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The task of devising and interpreting tables is an integral part of statistical practice. Yet tables receive little attention as a topic of statistical research and statistical education. This neglect seems to be reflected in the quality of the tables that accompany scientific and nonscientific presentations. This article argues that in statistics tables are important tools for communicating information, and hence should receive more attention in statistical research, education, and practice. I discuss basic tables, Graphically enhanced tables and, in the context of OLAP, dynamic tables. Examples supporting the case should be of interest to statistical practitioners and educators alike.
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A Case for Simple Tables
The task of devising and interpreting tables is an integral part of
statistical practice. Yet tables receive little attention as a topic of
statistical research and statistical education. This neglect seems
to be reflected in the quality of the tables that accompany sci-
entific and nonscientific presentations. This article argues that
in statistics tables are important tools for communicating infor-
mation, and hence should receive more attention in statistical
research, education, and practice. I discuss basic tables, graph-
ically enhanced tables and, in the context of OLAP, dynamic
tables. Examples supporting the case should be of interest to
statistical practitioners and educators alike.
KEY WORDS: Data analysis; OLAP; Statistical graphics.
A statistical analysis starts with exploration and ends with
presentation. Exploration includes gaining an understanding of
the data sources, inspecting the data for data integrity as well
as unusual features that can guide the development of a formal
analysis, and staging the data for further analysis. Presentation
includes communicating the findings of the analysis to its in-
tended audience.
Tables and graphs figure prominently in both instances.
George Box’s (1988) dictum still holds true: “A first analysis of
experimental results should, I believe, invariably be conducted
using flexible data analytical techniques—looking at graphs and
simple statistics—that so far as possible allow the data to ‘speak
for themselves’. The unexpected phenomena that such an ap-
proach often uncovers can be of the greatest importance in shap-
ing and sometimes redirecting the course of an ongoing inves-
tigation. This is indeed the course of action initially taken by
most statistical practitioners. In the end, academic as well as
nonacademic articles and reports that communicate the findings
of statistical analyses usually include tables and graphs.
The benefits of the graphical display of information are widely
acknowledged, and statistical graphics has been an area of ac-
tive and lively research for many years. A substantial bibli-
ography of academic research articles, a dedicated major re-
search journal—namely the Journal of Graphical and Computa-
tional Statistics—and books (Tukey 1977; Mosteller and Tukey
1977; Chambers, Cleveland, Kleiner, and Tukey 1983; Cleve-
land 1994) that by now are seminal texts in the field of statistics,
Martin Koschat is Executive Vice President, Information Management, Time/
Warner Retail Sales and Marketing, Sports Illustrated Building, Seventh Floor,
135 West 50th Street, New York, NY 10020-1201 (E-mail: martin
koschat@ It is a pleasure to acknowledge the thoughtful comments and
helpful suggestions received from Andreas Buja, Anita Hussey, Naomi Robbins,
Raja Velu, the editor, associate editor, and two anonymous reviewers.
attest to a robust history and the continued vitality of statistical
graphics as an area of research.
In the statistical community, the benefits of tables as a format
for displaying information are acknowledged less enthusiasti-
cally. It would seem that the value of tables as a format for
displaying information is recognized mostly indirectly through
the frequency of use. Tables are indeed widely used, and in many
statistical reports and research papers more space is devoted to
tables than graphs. Yet this obvious prominence is barely re-
flected in the broader statistical discourse. Tables as displays of
information are rarely a topic in statistical education, are rarely
a point of discussion in statistical practice and, within the field
of statistics, do not appear to receive much, if any, attention from
An early and noteworthy contribution on the design of sta-
tistical tables to the statistical literature is Walker and Durost’s
(1936) text, considered by many to be a minor masterpiece.
Since then it would appear that scholarly articles on this topic
have been published mostly by a small handful of researchers,
notably Ehrenberg (1981, 1986), Wainer (1992, 1993, 1997a,
1998), and Tufte (2003). The latter two authors also acknowl-
edge tables as a valuable format for communicating information
in their books (Tufte1983, 1990, 1997; Wainer 1997b), but in the
statistical community these texts are mostly referred to for their
consideration of statistical graphics. Outside statistics, books on
document design occasionally have sections on the construction
of tables (Bigwood, Spore, and Seely 2003; Harris 2000; Shriver
The purpose of this article is to address the status quo and
contribute to a broader discussion on the value of tables and the
attention that tables should receive in education, practice, and
research. I perceive there to be a need, as well as an opportunity,
and will elaborate on both aspects in the next few sections. This is
obviously an ambitious objective fraught with some uncertainty
as to its outcome. A a more modest objective is the hope that
statistical practitioners and educators will find useful ideas in
the examples presented.
The prescription for the design of a table is straightforward.
Arrange numbers—and it is usually numbers—in parallel rows
and perpendicular columns.Atableisasimple structure for
arranging numbers. These two defining attributes—the well-
known structure and the use of numbers—also provide the major
rationale for using tables.
As to the structure of tables, it is probably moot to speculate
whether tables’ simple ingenuity is responsible for their wide
use or whether it is tables’ wide availability and use that have
created familiarity and wide acceptance. The fact remains that
even users untrained in any quantitative discipline readily com-
prehend that there is an implied commonality to all the numbers
in the same row, often pertaining to a “case, and all the num-
bers in the same column, often with the meaning of a “variable.
© 2005 American Statistical Association DOI: 10.1198/000313005X21429 The American Statistician, February 2005, Vol. 59, No. 1 31
Figure 1. Estimated revenue (in $U.S. ,000) in total (Total) and by major source, advertising (Ad) and circulation (Circ), for the 20 leading U.S.
consumer magazines for 2000 and 2001 as well as year-over-year (YOY) changes. Source: Advertising Age ( Generating Tool:
Microsoft Excel.
This universal understanding of a table’s structure, shared by few
other statistical constructs, is a simple but powerful argument for
using tables.
The act of displaying numbers often gets a bad rap. On the
one hand, there is the notion that the sensibilities of the quanti-
tatively challenged need to be protected; this is often a concern
in business research. On the other hand, analysts believe that
an analysis that starts with basic numbers necessarily needs to
yield something more profound than numbers; this is a notion
that is often encountered in the sciences. There are, however,
good arguments for using numbers in displays, and we mention
A numerical display often presents data in their original form
and subjects them to minimal intervention from an analyst pur-
suing a directed investigation. An analysis of data is, of course,
nothing but a directed investigation, and a thorough analysis may
well require the transformation of original data through graph-
ical constructs or the application of a sophisticated mathemati-
cal model. Although such transformations have significant and
well-known benefits, the appreciation for these benefits needs
to be balanced by a consideration for the bias that such trans-
formations may introduce and that is usually transparent to the
end-user of an analysis. The outcome of an analysis is as much
driven by the analytical assumptions and choices as by the data.
At a minimum, there is a distinct benefit to complementing the
presentation of a formal analysis by an informative presentation
of the underlying data in their original form or a meaningful and
unambiguous numerical summary of such data.
Data presented in numerical form can also be easily manip-
ulated and transformed. The reader can easily take the data as
input for a graph or a formal model of her choosing. Numerical
displays support and encourage such a flow. It is easy to translate
numbers into graphs or parameters of a model. It is much harder,
if not impossible, to take the reverse route.
Finally, numbers often—not always—require less of an ex-
planation than glyphs or modeled constructs. In many areas of
statistical application, such as business, analysts and analyti-
cal clients can easily put the number presented to them into
a context where the number has some immediate meaning. In
business, it helps, of course, that the raw numbers encountered
usually fall into a few standard and well-understood categories.
Further, business people often “live” their numbers, particularly
if these happen to be sales numbers. Not only do these people
fully comprehend the meaning of a sales number, they want to
see it!
I do not wish to suggest that the presentation of numerical
information is uniformly better than a graphical display or the
presentation of a formal analysis. I do want to emphasize, how-
ever, that there are often good reasons for displaying numerical
information in a simply structured format even if only to com-
plement graphs or to support the communication of model-based
In many computer programs for statistical analysis, raw and
derived data tend to be organized in rows and columns, thus nat-
urally forming tables. Analysts often succumb to the temptation
to grab such tables in their original form and to drop them into a
document or a presentation without further concern for making
appropriate adjustments. The results of this approach are often
painful to behold as Figure 1, a table taken from an actual pre-
32 General
Figure 2. The same three numbers are arranged as a row and as a
column. The column presentation is better suited to rank-order the three
sentation to the senior management of a major U.S. publishing
house, shows.
This table attempts to compare changes in the revenue break-
down of the 20 leading consumer magazines in the U.S. for two
consecutive years. The table fails on several basic dimensions.
For the table to be an effective display of data, its entries must be
easy to compare. Also, entries and comparisons that are of spe-
cial interest should be prominently displayed and easy to find.
Neither is the case in Figure 1. This simple observation leads
to a few basic principles useful in the construction of informa-
tive tables. These principles pertain to the choice of rows and
columns, the arrangement of rows and columns, the presenta-
tion of numbers, and the use of simple graphical elements for
structuring the display.
An application of these principles resulted in the table in Fig-
ure 3. This table now effectively shows the gestalt of the market
comprised of the 20 magazines. It quickly communicates the rel-
ative size of each magazine and the relative contribution of each
source to total revenues. Most importantly, it clearly commu-
nicates that 2001 was a difficult year for magazine publishing,
with the difficulty mostly due to a softening of the advertising
We use Figure 3 to briefly discuss each of the guiding princi-
3.1 Choice of Columns and Rows
A first step in the construction of a table requires a decision
on which entries to arrange in rows and which entries to arrange
in columns. In general, numerical comparisons are easier made
within columns than within rows. Figure 2 illustrates this point.
The same set of three numbers is arranged as a row and also as a
column showing that numbers are easier to compare and to rank
order in the column presentation.
Apart from supporting the case, the example also indicates
why this should be so. The ranking of two numbers is entirely
determined by the left-most digit in which the two numbers
differ. The assessment of which is larger is then reduced to the
inspection of a single digit. The determination of the position of
this distinguishing digit and the comparison is easily done in the
column presentation even for fairly large numbers, provided, of
course, all digits have been properly aligned.
In the consumer magazine example, one may surmise that
the comparison of different magazines is probably of greater
interest than the comparison of revenue sources. Hence I left the
column/row breakdown as it was in the original table.
3.2 Arrangement of Rows and Columns
Of equal importance, albeit for different reasons, are the
relative arrangement of rows and the relative arrangement of
columns. Rows whose entries one wishes to compare should
ideally be displayed close together, and the same holds true for
the arrangement of columns. Often the data themselves suggest
such a grouping. For example, it may be useful and informative
to arrange rows such that the entries in the principal column of
interest appear sorted. Also, because we tend to read from left
to right and top to bottom, a table’s left upper quadrant is likely
to receive most of a reader’s initial attention. Hence one should
consider arranging rows and columns such that the entries of
greatest interest fall into the left upper quadrant.
In the redesign of the table in Figure1Ilefttheoriginalorder
of the rows intact because it was determined by each magazine’s
total revenue in 2001, resulting in a display with the largest and
presumably most important magazines in the leading rows. I
rearranged the columns so that now all columns pertaining to a
particular revenuestream are grouped together, with the columns
related to total revenue, arguably the most important revenue
figure, moved farthest to the left.
3.3 Presentation of Numbers
It is usually not possible to arrange all entries that one should,
or might want to, compare in adjacent and vertically aligned
positions. In such instances the reader has to commit, however
briefly, at least one of the numbers to be compared to memory.
The number of distinct digits that most people retain easily after
a single pass is more or less limited to seven (Miller 1956). It is
therefore good practice to transform the data such that five digits
or fewer represent each table entry, if possible. Often this can be
accomplished by adjusting the scale and, by rounding, limiting
the number of digits retained. Thus, rounding is an important
step with an additional benefit. Usually the left-most digits of a
number are more important than the digits to the right. Retaining
too many digits hinders the reader from paying attention to the
more important digits.
Of course, readers perform some mental rounding on their
own, thus focusing on the relevant digits and retaining a com-
pact mental image of numbers. This task is made easier by the
addition of commas for everythree digits displayed. Generally, it
is the responsibility of the table’s designer to make this process-
ing as easy as possible by displaying numbers concisely. This
means creating a numerical display that retains as few digits as
possible but as many as necessary and that adds useful structure
such as commas to the digits retained.
In the examplehere, I expressedall revenuefigures in $ mill.—
rounded to the nearest million—added commas where appro-
priate, and took care to assure that corresponding digits were
aligned within each column.
3.4 Simple Graphical Elements
A table entry is characterized not only by its numerical value
but also by its position within the table. The effectiveness of a
table presentation depends in part on how easy it is to determine
an entry’s positions within the table and to link the entry to its
row and column labels. Two simple graphical elements, lining
and shading, help as the redesigned table in Figure 3 illustrates.
The American Statistician, February 2005, Vol. 59, No. 1 33
Figure 3. Estimated revenue (in $U.S. 000,000) in total and by source for the 20 leading U.S. consumer magazines for 2000 and 2001 as well
as year-over-year changes. Source: Advertising Age ( Generating tool: Microsoft Excel.
Lining refers to the judicious and parsimonious addition of
lines to the basic rectangular data display. The emphasis is on
“judicious” and “parsimonious” because, as in Figure 1, the
problem is often not that there are too few but that there are too
many lines. Separating each pair of adjacent rows and columns
by a line results in a useless grid that does nothing to help the
reader orient herself. On the other hand, the horizontal lines
added to the table in Figure 3 define horizontal bands of five
rows each, with two complementary benefits. On the one hand,
the bands are sufficiently wide and distinct to be easily trace-
able as the reader’s glance moves from left to right. On the other
hand, each band is sufficiently narrow to let each row’s posi-
tion be easily determined within the band. If the sole purpose
of lining is to help the reader orient herself, the number of rows
within each band should be between three and five.
Lines can also be used to group rows with common themes.
Assuming the number of rows within each group is sufficiently
small, the actual number of rows within each band will then
be determined by the subject matter context that defines the
groupings, and it may well vary by band.
Shading refers to changing the background color for selected
rows and columns. The table in Figure 3 has shaded columns
containing revenue changes. In this example, the benefits of
shading are two-fold. First, similar to lining, the shading of se-
lected columns creates bands that help determine the column
positions of individual entries. Second, shading creates groups
of rows or columns that the reader might wish to compare. This
is particularly useful in instances where the rows or columns one
wishes to compare are not adjacent to one another.
Shading is a frequently used typographical option (Wheildon
1990). It is a delicate addition to a table, and it has to be used
cautiously and judiciously. In particular, it is well known that the
readability of text deteriorates as the background tint gets too
dark. In general, lightly hued background colors are preferable
to gray. If color is not an option, careful attention needs to be
paid to choosing a gray scale that is dark enough to serve the
purpose of structuring the table and that is also light enough so
that numbers can be read easily.
There are other typographical elements one can consider with
benefits perhaps similar to or complementary to lining and shad-
ing. Adherents of the “minimal use of ink” school of thought
might argue that in Figure 3 an effect similar to lining could
have been achieved simply by increasing the line spacing every
five rows. Arguably, an effect similar to shading could have been
achieved by, instead of shading selected cells, choosing a font
distinct in type, style, or size for the entries in these cells. Figure
6 includes an example of such alternatives.
I would like to emphasize that the implementation of these
suggestions are well within the reach of most statistical ana-
lysts. Although one could argue that Microsoft Excel is single-
handedly responsible for creating most of the abominations
looking like the table in Figure 1, one also needs to acknowl-
edge that this spreadsheet program has the flexibility to help an
analyst prepare tables like the one in Figure 3 with a modest
amount of effort.
34 General
Figure 4. Number of data displays in three arbitrarily chosen issues published in 2003 by major business publications sold in the U.S. Within
each row—but not across rows!—circles are proportional in area to frequency counts. Generating tool: S-Plus 6 (2001).
The suggestions of this section are not intended as a compre-
hensive style guide for the design of tables. Some of the ref-
erences mentioned in the introduction present interesting and,
in specific applications, useful design ideas that have not been
covered here. For example, Tufte (2003) shows a table on can-
cer survival rates that breaks the standard rectangular layout
of a simple table to good effect. Also, our suggestions do not
provide automatic rules that absolve the analyst from reconcil-
ing often-conflicting considerations. However, in my experience
these guidelines together with a modest amount of effort sig-
nificantly improve on the design of the default tables copiously
generated by spreadsheet and other data manipulation programs.
I hope that these suggestions be used as part of a broader effort
Figure 5. Breakdown of annual sell-through efficiencies
(sales/inventory) for a monthly magazine for the 111 stores of a small
Midwestern supermarket chain. Generating tool: S-Plus 6 (2001).
that is informed by a variant of the Socratic Principle: The un-
examined table is not worth showing!
A well-crafted table provides a concise presentation of data.
If done well, it guides the viewer in the exploration of numerical
information, in the process revealing valuable structural insights.
I have already noted the value of simple graphical additions for
structuring tables. A table’s ability to communicate the gestalt
of data can be further enhanced by the addition of graphical ele-
ments that themselves contain information. Such enhancements
may be additions to table entries or additions to the table as a
whole. A few examples will illustrate the point. (I do not lay any
claim to originality and vaguely—hence no references—recall
having seen variants of the graphics used in these examples else-
Figure 4 includes a table that records for each of four popular
business publications sold in the U.S. the frequency of several
data displays used by the magazines. Within each row—but not
across rows!—a circle proportional in area to its numerical value
surrounds each entry. In conjunction with horizontal shading, the
circles induce a comparison of the share of each display within
each magazine. The graphical enhancement quickly communi-
cates the relative prominence of each display in each publication.
Also, note the angled column labels. Such angled labels can be
fairly long, but are still readable without a need for turning the
The second example is motivated by common statistical prac-
tice. In order to capture the variation in a magazine’s sell-through
efficiency—calculated as the ratio of sales and inventory—
across the stores of a small supermarket chain, we broke up the
efficiency range into contiguous intervals of equal length and
counted the number of stores falling into each interval. Figure 5
shows the resulting table that has been enhanced by horizontal
bars proportional in length to the respective frequency counts.
The resulting bar chart is, of course, nothing but the rotated
mirror image of a standard histogram.
The American Statistician, February 2005, Vol. 59, No. 1 35
Figure 6. Numerical summaries of p185
concentration for paired partitions of subgroups of patients and control subjects (source: Holzer
et al. 2005). Also displayed are simultaneous confidence intervals based on the Bonferroni principle (Bickel and Doksum 2000). Generating tool:
S-Plus 6 (2001).
The tabulation made explicit in Figure 5 is an essential step in
the construction of a histogram. This frequency tabulation, when
first proposed more than 300 years ago by John Graunt (see, e.g.,
Tapia and Thompson 1978), was duly acknowledged as an im-
portant and original scientific accomplishment. In a standard
histogram this step is implicit with the result that, in my expe-
rience, most readers—even those who have received statistical
training in the form of an introductory statistics course—have
difficultiesinterpreting them. On the other hand, fewpeople have
difficulties interpreting the table in Figure 5 and subsequently
interpreting the accompanying bar chart.
The final example shows how tables that present statistical
results can be graphically enhanced. Summaries resulting from
statistical models are typically presented as tables containing
parameter estimates and associated standard errors. The intent
of such a presentation is often to invite a comparison of selected
parameters to determine whether there is evidence that they are
A simple and robust method for the comparison of pairs of
parameters is based on an inspection of the corresponding confi-
dence intervals. If these intervals do not overlap it may be taken
as evidence that the underlying parameters are indeed different.
Hence there is value in adding confidence intervals to the dis-
play. Adding the confidence intervals as two additional numeric
columns, however, results in an imperfect display. Because a
comparison of confidence intervals requires a comparison of the
upper endpoint of one with the lower endpoint of the other in-
terval, a comparison of numbers in different columns would be
required. A graphical addition to the base table provides a useful
enhancement of the numerical display.
Figure 6 shows a table that includes the number of cases, the
means and the standard errors for the blood serum levels of a
cancer marker for subgroups of a sample comprised of patients
and control individuals. In this table I used different font size
and different line spacing to structure the row presentation. In
the presentation of the numerical table entries I also broke the
horizontal alignment. The vertical offset of the numbers within
each row improves item identification by making it easier to
determine where one number ends and the next number starts.
The distinguishing feature of this table is the addition of si-
multaneous confidence intervals based on the Bonferroni prin-
ciple. The joint coverage probability of these intervals exceeds
99%. As can be seen, these intervals all overlap and therefore
do not provide evidence that these means differ from each other.
(Denote by U
,k =1,...,Kconfidence intervals whose prob-
ability of jointly covering corresponding parameters p
or exceeds 1 α. Note, that if the individual coverage probabil-
ity of each interval is at least 1 α/K, the Bonferroni principle
guarantees that the joint coverage probability equals or exceeds
1 α. The decision rule “Do not reject H
: p
= ··· = p
if the U
contain at least one common point; otherwise reject”
defines a level α test.)
Here the table and the graphical addition effectively comple-
ment each other. On the one hand, the graphical display helps
make the desired comparison. On the other hand, the table read-
ily displays all the information a reader would need should she
choose to change the confidence level and with it the confidence
As the volume and the complexity of data increase, a single
simple table will usually no longer do the data justice. One could
attempt to deal with the problem by devising some appropriately
complex multiway layout. Such an approach may, however, de-
feat its intended purpose by resulting in a data display that lacks
36 General
Figure 7. Example of an OLAP cube with three dimensions and two variables for each dimension. The contents of the cells may be measurements
such as inventory, sales or revenues.
the self-explanatory immediacy of a simple table. An alternative
tactic consists of providing an environment that permits an an-
alyst to create and organize a multitude of simple tables in real
time. Through selection and modification, the analyst can then
guide herself to a small set of tables that capture and highlight
interesting structure in the data.
Such a proposal leads to interesting questions for statistical
methodological research. Fortunately the implementation of this
proposal does not have to await the outcome of such research be-
cause systems that are viable for analytical practice are already
available. Within decision-support software, there are programs
developed around the concept of OLAP (on-line application pro-
gramming). From the user’s point of view the three essential el-
ements of OLAP are its structuring of data, a user interface that
exploits this structure, and the speed with which the data can be
accessed and manipulated.
In OLAP, data—usually summary measurements—are orga-
nized in a multiway layout along dimensions where each dimen-
sion consists of hierarchically structured or nested categorical
variables, each with a distinct set of levels (Shoshani 2003).
Figure 7 provides an example, again from magazine publishing,
Figure 8a. Commercial OLAP interface (Powerplay by Cognos 2003)
to a cube that captures magazine unit sales broken down by the variables
associated with the four dimensions “Class of Trade, “Time,” “Publisher,
and “Edit.
that illustrates the concept. The measurement could be inventory,
unit sales, revenues, or profits for a particular magazine during
a fiscal quarter in a particular retail chain. These measurements
are organized along the dimensions “Class of Trade, “Time,
and “Editorial Category. In our example, each dimension has
two variables. In general, there can be more than two variables
per dimension, and there can be more than three dimensions. In
OLAP, data organized in this manner are, for obvious reasons,
referred to as a “cube.
Through a dedicated interface, the analyst interacts with a
cube by selecting subranges and summarizing over subranges
to generate subcubes. The analytical objective is to generate
informative subcubes of low dimensionality. Preferred low-
dimensional subcubes are tables.
In an application like the one described in Figure 7 the objec-
tive might be to track changes in magazine sales over a particular
time period for a particular class of trade and, if changes occur,
to identify the sources of these changes. The analyst often starts
with a high level summary view and, by selecting and filtering,
“drills down” to interesting and revealing views.
The analyst interacts with the data with the help of a user
interface that is designed around the data structure. Figure 8(a)
shows an example of a Web-enabled user interface from a com-
mercial OLAP tool (PowerPlay by Cognos, 2003). A table is
created using three control buttons for choosing the measure
one wishes to display, as well as the horizontal and vertical vari-
ables by which the measure is broken down. By clicking on one
of these control buttons the user is presented with a menu from
which to choose the dimension, as well as the variable of each
dimension. In addition, each dimension has a control button that
allows filtering by the levels of each associated variable.
Working off a cube with a structure similar to the one shown
in Figure 7 and starting with the view shown in Figure 8(a) I
decided to investigate what over the course of 2002 happened in
bookstores. I filtered on the “Time” dimension to select the year
2002 and the “Class of Trade” dimension to select Bookstores.
I chose editorial categories and quarters as the horizontal and
vertical variables to produce the sale table shown in Figure 8(b).
The American Statistician, February 2005, Vol. 59, No. 1 37
Figure 8b. Unit sales in 2002 in Bookstores broken down by editorial
categories and quarters.
It is noteworthy to emphasize the speed with which this activity
could be performed; it took less than a minute.
A quick inspection of this table reveals a marked drop in unit
sales during the second quarter. At this point one may wish to
investigate whether this sales fall-off can be traced to specific
sources such as particular editorial categories. This is easily done
with some minor adjustments to the sale table. Figure 8(c) shows
the sale table with the rows ranked by annual sales volume and
the raw sales numbers replaced by each column’s row percent-
age, which is nothing other than each editorial category’s market
share. Because these shares did not change much across these
four quarters, we can conclude that all editorial categories ex-
perienced comparable sales losses during the second quarter.
The analysts may now wish to investigate whether the ob-
served sales drop was a singular event or a reflection of inherent
seasonality. Displaying quarterly data for more than one year
helps address this question. Alternatively, the analyst might want
to investigate whether the sales fall-off can be traced to specific
stores. If the underlying cube contains store-level detail she can
easily drill down to perform the required store level analysis.
Again, with a suitable OLAP interface such as the one shown
in these examples the necessary comparisons can be easily per-
formed in real-time.
The OLAP interface example shown here is just one commer-
cially available example. Several of the established statistical
software vendors now offer OLAP add-on functionality to their
statistical offerings. Also, the ubiquitous Microsoft Excel with
its Pivot Table provides OLAP functionality.
The criteria for designing a good OLAP interface are similar
to the ones laid out by Koschat and Swayne (1996) in a slightly
different context. The interface should include a visual represen-
Figure 8c. Market share for each editorial category by quarter.
tation of every dimension and every variable so that variables
and levels are easily available for selection, and so that feedback
is provided to the analyst about the current presentation of data.
On the other hand, the interface should not be cluttered with
irrelevant information. Additionally, if analysts go down ana-
lytical dead ends, it should always be easy for them to retrace
their steps (Norman 1988). Overall, the interface design should
strive for the principles of “direct manipulation” (Schneiderman
1998), which give the analyst a sensation of interacting directly
with the data rather than with a computer or a system.
At the outset, I do not wish to suggest that the mere display of
data is a substitute for a comprehensive statistical analysis, which
includes statistical inference. Also, I do not claim that a table
is always the preferred display of data. I would like to argue,
however, that tables should receive more attention within the
statistical discipline. In particular, tables can and should receive
more thoughtful consideration in statistical practice, statistical
education, and statistical research.
It would appear that not all is well in the practice of displaying
information. Figure 4 (p. 35) provides an interesting commen-
tary on the type of data displays that the readership of a broad
cross-section of business publications is exposed to. These read-
ers tend to be well educated and should comprise a fairly sophis-
ticated audience. Starting with graphs, their seeming abundance
is misleading because most graphs tend to be time series plots
and time series bar charts. This is a rather modest subset of the
options available for a meaningful graphical display of informa-
tion. About the only positive in Figure 4 is the observation that
nowadays the number of pie charts used by all publications is
mercifully small.
Even though the count of tables is uniformly smaller than
that of graphs, the amount of space devoted to the display of
tables is larger than that devoted to the display of graphs. The
quality of the table displays varies. Not surprisingly, tables that
appear as a regular feature in most issues of a magazine such as
stock tables are usually carefully laid out while one-of-a-kind
tables are often hard to parse with plenty of room for improve-
ment. Overall, it seems that the everyday practice of data display
lags the lofty expectations established by a quarter of a century
worth of systematic research on displaying and communicating
Information displays devised for what might be expected to
be a more knowledgeable audience—namely statisticians—do
not necessarily fare better either. Gelman, Pasorica, and Dodhia
(2002) point out that the principal means of displaying scientific
findings in statistical research journals are often-perfunctory ta-
bles, and the authors suggest graphical displays as an alternative.
It is tempting to blame statistics education for this less-than-
perfect state of affairs, and it is hard to see how one can resist this
temptation. I recently had the opportunity to review 11 popular
texts used in introductory business statistics courses for their
treatment of the display of information. These texts spent be-
tween 1.2% and 8.9% of their pages on this topic. Most texts
simply confined themselves to introducing the standard graphi-
cal displays. Only two of the texts actually discussed principles
of good graphical design. These two texts also explicitly ac-
knowledged tables as a means of communicating information
38 General
but did not provide any guidance on how to construct infor-
mative tables. A third text had a short but serviceable section
on Pivot Tables. This state of affairs, in my opinion, calls for
It is argued that graphs and certainly tables are simple con-
structs akin to hammers and chisels that do not warrant the same
amount of time as the more complicated tools of inference which
are, after all, also part of the curriculum. It is certainly true that
the principle underlying the design of graphs and tables (like that
of hammers and chisels) is quickly communicated. The proper
use of these simple tools on the other hand may well require
dedicated and sustained training. Significant benefits could be
reaped if in introductory statistics courses more time were spent
on the proper design and use of graphs and tables. It is too much
to hope that such a focus would yield the next John Tukey, Ed-
ward Tufte, or Howard Wainer (or Leonardo, for that matter) but
it stands to reason that it would improve the quality of data dis-
plays that are routinely generated in statistical practice, and that
it would beneficially raise expectations regarding a meaningful
display of information.
I suspect that, for obvious reasons, academic educators grav-
itate in their choice of course topics toward those that readily
tie into active research. It therefore needs to be emphasized that
dynamic tables are the focus of active research. Though OLAP
is not the outgrowth of traditional statistical research, it is a sub-
ject of active research that readily connects with the statistical
research tradition. This connection is the result of OLAP’s phi-
losophy of data access, as well as the structure of the underlying
Regarding the philosophy of data access and presentation,
there is a well-established and successful precedent in statistical
graphics of representing complex data by using simple displays
that can be easily manipulated. The philosophy driving this re-
search is captured in the visualization programs XGobi (Swayne,
Buja, and Cook 1998) and GGobi (Swayne, Temple Lang, Buja,
and Cook 2003). In these programs the user generates through
a user-friendly and intuitive GUI in real time multiple simple
views, such as dot plots, scatterplots and time series plots, that
represent different views of complex, high-dimensional data.
The information contained in these simple views is enhanced by
functions such as linking and painting, which allow the simple
displays to jointly represent a more complex whole. In addition,
the programs include search utilities based on concepts such as
the Grand Tour (Asimov 1985) and Projection Pursuit (Cook,
Buja, Cabrera, and Hurley 1995) which help steer the analyst
towards views that could be interesting in the specific analytical
While these programs are concerned with graphs, an OLAP
tool deals with tables. Other than that, there is a remarkable
commonality between these utilities. It would seem obvious that
many of the thoughts and insights that informed the development
of dynamic graphics could be beneficially applied to the develop-
ment of OLAP. For example, it should be an interesting research
proposal to devise search utilities similar to the Grand Tour or
Projection Pursuit that help an analyst steer toward interesting
and meaningful tables.
OLAP has a second point of contact with statistical research.
Linear models (e.g., Rao and Toutenburg 1999) provided a rich
platform for inference in the analysis of multiway layouts, the
underlying data structure of OLAP. This holds out the prospect
for utilities that, bundled with OLAP, may help reconcile an
analyst’s belief that there is structure in a particular view of the
data with formal statistical inference. This reconciliation has
traditionally been a challenge for statistical graphical analysis.
Thus OLAP provides not only an interface to data but it has the
potential of supporting a broad array of tools for interfacing with
information. This ability to interface with information quickly
and efficiently and to produce views of the data in a format that
is easily understood will be critical in the years to come. In most
organizations the amount of data generated grows at a rapid clip
while the number of analysts dealing with this data flow tends
to stay constant. Analysts and statisticians will need to become
more efficient. Concepts such as OLAP will be critical in this
effort and it stands to reason that OLAP should therefore be an
integral part of a statistics curriculum.
[Received March 2004. Revised October 2004.]
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40 General
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... Tables are only occasionally considered in the literature on statistics education research, and when they are then it is mostly about numeric tables, their construction, and interpretation from statistical perspectives (see for example [Koschat 2005;Pfannkuch and Wild 2004]). The five-step framework for interpreting tables and graphs in [Kemp and Kissane 2010] is essentially about identifying different aspects of the table, its statistical context, quality indicators, the meaning of numbers, their differences, and identifying reasons for changes. ...
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Tabular representation of information can be traced to ancient times. They were and are used in a wide variety of ways. While every table is essentially a list, their interpretation varies from the simpler to the very complex. This aspect is explored in some detail by the present author. Based on the considerations, she suggests a number of ways of improving the study of tables in the classroom. Possible ideas of pedagogic consistency in the identification of difficulty of table use, and reduction of math anxiety are also touched upon.
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For the analysis of designed experiments, Taguchi uses performance criteria that he calls signal-to-noise (SN) ratios. Three such criteria are here denoted by SNT, SNL, and SNS. The criterion SNT was to be used in preference to the standard deviation for the problem of achieving, for some quality characteristic y, the smallest mean squared error about an operating target value. León, Shoemaker, and Kacker (1987) showed how SNT was appropriate to solve this problem only when σy was proportional to μy. On that assumption, the same result could be obtained more simply by conducting the analysis in terms of log y rather than y. A more general transformation approach is here introduced for other, commonly met kinds of dependence between σy and μy (including no dependence), and a lambda plot is presented that uses the data to suggest an appropriate transformation. The criteria SNL and SNS were for problems in which the objective was to make the response as large or as small as possible. It is argued here that these predecided “portmanteau” criteria can provide an inadequate summary of data and that, regarded as measures of location, they can be extremely inefficient. In preference to such performance criteria, the merits of simple methods of data analysis that can uncover information both expected and unexpected are urged. A reanalysis of an interesting experiment due to Quinlan (1985) illustrates the value of this approach and its contribution to the art of discovery. It is argued that improvement of quality will best be catalyzed by engineers using elementary data analysis with computer graphics rather than by those trained only to employ more rigid predecided criteria.
For the analysis of designed experiments, G. Taguchi uses performance criteria that he calls signal-to-noise (SN) ratios. Three such criteria are denoted by SN//T, SN//L, and SN//S. The criterion SN//T was to be used in preference to the standard deviation for the problem of achieving, for some quality characteristic y, the smallest mean squared error about an operating target value R. V. Leon, A. C. Shoemaker, and R. N. Kacker showed how SN//T was appropriate to solve this problem only when sigma //y was proportional to mu //y. A more general transformation approach is introduced for other, commonly met kinds of dependence between sigma //y and mu //S (including no dependence) and a lambda plot is presented that uses the data to suggest an appropriate transformation.
In reading a table of numbers it helps to focus first on the variation in a single row and a single column, preferably of summary figures such as averages or totals. This then provides a base for looking at the rest of the data.