Stem and Leaf Displays2.112001/06/292003/07/18 15:00:42.865 GMT-5DavidLanelane@rice.eduDavidLanelane@rice.eduAdanGalvanjago@rice.eduEileenMeyermeyer@rice.edustem and leaf plotsstatisticsIntroduction to stem and leaf plots.
A stem and leaf display is a graphical method of
displaying data. It is particularly useful when your data are
not too numerous. In this section, we will explain how to
construct and interpret this kind of graph.
As usual, an example will get us started. Consider . It shows the number of
touchdown (TD) passes Touchdown Pass: In
American football, a touchdown pass occurs when a completed pass
results in a touchdown. The pass may be to a player in the end
zone or to a player who subsequently runs into the end zone. A
touchdown is worth 6 points and allows for a chance at one (and
by some rules two) additional point(s). thrown by each
of the 31 teams in the National Football League in the 2000
season.
Number of touchdown passes.
A stem and leaf display of the data is shown in the below. The left
portion of the table contains the stems. They are the numbers
3, 2, 1, and 0, arranged as a column to the left of the bars.
Think of these numbers as 10's digits. A stem of 3 (for
example) can be used to represent the 10's digit in any of the
numbers from 30 to 39. The numbers to the right of the bar are
leaves, and they represent the 1's digits. Every leaf in the
graph therefore stands for the result of adding the leaf to 10
times its stem.
Stem and leaf display showing the number of passing
touchdowns.3|23372|0011122238891|22444568888990|69
To make this clear, let us examine this more closely. In the top row, the
four leaves to the right of stem 3 are 2, 3, 3, and 7. Combined
with the stem, these leaves represent the numbers 32, 33, 33,
and 37, which are the numbers of TD passes for the first four
teams in the table. The next row has a stem of 2 and 12 leaves.
Together, they represent 12 data points, namely, two occurrences
of 20 TD passes, three occurrences of 21 TD passes, three
occurrences of 22 TD passes, one occurrence of 23 TD passes, two
occurrences of 28 TD passes, and one occurrence of 29 TD passes.
We leave it to you to figure out what the third row represents.
The fourth row has a stem of 0 and two leaves. It stands for the
last two entries, namely 9 TD passes and 6 TD passes. (The
latter two numbers may be thought of as 09 and 06.).
One purpose of a stem and leaf display is to clarify the shape
of the distribution. You can see many facts about TD passes
more easily in than in the
. For
example, by looking at the stems and the shape of the plot, you
can tell that most of the teams had between 10 and 29 passing
TDs, with a few having more and a few having less. The precise
numbers of TD passes can be determined by examining the leaves.
We can make our figure even more revealing by splitting each
stem into two parts. The below shows how to do this. The top
row is reserved for numbers from 35 to 39 and holds only the 37
TD passes made by the first team in the . The second row is reserved for
the numbers from 30 to 34 and holds the 32, 33, and 33 TD passes
made by the next three teams in the table. You can see for
yourself what the other rows represent.
Stem and leaf display with the stems split in two.3|73|2332|8892|0011122231|568888991|224440|69
The with stem
and leaf split in two is more revealing than the simpler before because the
simpler table lumps too many values into a single row. Whether
you should split stems in a display depends on the exact form of
your data. If rows get too long with single stems, you might
try splitting them into two or more parts.
There is a variation of stem and leaf displays that is useful
for comparing distributions. The two distributions are placed
back to back along a common column of stems. The result is a
back to back stem and leaf graph. The below shows such a
graph. It compares the numbers of TD passes in the 1998 and
2000 seasons. The stems are in the middle, the leaves to the
left are for the 1998 data, and the leaves to the right are for
the 2000 data. For example, the second-to-last row shows that
in 1998 there were teams with 11, 12, and 13 TD passes, and in
2000 there were two teams with 12 and three teams with 14 TD
passes.
Back to back stem and leaf display. The left side shows
the 1998 TD data and the right side shows the 2000 TD
data.19982000114373323233886528894433111020011122239877766651568888993211224447069
This helps us
see that the two seasons were similar, but that only in 1998 did
any teams throw more than 40 TD passes.
There are two things about the football data that make them easy
to graph with stems and leaves. First, the data are limited to
whole numbers that can be represented with a one-digit stem and
a one-digit leaf. Second, all the numbers are positive. If the
data include numbers with three or more digits, or contain
decimals, they can be rounded to two-digit accuracy. Negative
values are also easily handled. Let us look at another example.
shows data from a study
on aggressive thinking. Each value is the mean difference over
a series of trials between the time it took an experimental
subject to name aggressive words (like "punch") under two
conditions. In one condition the words were preceded by a
non-weapon word like "rabbit" or "bug." In the second
condition, the same words were preceded by a weapon word such as
"gun" or "knife." The issue addressed by the experiment was
whether a preceding weapon word would speed up (or prime)
pronunciation of the aggressive word, compared to a non-weapon
priming word. A positive difference implies greater priming of
the aggressive word by the weapon word. Negative differences
imply that the priming by the weapon word was less than for a
neutral word.
The effects of priming (thousandths of a second).
You see that the numbers range from 43.2 to -27.4. The first
value indicates that one subject was 43.2 milliseconds faster
pronouncing aggressive words when they were preceded by weapon
words than when preceded by neutral words. The value -27.4
indicates that another subject was 27.4 milliseconds slower
pronouncing aggressive words when they were preceded by weapon
words.
The data are displayed with stems and leaves in the . Since stem and
leaf displays can only portray two whole digits (one for the
stem and one for the leaf) the numbers are first rounded. Thus,
the value 43.2 is rounded to 43 and represented with a stem of 4
and a leaf of 3. Similarly, 42.9 is rounded to 43. To
represent negative numbers, we simply use negative stems. For
example, the bottom row of the figure represents the number -27.
The second-to-last row represents the numbers -10, -10, -15,
etc. Once again, we have rounded the original values from .
Stem and leaf display with negative numbers and rounding4|333|562|004561|001340|1245589-0|0679-1|005559-2|7
Observe that the figure contains a row headed by "0" and another
headed by"-0". The stem of 0 is for numbers between 0 and 9
whereas the stem of -0 is for numbers between 0 and -9. For
example, the fifth row of the table holds the numbers 1, 2, 4,
5, 5, 8, 9 and the sixth row holds 0, -6, -7, and -9. Values
that are exactly 0 before rounding should be split as evenly as
possible between the "0" and "-0" rows. In , none of the values are 0 before
rounding. The "0" that appears in the "-0" row comes from the
original value of -0.2 in the figure.
Although stem and leaf displays are unwieldy for large datasets,
they are often useful for datasets with up to 200 observations.
portrays the distribution of
populations of 185 US cities in 1998. To be included, a city
had to have between 100,000 and 500,000 residents.
Stem and leaf display of populations of US cities with
populations between 100,000 and 500,000.
Since a stem and leaf plot shows only two-place accuracy, we had
to round the numbers to the nearest 10,000. For example the
largest number (493,559) was rounded to 490,000 and then plotted
with a stem of 4 and a leaf of 9. The fourth highest number
(463,201) was rounded to 460,000 and plotted with a stem of 4
and a leaf of 6. Thus, the stems represent units of 100,000 and
the leaves represent units of 10,000. Notice that each stem
value is split into five parts: 0-1, 2-3, 4-5, 6-7, and 8-9.
Whether your data can be suitably represented by a stem and leaf
graph depends on whether they can be rounded without loss of
important information. Also, their extreme values must fit into
two successive digits, as the data in fit into the 10,000 and 100,000 places (for
leaves and stems, respectively). Deciding what kind of graph is
best suited to displaying your data thus requires good judgment.
Statistics is not just recipes!