- - - (1980). "We need both exploratory and confirmatory," The American Statistician, 34,
23-25.
Displays
John D. Emerson
Middlebury College
David C. Hoaglin
Harvard University and Abt Associates
The most common data structure is a b
structure of data may have characteristi
or studying the numbers. The stem-and
the numbers graphically in a way tha
features of the data. This basic but ve
most widely used; we call upon it in late
batches and to examine residuals.
The stem-and-Ieaf display enables us
notice such features as:
How nearly symmetric the batch is.
How spread out the numbers are.
Whether a few values are far remove
Whether there are concentrations of
Whether there are gaps in the data.
In exposing the analyst to these f
display has much in common with its clo
the digits of the data values themselves i
display offers advantages in some situat
easier to construct, and it takes a majo
IA. THE BASIC DISPLAY
The stem-and-leaf display (Tukey, 1970, 1972) provides a flexible and
effective technique for starting to look at a batch or sample of data. The
most significant digits of the data values themselves do most of the work of
sorting the batch into numerical order and displaying it.
To explain the display and how one constructs it, we begin with an
example. Royston and Abrams (1980) give the mean menstrual cycle length
and the mean preovulatory basal body temperature for 21 healthy women
who were using natural family planning. We reproduce these data in Table
1-1 and show one stem-and-leaf display for the cycle lengths in Figure 1-1.
If we follow the first data value, 22.9, we see that it appears in the display
as 2219. To construct this simplest form of stem-and-leaf display, we
proceed as follows. First we choose a suitable pair of adjacent digits in the
data-in the example, the ones digit and the tenths digit. Next we split each
data value between these two digits:
because the interaction between analyst and display is a personal one, we
face considerations of taste and esthetics, particularly in choosing the
number of intervals or the interval width. As often happens in exploratory
data analysis, we aim to provide flexibility through a number of variations.
We begin with the basic stem-and-leaf display and the steps in lts construction,
and then we add variations. For historical interest, we reproduce a
close predecessor of the stem-and-leaf display. We also briefly pursue the
connections with sorting to put the data in order. Choosing the number of
lines (or the interval width) provides a point of contact with work (some of
it more theoretical) on these choices for histograms.
26.6
26.8
26.9
26.9
27.5
27.6
27.6
28.0
28.4
28.4
28.5
28.8
28.8
29.4
29.9
30.0
30.3
31.2
31.8
3
4
5
6
7
8
9
IO
11
12
13
14
15
16
17
18
19
20
21
Source: J. P. Royston
(1980). "An objective me
shift in basal body tem
Biometrics, 36, 217-224 (
221, used by permission)
Then we allocate a separate line in the
leading digits (the stem)-for Figure 1-1
from 22 to 31. Finally we write down t
each data value on the line correspondin
The finished display, Figure 1-2, inclu
are in units of .1 day, as well as a colum
shortly) to the left of the stems. When t
sorted, the initial display will not usually
As an option, the final display can then9
leaf
and
and
22
stemsplit
~
221 922.9
data value
31 2 8 cycle length. 2 31 2 8
for mean menstru
automatically when a computer produces the display (Velleman and
Hoaglin, 1981). In overall appearance the display resembles a histogram
with an interval width of I day; the leaves add numerical detail, and in this
instance they preserve all the information in the data.
Figures I-I and 1-2 show that two-thirds of the women have mean cycle
length between 26.3 and 28.8 days. All but one of the other third have
longer mean cycle lengths. One woman, who appears unusual, has a mean
cycle length of only 22.9 days. We r~gard such an unusual data point as an
outlier, and we might well treat it separately from the other data points. If
we do, then a stem-and-leaf display for the remaining cycle lengths can give
more detail by choosing the stems differently. We return to this example in
Section lB.
Depths of Data Values
A data value can be assigned a rank by counting in from each end of the
ordered batch. For example, in Figure 1-2, 26.3 has rank 2 when counting
up from 22.9 and rank 20 when counting down from 31.8. The depth of the
data value is the smaller of these two ranks, 2 in the example. Because a
number of summary values (such as the median and the quartiles or fourths,
see Chapter 2) can easily be defined in terms of their depths, it is helpful to
present a set of depths with the display. Except for one middle line, the
number in the depth column is the maximum depth associated with data
values on that line. Thus the depth of 29.4 is 6.
The "middle line" includes the median, and the depth column shows in
parentheses the number of leaves on this line. In the example, this number is
6. (When the batch size is even and the median falls between lines, we do
not need this special feature.) If the display has been prepared by hand,
adding the count on the middle line an
lines provides a ~imf,lc check that no da
example, 9 + 6 + (, -::, 7.1, the total num
dentally, care is neethJ ."hen determinin
median: in the example 0nl: Iľight tend,
depth of 30.3.) Chapters :2. ~ Cij 3 usc t
summaries of data.
Organizing the Display
An effective choice of the number of
involves the number of data values in the
as well as some judgment. To get start
lines:
where n is the number of data values
exceeding x. This rule seems to give v
displays over the range 20 ~ n ~ 300, w
example, which has n = 21, it gives
L = [10 X 1
= [10 X 1
= 13.
Thus unless we decide to treat the value
the 1O-line stem-and-Ieaf display of Figu
way to do this uses a power of 10 as the interval width. (We discuss other
interval widths in Section IB.) Thus we divide R, the range of the batch, by
L and round the quotient up, if necessary, to the nearest power of 10. In the
example, the range R = 31.8 - 22.9 = 8.9 and L = 13, so that R/L = .68.
Rounding up to the nearest power of 10 gives the value 1 as the interval
width. This is the value used in Figures 1-1 and 1-2, which actually require
only 10 lines.
lB. SOME VARIAnONS
A segment of stems for the basic stem-and-leaf display might look like this:
digits that come after those that serve
low-order digits of data values are prefe
This practice makes it easier to recover
ing to a leaf in the display. Thus the
truncated at the decimal point and ap
display; the values at 55.7 are not roun
data values, we simply locate the three
first two digits are 55.
Sometimes the display is still too cr
too straggly with one line per stem at t
Data: Hardness of aluminum die
53.0 70.2 84.3 55.3 7
82.5 67.3 69.5 73.0 5
74.4 54.1 77.8 52.4 6
55.7 70.5 87.5 50.7 7
o
1
2
3
and each line may receive leaves 0 through 9. But sometimes this format is
too crowded, having too many leaves per line. One effective response is to
split lines and repeat each stem:
putting leaves 0 through 4 on the * line and 5 through 9 on the . line. In
such a display, the interval width is 5 times a power of 10.
EXAMPLE: HARDNESS OF ALUMINUM DIE CASTINGS
(II = 30)
Depths
11
(5)
\4
6
I
Displays:
(unit = 1)
50123 3 345 5 5 9
6 3 4 7 9 9
700 1 2 347 8
8 224 5 7
9 5
Shewhart (1931, p. 42) gives the hardness of 60 aluminum die castings. For
the first 30 of these, the data values and two stem-and-leaf displays appear
Figure 1-3. Splitting to get two lines per s
Economic Control of Quality of Manufactu
Nostrand, Inc. [data from Table 3 (specimen
Stem-and-leaf: Times at which patients' ob
Depths Censoring
Figure 1-4. A stem-and-leaf display with fiv
(1982). "Nonparametric estimation for parti
data," Biometrics, 38, 417-431 (data from Ta
devote Chapter 7) are centered at 0, and
signs. When we start to display such da
stem. The only tricky detail is that value
or both the - 0 stem and the +0 ste
roughly equally between the two.
To illustrate this variation, we use the
(by the "resistant line" method discu
2
3. 0
t 2
f 4
7
5
O. I
2
4
s 6
O 8
1.00
2
4
6
I ˇ 8
2. 0
9
7
5
I
10
6
18
30
37
(12)
34
26
20
15
12
II
EXAMPLE: TUMOR PROGRESSION IN PATIENTS WITH GLIOBLASTOMA
Dinse (1982) gives the times until tumor progression for 172 patients with
glioblastoma, a brain tumor. The times for 83 of the patients are censored
times in that the tumors for these patients had not progressed at the time
the study was concluded. These times are indicated by "+" in the data
portion of Figure 1-4, which also shows them in a stem-and-leaf display
with 5 lines per stem. The display has a total of 19 lines, and each line has
width 2 months. Note that for this example, L = [l010gI083] = 19 lines,
R = 37 - 1 = 36 months, and R/L = 36/19 = 1.89. When this quotient is
rounded up to I, 2, or 5 times the nearest power of 10, we obtain 2 X 100,
or 2, as the interval width. This is consistent with the width adopted for the
display.
In this display we immediately see the asymmetry of the main part of the
data, the clump of values from 30 months to 37 months, and the single value
at 25 months. The large number of small censoring times most likely
indicates that these patients have not been participating in the study long
enough to suffer a progression of their tumors. The practice of recording
these times in whole months, so that times within the first month are
recorded as 1, explains why there are no values at 0, and it may mean that
progression times and censoring times are both rounded up. The times at 25
months and beyond do not fit into a regular pattern with the rest of the
censoring times; they may represent an initial bulge in the rate of entry of
patients into the study.
with leaves 0 and 1 on the * line, 2 (two) and 3 (three) on the t line, 4 (four)
and 5 (five) on the f line, 6 (six) and 7 (seven) on the s line, and 8 and 9 on
the . line. As a reminder in starting to place leaves, the three lettered lines
contain leaves whose words begin with that letter. Here the interval width is
2 times a power of 10.
Another variation in the stem-and-leaf display accommodates data that
include both positive and negative values. Generally, residuals (to which we
Resistance
(un
low
26
27
28
29
30
31
Depths
(n = 21) (un
I low
2 26 *
6 26 .
27*
9 27'
(3) 28 *
9 28 .
6 29 *
5 29 .
4 30 *
30 .
2 31 *
I 31 ˇ
Depths
(n = 21)
I
6
9
(6)
6
4
2
One Line per Stem wit
Two Lines per Stem wi
Figure 1-6. Two stem-and-Ieaf dis
such lines in parentheses, as well as sep
mas. The first display in Figure 1-6 does
lines in Figure 1-2, we focus more attent
expense of no longer showing just how fa
Figure 1-5. Stem-and-Ieaf display of residuals from a
line for mean basal body temperature against mean
cycle length.
unit = .oloe
-3 0
-2 0
-13558
-0 0 2 4 7 9
o 3 679
I I 2 5
203 8
Figure 1-5 shows the stem-and-leaf display, with the residuals in units of
.01°C. We notice some tendency for the values to pile up around zero, and
the value at -.30 attracts some attention. Together with the one at +.28, it
probably deserves a closer look.
In Figure 1-5 and in other stem-and-leaf displays that involve both
positive and negative values, we have chosen, perhaps arbitrarily, to have
the data values increase from the top of the display toward the bottom. We
could equally well handle plus and minus by having the entries increase
from bottom to top, and others may prefer this direction. In anyone book
or paper, however, we usually adopt one of these directions.
Resistant methods are little affected by a small fraction of unusual data
values and are an important part of exploratory data analysis. Thus it is
unwise for the scale of a stem-and-Ieaf display to depend on the largest and
smallest data values. (The stem-and-leaf display of the mean cycle lengths in
Figure 1-2 is clearly influenced by the unusually small value at 22.9 days.)
Instead, we often begin by setting aside any unusual data values, and we
then base the choice of scale for the display on the rest of the data. (One
rule of thumb for setting aside low and high values appears in Section 2e.)
We list those outlying values on the lines labeled "low" and "high," beyond
the set of stems. To emphasize further the separate treatment of these
values, we may place parentheses around the lists. A comma after each
Depths
(n = 21)
I
2
6
(5)
10
6
3
Ie. AN mSTORICAL NOTE
OIJIJO 1/ J,.. ~:13 ~N'
0 /I J.JoJ- .3
'"J
I 3
Note: When necessary, allow two o
hand column.
Figure 1-7. Dudley's transcription of data i
tion of Industrial Measurements by John W
McGraw-Hill Book Company, Inc. Used w
Company.
The role and development of graph
ceived considerable attention in recent y
contributed several novel displays. Fienb
use of graphical methods and includes
tions. [See also Wainer and Thissen (198
L
and we round R/L up to .5-a display with two lines per stem. The second
part of Figure 1-6 shows the result, which probably has too many lines for
some viewers. Both displays list the 22.9 separately, and the choice between
them is a matter of taste.
With its variations, the stem-and-leaf display has proved to be a versatile
technique for the analysťs first look at a batch of numbers. The three ways
of factoring 10 (1 X 10, 2 X 5, and 5 X 2) provide adequate control over
scaling (1,2, or 5 lines per stem) especially when combined with "low" and
"high" lines for unusual data values. Setting aside potential outliers in this
way often leads to a more detailed and more effective display; it also
focuses attention on the unusual data values so that we do what we can to
probe their surrounding circumstances for clues.
The histogram, the better known relative of the stem-and-leaf display, has
been in use for many years. Beniger and Robyn (1978) trace the origin of
the term to Karl Pearson in 1895, and they mention earlier examples going
back to the bar chart published by William Playfair in 1786 in his Commercial
and Political Atlas.
Of course, the primary difference between the histogram and the stemand-leaf
display is the use of a digit from each data value to form the
display. Without any systematic search for possible predecessors of the
stem-and-leaf display, we report finding a digit-based display (Figure 1-7) in
the text by Dudley (1946, p. 22). In the way it groups matching "leaves"
together and spreads the groups out over the line, this technique serves more
as a sorting and tallying device than as a semigraphic display. Still, its
similarity to the stem-and-leaf display emphasizes the convenience of working
with digits from the data values.
ID. SORTING
Because many exploratory techniques wo
the simplest way of gaining resistancesion
of sorting, the mechanical process o
(usually, from smallest to largest). Re
details may skip this section without los
In hand work, as we mentioned in Se
leaf display accomplishes much of the
remains is to rearrange the leaves on ea
"bucket sort" or "radix sort," with
"buckets." The skeleton structure of stem
the data establishes the range, and then