota  42 
Tennessee  40 
Utah  19 
Vermont  30 
Virginia  35 
Washington  48 
West Virginia  20 
Wisconsin  36 
Wyoming  5 
Source: http://www.zerowasteamerica.org/MunicipalWasteManagementReport1998.htm
 Create a dot plot for this data.
 Discuss the shape, center, and spread of this distribution.
 Create a stemandleaf plot for the data.
 Use your stemandleaf plot to find the median percentage for this data.
For 58, identify the important features of the shape of the distribution.
For 912, refer to the following dot plots:
 Identify the overall shape of each distribution.
 How would you characterize the center(s) of these distributions?
 Which of these distributions has the smallest standard deviation?
 Which of these distributions has the largest standard deviation?
 What characteristics of a data set make it easier or harder to represent using dot plots, stemandleaf plots, or histograms?

Here are the ages, arranged order, for the CEOs of the 60 topranked small companies in America in 1993
http://lib.stat.cmu.edu/DASL/Datafiles/ceodat.html
32, 33, 36, 37, 38, 40, 41, 43, 43, 44, 44, 45, 45, 45, 45,46, 46, 47, 47, 47, 48, 48, 48, 48, 49, 50, 50, 50, 50, 50, 50, 51, 51, 52, 53, 53, 53, 55, 55, 55, 56, 56, 56, 56, 57, 57, 58, 58, 59, 60, 61, 61, 61, 62, 62, 63, 69, 69, 70, 74
 Create a stemandleaf plot for these ages.
 Create a dot plot for these ages.
 Describe the shape of this dataset.
 Are there any outliers in this dataset?
 Give an example in which the same measurement taken on the same individual would be considered to be an outlier in one dataset but not in another dataset.
 Does a stem and leaf plot provide enough information to determine if there are any outliers in the dataset? Explain.
 Does a five number summary provide enough information to determine if there are any outliers in the data set? Explain.

A set of 17 exam scores is 67, 94, 88, 76, 85, 93, 55, 87, 80, 81, 80, 61, 90 ,84, 75, 93, 75
 Draw a stemandleaf plot of the scores.
 Draw a dotplot of the scores.

Make a stem and leaf plot of the mean high temperature in December (Farenheit) in 15 cities in California. The “stem” gives the first digit of a temperature, while the “leaf” gives the second digit. You can find the data at:
http://countrystudies.us/unitedstates/weather/California/beverlyhills.htm)
 Describe the shape of the dataset. Is it skewed or is it symmetric?
 What is the highest temperature in the dataset?
 What is the lowest temperature in the dataset?
 What percent of the 15 cities have a mean high December temperature in the 60s?