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Stem & Leaf Plots Extended for Text Visualizations
Richard Brath
Computer Science and Informatics
London South Bank University & Uncharted Software
Toronto, Canada
richard.brath@alumni.utoronto.ca
Ebad Banissi
Computer Science and Informatics
London South Bank University
London, U.K.
banisse@lsbu.ac.uk
Abstract—Stem and leaf plots are data dense visualizations
that organize large amounts of micro-level numeric data to
form larger macro-level visual distributions. These plots can be
extended with font attributes and different token lengths for
new applications such as n-grams analysis, character
attributes, set analysis and text repetition.
Keywords - stem & leaf, text visualization, exploratory data
analysis.
I. INTRODUCTION
Stem and leaf plots were described by John Tukey [1]
and popularized by Edward Tufte in The Visual Display of
Quantitative Information [2]. The stem and leaf plot provides
a macro-level visual distribution and a micro-level view of
the individual data points. Identified benefits [3] of stem and
leaf displays include 1) more information is retained than a
bar chart; 2) reveals fine structure while showing the
distribution and 3) allows easy hand-calculation of measures
based on ordered values (e.g. median, quartiles). Cox also
points out limitations including: 1) problems with large
datasets, 2) whether extra digits are useful to the task and 3)
comparison can be awkward.
While the basics of stem and leaf plots may be known to
the visualization community (e.g. [4,5]), the contribution of
this paper is to:
1) Collect existing extensions;
2) Identify extensions and applications to text
visualization.
3) Extend stem and leaf plots of text to operate with
tokens of individual characters, words or phrases.
II. BACKGROUND
Stacked alphanumeric values to indicate distributions of
data predate their use by Tufte and Tukey. Modern uses of
stem and leaf plots in the wild indicate many extensions
including the use of colour (fonts, backgrounds) or added
indicators such as shading, lines or markers.
A. Statistics
Historic examples of stacked alphanumerics forming
distributions occur prior to Tukey. For example, in this 1937
chart (a subset shown in fig. 1, from [6]), individual U.S.
states are represented by numbers and stacked according to
percent of the population receiving debt relief during the
Figure 1. Portion of a 1937 chart showing percent of population per U.S.
state receiving relief during the Great Depression.
great depression (note that 2-letter state codes did not exist
until 25 years later).
The numerics denoting each state are surrounded by
circles: the circles make both one digit and two digit glyphs a
consistent size; facilitate disambiguation in a sequence of
numbers; and make the number of items within a stack more
visually apparent than a string of digits. The diagonal line
texture denotes the inner quartiles (i.e. the centre half of the
population) and the arrow indicates the median.
Stem and leaf plots in statistics have evolved with
various added features. Fig. 2 (from [7]) shows two parallel
stem and leaf plots indicating two subpopulations (e.g. male
vs. female). On the left distribution, the inner set of numbers
adjacent to vertical line indicate the stem on the left (i.e. the
most significant digits) and the next significant digit to the
right, with each successive digit indicating an additional
observation. For example 15|459 indicates data observations
with values of 154, 155 and 159 (e.g. the heights of three
individuals observed). Visually scanning the distribution
Figure 2. Paired stem and leaf plots with additional metrics.
Figure 3. Market profile charts from various software providers. Stem
represents discrete prices, while letters indicate times during which a trade
occurred at that price level.
Figure 4. Stem and leaf plot of train times. Note text colour, background
colour, background shape, added dots and outline square indicate
attributes of trains
shows that the longest bar is beside the stem 18, indicating
that observations between 180-189 occur most frequently.
To the left of the stem is a second set of numbers
indicating the cumulative count of members from the
extremes to the median denoted in round brackets [8], where
the bracket number indicates the number of observations at
the median. Note that the second parallel distribution is
aligned to the same vertical axis as the first distribution. This
facilitates comparison between the two populations. Note
that the two stem and leaf plots are aligned to the same
vertical axis and thus stem labels are indicated only in the
left stem and leaf plot.
B. Finance
In the late 1800’s alphanumeric charting techniques
began as a matrix of prices over time initially as figure charts
[9,10] and evolved into point and figure charts, using X’s to
indicate rising prices, O’s to indicate falling prices. Coloured
characters predate the use of computers, e.g. Livermore [11]
used colour pencils to record price levels in his alphanumeric
charts.
Market profile charts (fig. 3) closely resemble stem and
leaf plots with stems representing discrete price levels and
leaf characters indicate the time of day that a commodity
trades at that level. A common encoding uses A-X, a-x to
indicate half hour intervals starting at midnight with
uppercase indicating trades in the morning, lowercase for
afternoon. Characters are aligned vertically by price and
stacked horizontally forming a histogram, enabling a macro-
reading (the distribution) and a micro-reading (the individual
characters).
Many organizations provide market profile software (e.g.
CBOT, CQG, TradeStation, Reuters, ProRealTime, etc) with
many variants used to differentiate individual characters,
rows and ranges of rows. These include, per character: 1)
character colour, 2) background colour, 3) upper/lowercase;
per row: 4) background line, 5) background outline (e.g.
rounded rectangle), 6) added mark (e.g. coloured triangle at
start or end of row; or a glyph such as >); per vertical range,
such as 7) shaded background, 8) line beside representing
extent of range; and separately 9) an added distribution
representing a second metric either back-to-back or side-by-
side.
C. Timetables
One popular modern use of stem and leaf plots is
timetables, such as commuter rail schedules. The stem
indicates the hour of departure and each leaf indicates the
number of minutes past the hour. These timetables may have
additional information encoded for each specific train by
providing additional graphical attributes per leaf. In fig. 4,
information is added by the font colour; background shading
and shape of shading (e.g. box, bar, triangle) underneath the
leaf; a blue dot above the leaf; or an outline around the leaf.
Note how legibility can be reduced with particular
combinations, such as red text over an orange bar: the lack of
contrast between the red and orange and the overlapping
shapes interfere with the legibility of the text.
III. EXTENSIONS TO TEXT VISUALIZATION
Stem and leaf plots can be extended to text visualization.
There are several potential enhancements:
A) Font attributes: In addition to visual attributes such
as colour, typographic attributes such as bold, italic,
underline and so forth, can be used to add data. The use of
font attributes has been discussed earlier in [12]. The
contribution of this paper is to extend font attributes to stem
and leaf plots.
B) Token - character, word or phrase: Numerically
oriented stem and leaf plots typically use one or two
characters. However, in the context of text visualization, the
scope of the textual unit (i.e. token) can vary depending on
the application: e.g. individual characters, words, or phrases.
Figure 5. Stem and leaf plot with statistical values indicated via font
formats.
Figure 6. Bigrams in the English language. Font weight indicates
frequency of occurence.
Figure 7. U.S. States: stem indicates poverty rate, font weight indicates
life expectancy and colour indicates murder rate.
A key contribution of this paper is the extension of stem and
leaf plots to text visualization where the tokens may be at the
level of characters, words or phrases.
A. Font Attributes for Statistics
In the earlier stem and leaf plots shown, additional data is
encoded into the display using attributes such as foreground
colour, background colour, background shape and dots as
shown in figure 4. Instead, font attributes can be used. Figure
5 shows the plot of mountain heights (from [2]), with
quartiles in bold, median in bold italic; standard deviation
with underline, and mean with underline italic.
B. Character Examples: Letter frequencies and state stats
Instead of stems and leaves representing numerical
values, a simple extension for text analytics is to use stems
and leaves to represent alphabetic values. Bigrams (more
generally ngrams) are sequences of adjacent letters used to
provide the conditional probability of a token given the
preceding token. Frequency of bigrams can be used for
statistical language identification, prediction for auto-
completion and cryptography.
Figure 6 shows English language bigrams that occur
more than 0.5% of the time based on bigrams calculated
from the Leipzig Corpora Collection (corpora.uni-
leipzig.de). The stem indicates the first letter of the bigram,
the leaf indicates the second letter. Font weight indicates the
bigram frequency, e.g. TH is among the most frequent
bigrams in English.
In this case, the stem and leaf approach is used as a
layout to organize the first and second token. Leaf stack
length indicates the frequency of the initial token, e.g.
bigrams starting with E are most common in bigrams
occurring more than 0.5%. Additional data has been added
by the font weight, in this case, indicating the frequency
associated with each specific bigram.
Leaves do not need to be restricted to a single
alphanumeric character. Figure 7 shows U.S. states with
stem indicating poverty rate and additional data via hue and
font weight. Multi-attribute correlations are visible, e.g.
higher murder rates (red) and lower life expectancy (non-
bold) are associated with higher poverty rates (top portion of
plot).
C. Word Examples: Character Traits and Families
Leaves do not need to be restricted to tokens of the same
length, e.g. words can be used instead. Figure 8 illustrates a
character trait analysis by identifying adjectives that occur
within +/- 3 words from a character in Grimms’ Fairy Tales,
with the stem indicating the character and the leaves
indicating descriptors. Adjectives are ordered left to right
based on frequency with font weight indicating the level of
frequency (note that kings tend to be old while princesses are
beautiful).
Figure 8. Adjectives associated with chaacters from Grimms’ Fairy Tales
Figure 9. Titanic third class families. Sterm indicates surname, leaf for given
name, bold indicates death, italics for women, allcaps for children.
Figure 10. Titanic first class families, women left, men right.
Figure 11. Closeup of stock performance for 150 industries: see fig. 14
for full view.
One challenge with variable length tokens is spacing: if
the space provided per word is based on the longest word,
then there will be a lot of wasted whitespace. Also, uneven
whitespace across a string of adjectives is more difficult to
read than word spacing based on typical text spacing (e.g.
see [13]). Here, words are placed in sequence with an
expected single space between words. A horizontal scale
indicates the approximate number of words, based on
average word lengths in the plot. Longer word lists should
experience reversion to the mean: at the grid line for 10
average words, the number of words for king is
approximately 9.3, for princess 9, for wife 10 - i.e. an error
rate of only 10% in this example.
Another challenge is finding multiple visual attributes
that can be combined together and remain legible in any
combination (e.g. orange bars in fig. 4 interfere with text
legibility). Font attributes can be combined together while
retaining legibility [12]. Figure 9 uses font attributes on a
subset of third class Titanic passengers. The stem indicates
surname and the leaf indicates given name. In this particular
example, 1) font weight represents survival: e.g. bold
indicates death, 2) italics represent gender: e.g. italic
indicates female, 3) capitalization represents age: e.g. all
caps indicates children, and 4) font family represents class: a
plain font for third class, a serif font for first class (fig. 10).
Since there are many binary attributes indicating
membership for different sets, back-to-back stem-and-leaf
plots can be used to more clearly show membership for a
specific attribute. For example, fig. 10, shows a subset of
first class families, with females on the left and males on the
right, clearly indicating higher survivorship among women
(i.e. there is more bold on the right side of the plot).
Although there are survivors among the first class men,
capitalization reveals that all the dead men are adults
(“women and children first”).
The greater proportion of heavyweight font for third class
passengers (fig. 9) vs. first class passengers (fig. 10) is
clearly visible: far fewer first class passengers die. Similarly
only one first class child (all caps) death is visible in the
subset shown while many of the third class deaths are
children (allcaps bold).
D. Phrase Examples: Sectors and Companies
Figure 11 shows a stem and leaf plot for the performance
of 500 stocks aggregated into 150 different industries. In this
example, a different strategy is used for variable label length:
instead of placing each word in sequence, a fixed width is
provided for each label. Long labels are compressed using a
narrow version of the particular font with narrow inter-
character spacing (i.e. tracking), while short labels use a
wide version of the same font with a wide inter-character
spacing.
Another approach to scalability with long names and
many items is to rotate the stem and leaf plot 90 degrees as
shown fully in fig. 15 and closeup in fig. 12. This plot shows
a horizontal distribution of earnings performance of 500
companies. In this orientation, the plot more closely
resembles a bar chart. There is no layout error in counts as
the height is consistent for each item. Phrase length is
irrelevant – some phrases can be short (e.g. The Gap) and
some can be long (e.g. Molson Coors Brewing Company). In
Figure 12. Distribution of companies by performance. Stack height
indicates number of companies within a range of performance.
Figure 13. Common phrases in the Book of Psalms (centerline) with
preceding phrases above, following phrases below.
this example, font color indicates sector (e.g. Tech,
Financial, Retail), font weight indicates stock trading
volume.
The horizontal approach can be extended to longer
phrases. Figure 13 and 16 are subsets of a visualization of
common phrases repeated in the Book of Psalms from The
King James Version of the Bible (www.gutenberg.
org/ebooks/10). The source text is split on punctuation marks
into phrases. Commonly repeated phrases are shown on the
centerline. Phrases immediately prior the common phrase are
above the centerline, phrases immediately following are
below. Font weight indicates frequency.
For example, the phrase “O give thanks unto the Lord”,
is very commonly preceded by “Praise ye the Lord”,
although on one occasion is preceded by “I will exalt thee”.
Notice the common phrase “I will praise thee” is frequently
followed by “O Lord”, while on one occasion it is followed
by “O Lord my God” – one may wonder if the latter is a
transcription anomaly and then investigate the original
document.
IV. CONCLUSIONS
The contribution of this paper includes many novel
variants of stem and leaf plots, including 1) use of font
attributes to indicate data; 2) text markers to indicate either
categoric or numeric values for either stems or leaves; 3) text
markers which range from single character, to words to
phrases; 4) horizontal and vertical orientations.
Applications include text analytics such as n-gram
analysis, character analysis and set analysis.
One issue identified is variable length leaves, and
multiple strategies were identified: 1) tokens packed together
with a horizontal axis based on average token size; 2)
consistent token size and use of multiple font widths to
accommodate for variance in token width; and 3) a vertical
orientation so token width is not relevant.
None of the examples here discuss interaction. The
interaction of stem and leaf plots with other well-known
visualization techniques (e.g. linked interaction [14]) or
emerging visualization techniques (e.g. object constancy [15]
or sedimentation [16]) should be considered.
Scalability is not addressed, although fig. 15 shows
hundreds of phrases in a stem & leaf plot while retaining
legibility, and implies thousands of characters can be
depicted legibly. Interactive techniques such as zooming and
tooltips could allow for much larger stem and leaf plots
showing macro patterns zoomed out and details on
interaction.
Finally stem and leaf plots are less common than other
techniques such as histograms and scatterplots and more
prone to errors in interpretation (e.g. [17]). Some of the
techniques shown here could be utilized to improve
interpretation for novice users, for example, redundant
encoding of the primary measure of the distribution using
color or font-weight could be evaluated to determine
potential improvement in performance.
REFERENCES
[1] J. Tukey. “Some graphic and semigraphic displays.” Statistical Papers
in Honor of George W. Snedecor 5. 1966.
[2] E. Tufte The Visual Display of Quantitative Information. Graphics
Press, 1983.
[3] N. Cox. “Speaking stata: Turning over a new leaf.” The Stata Journal.
2013.
[4] J. Heer, M. Bostock, V. Ogievetsky. “A tour through the visualization
zoo.” Communications of the ACM 53, 6. 2010, 59–67.
[5] L. Wilkinson. The Grammar of Graphics, Second Edition. Springer,
2005.
[6] W. C. Brinton. Graphic Presentation. Brinton Associates, 1939.
[7] J. D. Emerson, D. C. Hoaglin. “Stem-and-leaf displays.”
Understanding robust and exploratory data analysis. 1983. 7–32.
[8] I. Salgado Ugarte. “Exploratory analysis of some measures of the
asymmetric otoliths of stone flounder kareius bicoloratus (pisces:
Pleuronectidae) in Tokyo bay.” Anales del Instituto de Ciencias del
mar y limnología. 18 (2). 1991, 261-278.
[9] Hoyle: The Game in Wall Street. Ogilvie Publishing, New York, NY,
1898.
[10] Wyckoff R.: Studies in Tape Reading. Noble Offset Printers, New
York, NY, 1910.
[11] J. Livermore. How to Trade in Stocks. Quinn & Boden Company,
Rahway, NJ, 1940.
[12] R. Brath, E. Banissi. “The design space of typeface.” In Visualization
and Computer Graphics, IEEE Transactions on. 2013.
[13] W. Tracy. Letters of Credit. Letters of Credit: A View of Type
Design. Jaffrey, NH: David R. Godine Publisher, 2003.
[14] R. A. Becker and W. S. Cleveland. “Brushing scatterplots.”
Technometrics 29, 2. 1987. 127–142.
[15] J. Heer and G. Robertson. “Animated transitions in statistical data
graphics.” Visualization and Computer Graphics, IEEE Transactions
on 13, 6. 2007. 1240–1247.
[16] S. Huron, R. Vuillemot, J.D. Fekete. “Visual Sedimentation.” IEEE
Transactions on Visualization and ComputerGraphics. Nov. 2013.
[17] R. S. Baker, A. T. Corbett, K. R. Koedinger. “Toward a model of
learning data representations.” Carnegie Mellon University Research
Showcase, 2001.
Figure 14. Distribution of stock market returns across 150 different industries.
Earnings Surprise for the Standard & Poors 500 Largest U.S. Stocks by Stock Market Capitalization
Horizontal bins indicate magnitude of earning surpse, colour indicates sector, font weight indicates trading volume
Figure 15. Distribution of earnings surprise for 500 stocks.
Figure 16. Subset of common phrases in the Book of Psalms (centerline), with preceding phrases (above) and following phrases (below). Phrase frequency
indicated by font weight.
