Automation and evaluation of graduated dot maps
Nicholas D. Arnold
, Bernhard Jenny
and Denis White
College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, OR, USA;
of Information Technology, Monash University, Melbourne, Australia
Dot mapping is a traditional method for visualizing quantitative data,
but current automated dot mapping techniques are limited. The
most common automated method places dots pseudo-randomly
within enumeration areas, which can result in overlapping dots and
very dense dot clusters for areas with large values. These issues aﬀect
users’ability to estimate values. Graduated dot maps use dots with
diﬀerent sizes that represent diﬀerent values. With graduated dot
maps the number of dots on a map is smaller, reducing the likelihood
of overlapping dots. This research introduces an automated method
of generating graduated dot maps that arranges dots with blue-noise
patterns to avoid overlap and uses clustering algorithms to replace
densely packed dots with those of larger sizes. A user study compar-
ing graduated dot maps, pseudo-random dot maps, blue-noise dot
maps and proportional circle maps with almost 300 participants was
conducted. Results indicate that map users can more accurately
extract values from graduated dot maps than from the other map
types. This is likely due to the smaller number of dots per enumera-
tion area in graduated dot maps. Map users also appear to prefer
graduated dot maps over other map types.
Received 28 April 2017
Accepted 22 July 2017
Dot map; graduated dot
map; blue-noise dot map;
Dot mapping is a method of cartographic symbolization for presenting quantitative
information. Dot maps are best used to display data of raw totals for an enumeration
area when the objective is to show that the underlying phenomenon is not uniform
throughout that area (Slocum et al.2009). The primary purpose of dot maps is to depict
variation in spatial density patterns by varying distances between dots.
The selection of dot size is an important consideration when designing a dot map. Dots
that are too small can make a distribution seem sparse and insigniﬁcant, and dots that are
very large can make a distribution seem excessively dense (Robinson et al.1995, p. 498). The
selection of a dot unit value –the numerical value represented by each dot –is equally as
important. If the unit value is too large, no dots will be placed in areas with low quantities,
and if the unit value is too small, dots overlap to form large dark regions (Mackay 1949).
Traditionally, there are two divergent schools of thought (Monkhouse and Wilkinson 1978,
p. 27): some posit that dot sizes and values should be chosen such that the dots begin to
coalesce in the area with the highest density of dots (Dent et al.2009,p.125).Theyreasonthat
CONTACT Bernhard Jenny email@example.com
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE, 2017
© 2017 Informa UK Limited, trading as Taylor & Francis Group
the primary purpose of dot maps is to depict variation in spatial density patterns by varying
distances between dots. Others argue that a complementary purpose of dot maps is to enable
readers to extract raw totals, and therefore, dots should not touch so that their number is easy
to estimate or count (Imhof 1972,Dentet al.2009,Hey2012). Extracting raw totals is only
possible if (a) the dot unit value is simple to sum and multiply and (b) dots can be counted or
accurately estimated for each enumeration unit. Dot overlaps can often not be avoided when
outliers with high values are present, and dots are often too numerous to be counted.
Extracting raw totals from dot maps is therefore often very diﬃcult or impossible.
Dot mapping algorithms in commonly available software do not allow the cartogra-
phers to control dot overlap. Most automated methods rely on pseudo-randomly placing
dots, which can lead to artiﬁcial local clusters of overlapping or coalescing dots. This
clustering can be misleading, as it suggests a spatial pattern in the data that does not exist.
A more regular dot distribution is preferable, if no additional information is available
about the distribution of the mapped phenomenon inside individual enumeration areas.
Graduated dot maps improve upon conventional dot maps by addressing issues of
countability of large numbers of dots and dot overlap. Graduated dot maps use a
number of increasing dot sizes, each of which corresponds to a larger unit value.
When creating a graduated dot map, the cartographer selects the number of dot classes,
the dot values and the dot sizes. As an example, the graduated dot map in Figure 1 uses
three dot sizes to visualize a spatial distribution that varies between very dense and very
sparse areas. The large dots clearly illustrate pockets of high concentration, while the
small dots eﬀectively illustrate the sparse presence of the phenomenon along valley
bottoms, for example in the southeast of the map in Figure 1.
Conceptually, there are two key advantages to graduated dot maps. First, both
small and large enumeration values can be mapped simultaneously, because small
dots are placed in sparse areas, and large dots do not overlap and remain distin-
guishable, even in the densest areas. Second, graduated dot maps depict both
density patterns and raw totals. Because they have fewer dots than conventional
dot maps, there are fewer dots to examine and sum when extracting raw totals.
Figure 1. Graduated dot map with three dot sizes for 200, 1000 and 5000 swine (Extent shown
approx. 220 ×105 km. ©Atlas of Switzerland (1977), sheet 51, www.atlasofswitzerland.ch).
2N. D. ARNOLD ET AL.
Reading of raw totals can be expected to be more accurate from graduated dot maps
than conventional dot maps.
Despite these advantages, lack of both available software and algorithmic methodology
described in the literature hindered widespread adoption. Most cartographic textbooks that
discuss dot maps do not consider graduated dot maps (for example, Robinson et al.1995,
Dent et al.2009,Slocumet al.2009,Tyner2010) or do so only brieﬂy(Hakeet al.2002,Kraak
and Ormeling 2011). An exception is Imhof (1972), who discusses design considerations for
the combination of graduated dot maps with area features and proportional diagrams, but
does not provide a methodology or algorithm for creating graduated dot maps.
The goal of this research is to propose a methodology for creating graduated dot maps, to
evaluate their performance compared to conventional dot maps and area-proportional circle
maps, and to test whether users prefer graduateddotmapstoothervisualizationtechniques.
The proposed algorithmic method for creating graduated dot maps does not pseudo-
randomly place dots, but arranges dots in a distribution that exhibits blue-noise char-
acteristics, wherein dots have ‘a large mutual distance and no apparent regularity artifacts’
(Balzer et al.2009, p. 86:7). Regions of dense coalescing dots are identiﬁed using a
clustering algorithm and are replaced with dots of a larger size and unit value. Details of
this method, including blue-noise and clustering algorithms, are presented in Section 3.
A user study with almost 300 participants was conducted to evaluate graduated dot
maps compared to random dot maps, dot maps with blue-noise patterns and area-
proportional circle maps. The user study is presented in Section 4. Results of the user
study indicate that graduated dot maps are the preferred method and outperform
conventional dot maps.
2. Literature review
2.1. Graduated dot maps
It is not clear when the ﬁrst graduated dot map was produced. Robinson (1982)
discusses a map by Petermann (1857) that shows the population of Transylvania
using a mixture of area-proportional circles and dots, which illustrate the population
distribution of towns and farmsteads. However, Petermann’smapisahybrid
between a dot map and an area-proportional symbol map and not a pure graduated
dot map. Various cartographers have used graduated dot maps since then. An
exemplary set of maps is included in the Atlas of Switzerland (1977) showing farm
animals (Figure 1).
Various manual and automated techniques have been proposed for conventional dot
mapping, but no digital technique for generating graduated dot maps has been described.
2.2. Dot map readability
User studies about dot maps have largely focused on user perception of dots. According to
Taves (1941), when small numbers of dots were present, users were able to accurately
estimate dots; however, when seven or more dots were present, user accuracy decreased
and dots were underestimated. Olson (1975)andProvin(1977)testedusers’ability to estimate
dot density, and found that underestimation of dot number is all but universal. Mashoka et al.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 3
(1986)comparedreadabilityandpreferenceofdot maps versus proportional circle maps.
Their ﬁndings further demonstrated that users underestimate numbers of dots and that
proportional circle maps were favored over dot maps for their accuracy and simplicity.
It is to note that accurate estimation of dot densities requires an equal-area projection
for maps at small scales to avoid misleading distortion of dot spacing (Jenny et al.2017).
2.3. Design principles for dot maps
When dots are placed manually, cartographers generally use one of three approaches:
uniform, geographically weighted and geographically based (Slocum et al.2009,p.322).In
the uniform technique, dots are placed uniformly across the enumeration area (Mackay 1949).
The uniform distribution technique creates choropleth-like maps, where dot patterns create
the impression of shaded enumeration areas. It is also valuable for bivariate or multivariate
maps when area color is used to depict diﬀerent information. This method has the disadvan-
tage that boundaries of enumeration areas are easily detected, and the geographic distribu-
tion inside enumeration areas is not represented. In the geographically weighted approach,
dots are placed such that they are shifted closertoneighboringenumerationareasofhigher
value, which creates the impression of a continuous phenomenon being mapped because
enumeration area boundaries are less visible. The geographically based approach places dots
as a function of ancillary data such as land coverinformation.Althoughthesemethodswere
common in manual cartography, automated techniques that apply these three approaches
are not widely available.
Multi-color dot maps were suggested by Jenks (1953) and Thomas (1955). This
technique eﬀectively shows multiple distributions with dots of diﬀerent colors, as
shown by Rogers and Groop (1981).
Prior to 1949, cartographers selected dot size, unit value and placement without tools
to assist them. In a landmark study, Mackay (1949) developed a nomograph to assist in
the selection of dot size and value. With the nomograph, a ratio of dot size to unit value
is identiﬁed that Mackay calls ‘the zone of coalescence’, which enables cartographers to
estimate the point at which dots will begin to coalesce, given the size and number of
dots per square inch. Although the nomograph has been a widely used tool in carto-
graphy, Kimerling (2009) points out that the nomograph has ‘serious drawbacks in the
modern age of computer cartography’. He extends Mackay’s nomograph to include an
automated method to deﬁne the amount of dot overlap.
2.4. Digital methods for creating dot maps
The most common automated method of producing dot maps is to pseudo-randomly
place dots in an enumeration area. This method uses a random number generator to
calculate coordinates of dot locations. Random dot placement is not a common
approach in manual cartography, as it can lead to unrealistic clusters and gaps in the
dot pattern that imply nonexistent spatial patterns (Slocum et al.2009). Dent et al.
(2009) recommend the use of zones of exclusion, which follows the geographically
based approach for dot placement. Zones of exclusion are created with ancillary data
to deﬁne regions where dots are not to be placed.
4N. D. ARNOLD ET AL.
If the size of dots is minimized and their number maximized, dot maps are perceived as
relative shades of gray, resulting in dot-density shading. Lavin (1986) introduced this
technique based on Jenks’(1953) pointillism technique. Dot-density shading does not
assign a speciﬁc unit value to dots; information can only be derived through dot numer-
ousness and spacing. Lavin’s method is well suited for geographically continuous distribu-
tions, but it is not intended for extracting data values, because dots cannot be perceived as
discrete symbols. The texture is very coarse, and the technique is not commonly used.
Based on the suggestion that overlap interferes with countability, Hey (2012) pro-
posed a method to produce dot maps with spiral patterns that do not have overlapping
dots. An Archimedean spiral pattern is used to determine the placement of ‘spiral arms’,
wherein multiple curves of dots are placed such that dots radiate out from a single
point. Although dots do not overlap, the dot clusters still have a very regular appear-
ance. Hey and Bill (2014)reﬁned the spiral-inspired method by introducing a new dot
arrangement, addressing the regular appearance. The method deﬁnes large dots as
regional boundaries for potential dot positions. Larger dots are to reserve the space in
which smaller real dots may wander. When calculating the ﬁnal dot positions on the
map, dots are shifted within the reserved space for potential dots to reduce pattern
regularity. Dots are allowed to touch but do not overlap.
De Berg et al.(2004) studied a problem relating to dot numerousness that has utility
in dot mapping: given a point set representing a certain distribution, how can it be
automatically simpliﬁed, generating a smaller point set? They tested (1) iterative algo-
rithms and (2) clustering algorithms to simplify a point set and generate an approxima-
tion of the original dots with the smallest error. The tested clustering algorithms require
the desired number of clusters as an input parameter. For our research this is not
practical, as the method of ﬁnding clusters in dots should not require the user to
predeﬁne the number of clusters.
Graphical design principles for generalizing dot maps have been studied by Spiess
(1990), and Yan and Weibel (2008) have developed an algorithm for point cluster
generalization. Yan and Weibel treat four basic types of information including statistical,
metric, thematic and topological information. The primary objective is to ensure that the
four types of information are transmitted from the original data to the generalized
result. Based on Voronoi diagrams, the method follows three basic procedures: (1)
compute a distribution range, which deﬁnes the area that dots are potentially placed
within; (2) delete dots based on their selection probability; and (3) determine the
number of dots in the ﬁnal set. The algorithm presents a potential approach to general-
izing clusters in graduated dot mapping applications.
3. Method for creating graduated dot maps
3.1. Method overview
The proposed method for creating graduated dot maps starts by creating a dot map with a
pseudo-random distribution. These dots are generated by calculating the number of dots
per enumeration area and randomly placed within their respective polygon. The pseudo-
randomly placed dots are then rearranged in a blue-noise pattern with the capacity-
constrained Voronoi tessellation (CCVT) algorithm by Balzer et al.(2009). CCVT disperses
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 5
dense groups of dots such that their distribution is uniform but randomized, while main-
taining the density distribution of the original dots. Blue-noise patterns and the CCVT
algorithm are discussed in the following subsection. Blue-noise dots can still overlap in
areas where dots are densely clustered. Next, the density-based spatial clustering of
applications with noise (DBSCAN) algorithm introduced by Ester et al.(1996) is used to
identify dense clusters of dots. The dots identiﬁed by the DBSCAN algorithm are removed
from the set of dots and are used in another iteration of the CCVT blue-noise algorithm to
create the next class of dots of a larger size and unit value. This process is repeated, as many
times as there are sizes of dots. In the end, the diﬀerent classes of dots are combined for the
ﬁnal map. The DBSCAN clustering algorithm and its combination with the blue-noise
algorithm are described in two separate subsections.
The user of this method needs to provide the number of dot classes, the dot sizes and
the unit values. The dot diameters should be suﬃciently diﬀerent to create a clear visual
diﬀerence between classes.
3.2. Creating a blue-noise dot pattern
The term blue noise refers to an isotropic, yet unstructured distribution of points (de
Goes et al.2012). This distribution exhibits a spectral density distribution with minimal
low frequency components and no spikes in power, resulting in dots that have ‘a large
mutual distance and no apparent regularity artifacts’(Balzer et al.2009). Blue-noise
sampling distributions can be generated with various approaches, and they have
many applications in computer graphics, such as photorealistic rendering, computer-
generated artistic stippling or texture synthesis (Pharr and Humphreys 2004, Lagae and
Dutré 2008, Yan et al.2015). Blue-noise sampling distributions have useful perceptual
characteristics that we utilize for creating dot maps (Figure 2). Because blue-noise
distributions have well-dispersed dots, we are able to avoid artiﬁcial local clusters of
dots that can be created by pseudo-random placement.
We use the CCVT algorithm, as proposed by Balzer et al.(2009), to produce blue-noise
patterns on dot maps. This particular algorithm provides three important functions for
the creation of graduated dot maps because it (1) reduces the number of dots; (2)
optimizes the distribution of dots; and (3) maintains the density distribution of the
original dots. The property of reducing the number of dots is important because it is
used to develop the next larger dot class. The CCVT partitions space into Voronoi
regions from an initial random distribution and iteratively optimizes the placement of
dots. Balzer et al.(2009, p. 86.1) note that the number of iterations has a direct eﬀect on
the quality of the distribution of dots, pointing out that ‘if the method is not stopped at
a suitable iteration step, the resulting point distributions will develop regularity artifacts’.
To avoid these regularities and stop the algorithm, Balzer et al.(2009) introduce a
‘capacity-constraint’, which simultaneously reduces the number of dots for a region
while maintaining the original density of the region. The capacity-constraint, which is
the factor by which the number of dots is reduced, is a modiﬁable parameter kthat we
use to reduce the number of dots.
6N. D. ARNOLD ET AL.
3.3. Identifying dot clusters
The DBSCAN clustering algorithm was proposed by Ester et al.(1996)andhasseveral
advantages over other clustering algorithms for this application. Many clustering algorithms
require the number of desired clusters as an input parameter, but in the case of identifying
spatial dot clusters, the number of clusters is unknown. Another advantage of DBSCAN is that
the algorithm can discover clusters of arbitrary shape. DBSCAN is a density-based clustering
algorithm. It identiﬁes clusters of points based on their mutual distance and the number of
nearby neighboring points. The result of the algorithm is a set of point clusters and a set of
noise points. Noise points are in low-density regions and are not part of any cluster.
DBSCAN requires two parameters: a search radius εand the minimum number of dots
μrequired to form a cluster. A point is added to a cluster if its distance to a cluster point
is smaller than the search radius ε.Ifεis identical to the dot diameter, dots may touch
but not overlap. εcan be adjusted to allow overlapping or spaced dots.
The minimum number of dots μdetermines the minimum number of dots required to
constitute a cluster. For example, if the minimum number of dots μis four, then three
dots that coalesce or overlap will not be identiﬁed as a cluster. We set μto two, which
means that only two dots are required to form a cluster, disallowing any dot overlap.
The DBSCAN clustering algorithm uses the concept of core points (Ester et al.1996). A
point is a core point if it is among at least μpoints with mutual distances shorter than ε.
A random point is selected as the starting point and if there are at least μ−1 points
closer than ε, the points are identiﬁed as core points, and the ﬁrst cluster is identiﬁed.
Otherwise, the point is considered noise. The algorithm iterates through all points until
each point is identiﬁed as either a cluster or noise point. Points can be classiﬁed as noise
or they can be added to existing or new clusters.
3.4. Combining CCVT blue-noise and DBSCAN clustering algorithms
The CCVT blue-noise algorithm and the DBSCAN clustering algorithm are used iteratively
to produce a graduated dot map. The input for this combined algorithm consists of m
enumeration areas with values v
to be mapped, and ndot unit values (d
) ordered in
increasing order. The output is nsets (s
) of dot coordinates.
Figure 2. Dots with a pseudo-random distribution (left) and with blue-noise pattern (right).
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 7
An initializing step produces pseudo-random dot locations for each enumeration area.
The CCVT blue-noise algorithm is then used to reduce the number of dots by a factor k
that the dot unit value d
for the dots in set s
. For example, if the unit value of
the smallest class of dots (d
) is to be 2000 units and k
is 2, the dot value d
is 1000. The
number of pseudo-randomly placed dots per enumeration area is v
The following procedure is then iteratively applied to create the noutput dot sets: the
CCVT blue-noise algorithm is run on s
with the reduction factor k
. This results in a
reduction of the number of dots in s
by a factor of k
. Dots in s
characteristics and are potentially arranged in dense clusters. The DBSCAN clustering
algorithm is run on the dots in set s
, which marks each dot as either belonging to a
cluster or occurring as a noise dot. The default search radius ε
for the DBSCAN
clustering algorithm is identical to the diameter of the dots in set s
. Dots that are
part of a cluster are removed from s
and added to s
. The ﬁnal set s
noise dots. The reduction factor for the next iteration is computed with k
This procedure is executed ntimes, creating the sets s
. Note that for the last set the
DBSCAN clustering algorithm is not run and no clustered dots are removed. The user
selects the number nof dot classes, the dot unit values d
and the dot size for each class.
The amount of overlap or distance among dots of one class can be adjusted by
adjusting the DBSCAN search radius ε
for each class. If the search radius ε
to the diameter of the dots of set s
, the application of the DBSCAN clustering algorithm
guarantees that dots of the same class do not overlap.
Although it is possible to produce many classes of dot sizes, we have found that three
classes is a good number to avoid confounding users’ability to diﬀerentiate between
dot sizes. However, we have not evaluated this choice in the user study. The reduction
factor for the initializing step k
is the only parameter that users cannot be expected to
specify. In our experiments, we used k
= 2, which resulted in visually satisfactory results.
Figures 3–5illustrate the algorithm. In the initialization step, the set s
randomly distributed dots is produced (Figure 3 left). The CCVT blue-noise algorithm
then reduces the number of dots in s
by half and stores them in s
; the unit value is d
and the distribution has blue-noise characteristics (Figure 3 right). The iterative proce-
dure of identifying clusters and replacing clustered dots with fewer, larger dots begins
with using the DBSCAN clustering algorithm to identify clusters from s
(Figure 4 left).
Once these dots are identiﬁed, they are removed from s
and added to s
. Dots from s
are used for the next iteration. Figure 4 (right) shows the result of the next iteration:
the dots in s
from the ﬁrst iteration were retained and the dots from s
through the algorithm again. Figure 5 shows the ﬁnal result with three dot size classes.
The ﬁnal map in Figure 5 uses fewer points in high-density areas than Figure 4 (right),
but the enlargement of the dots visually compensates for the smaller number of dots.
This allows Figure 5 to still show the geometry of high-density areas. Note that the
smallest dots in Figure 5 are identical to the black dots in the left map of Figure 4.A
graduated dot map therefore shows the same number of small dots in areas with sparse
data as a pseudo-random dot map, and the smallest dots in a graduated dot map have
the same spatial distribution as a single-class blue-noise dot map.
8N. D. ARNOLD ET AL.
4. User evaluation
The objectives of a user study were to evaluate estimation accuracy and user prefer-
ences of graduated dot maps compared to other map types. Because graduated dot
maps reduce the number of dots, we hypothesize that users will more accurately
estimate values for graduated dot maps than the other map types, and that users will
prefer graduated dot maps over the other map types. The user study compared four
map types: dot maps with a pseudo-random dot distribution, dot maps with a blue-
noise dot distribution, graduated dot maps and area-proportional circle maps.
Perceptual scaling was applied to the area-proportional circle maps. Proposed by J.J.
Flannery, perceptual scaling allows for compensation of the expected value underesti-
mation (Flannery 1971, Dent et al.2009, Slocum et al.2009). The graduated dot maps in
Figure 3. Pseudo-random dots for the initialization step (left); CCVT algorithm reduces the number
of dots by a factor of two and creates a blue-noise distribution (right).
Figure 4. Clusters identiﬁed by the DBSCAN algorithm in red (left). Clustered points are replaced by
the CCVT blue-noise algorithm resulting in an intermediate map with two classes (right).
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 9
the user study were created with the described method, and census tract population
data for various regions of the United States were used. All maps and questions included
in this user study, as well as collected results, are documented in Arnold (2015).
The user study was built with the Qualtrics survey platform, and participants were
recruited via Amazon Mechanical Turk, a web-based crowdsourcing service where users
complete tasks or surveys and receive micro-payments. Heer and Bostock (2010) found
that crowdsourcing is viable for testing graphic perception and provides high-quality
responses. Respondents were paid $1.00 for completing the survey. The user study
consisted of a demographic survey, a short map-reading tutorial, a series of timed
map-reading tasks, a map preference survey and the question whether participants
attempted to count or estimate dots. Users were not permitted to go back to any
questions once a response was submitted. For the timed map-reading tasks, participants
were shown dot maps with pseudo-random distributions, dot maps with blue-noise
distributions, graduated dot maps and area-proportional circle maps. The preference
questions evaluated each of the map types for clarity and preference.
The demographic survey collected information regarding participants’gender, age,
country of residence and education level. Following the demographic survey, users
completed a tutorial explaining how to read conventional dot maps, graduated dot
maps and area-proportional symbol maps and showing a legend for each map type. Two
untimed example questions were shown to familiarize users with the questions and then
two timed questions were shown. Participants could repeat the tutorial if desired.
The map-reading tasks included a total of 45 dot maps, including 15 maps with
pseudo-random dots, 15 maps with blue-noise dots and 15 maps with graduated dots.
Because graduated dots are similar to area-proportional circles, ﬁve area-proportional
circle maps were included for comparison. Each dot map had one area highlighted in
gray, and users were asked to estimate the value represented by dots for this area. Each
area-proportional symbol map had one circle highlighted in gray, and users were asked
to estimate its value. All pseudo-random and blue-noise dot maps used the same unit
value (200). All graduated dot maps used the same three unit values (1000, 10,000 and
Figure 5. Final graduated dot map with three classes.
10 N. D. ARNOLD ET AL.
100,000). Participants were shown all maps of one type before being shown the maps of
other types. The order of type groups was randomized, and the maps within each type
group were randomized.
For each of the three dot map types, there were 10 maps with realistic administrative
boundaries (two examples are shown in Figure 6) and ﬁve maps overlaid with a regular
grid (an example is shown in Figure 7). To minimize learning eﬀects, the mapped values
were changed for each map, the enumeration areas of ﬁve maps were replaced with a
regular grid and maps with administrative boundaries were rotated to reduce potential
learning eﬀects. For each map, participants were given 10 s to view the map and were
asked to estimate the value of the gray area (see gray areas in Figures 6 and 7). Viewing
time was limited to 10 s such that participants had to estimate rather than count dots in
enumeration areas with many dots. After 10 s elapsed, the map disappeared and
respondents were required to enter their estimate. The legends for all maps of one
group were identical and were shown before the timed maps appeared and during the
= 1,000 = 10,000 = 100,000
= 200 = 1,000 = 100,000
Figure 6. A pseudo-random dot map (left) and a graduated dot map (right) for the user study.
Subjects are asked to estimate the value for the gray areas. Geometry is rotated for the second map.
= 1,000 = 10,000 = 100,000
Figure 7. Example of graduated dot map with grid overlay for the user study. Subjects are asked to
estimate the value for the gray square.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 11
timed task, which prevented participants from losing time to familiarize themselves with
the legend. Each legend for the single-size dot maps showed the individual dot value, as
well as a sample of three varying densities of dots (Figure 6 left). Samples of varying
densities were used because Provin (1977) tested the eﬀects of legends on estimating
dot values and noted that users showed a marked improvement in average estimates
when such legends were present.
Five additional test maps with easily readable values were placed throughout the survey.
Four test maps were dot maps with ﬁve to seven dots in the enumeration area of interest;
one test map showed area-proportional circles with the test circle being identical to one of
the circles in the legend. The test maps were used to eliminate responses from participants
who entered random values rather than attempting to estimate values.
For the ﬁrst map preference question, participants were shown two maps of each
type individually and asked to rate each map from 1 to 5 based on ‘Clarity and Legibility’
and ‘Preference & Appeal’. A rating of 5 for both questions indicates that the map is
‘Very Clear’and ‘Very Appealing’, whereas a rating of 1 for both questions indicates ‘Not
Clear’and ‘Not Appealing’. The second type of map preference question showed
participants two, 2 ×2 image matrices (Figure 8) containing each map type, and
participants were asked to rank the maps on each matrix. Responses ranged from 1 to
4, with 1 being their ‘favorite’and 4 being their ‘least favorite’.
4.1. User evaluation results
Of the 420 participants in the user study, 123 did not correctly respond to at least three
of the ﬁve trivial test maps. The results of these respondents were discarded, and only
the results from the 297 remaining participants were analyzed. Of the 297 participants,
149 were female, 147 were male, 58% were under the age of 35 and 68% had completed
some level of college education. A total of 264 participants were from the United States,
28 were from India and 5 were from other countries.
4.1.1. Dot map estimates
The dot map estimation tasks asked respondents to estimate the value for regions on
the map. There were three groups of maps: one group with 15 maps with randomly
placed dots, one group with 15 maps with blue-noise dot maps and one group with 15
graduated dot maps. For each of the three map groups, the accuracy of responses was
compared across respondents.
The blue-noise algorithm can move dots outside of their enumeration area. For
analyzing the accuracy of responses for blue-noise and graduated dot maps, we used
the number of dots actually placed inside enumeration units.
The Kruskal–Wallis one-way ANOVA test was used to determine if each map group
showed a signiﬁcant diﬀerence in the distribution of estimates among respondents. This
test was used because we analyzed three independent groups of dot maps (Kruskal and
Wallis 1952). p-Values for each map group were <0.001, indicating that the results are
signiﬁcant. The rate of error for each dot map was compared to the total number of dots
per test area (Figure 9) and then averaged by map and by type giving 45 error estimates.
The scores indicate that user estimations of blue-noise dot maps were more accurate
than random dot maps, and graduated dot maps were more accurate than blue-noise
12 N. D. ARNOLD ET AL.
dot maps. In addition, the accuracy of user estimation is correlated with the number of
dots per estimated value, with increasing numbers of dots resulting in increasing relative
error. That is, with all three types of dot maps, error rates are low for enumeration areas
Figure 8. Example of a 2 ×2 image matrix shown for map preference questions. Subjects were
asked which maps they preferred. Values mapped are diﬀerent for the four maps.
Figure 9. Number of dots per test area versus relative error for the 45 dot maps used in the user study.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 13
with small numbers of dots, and error rates are higher for areas with larger numbers of
dots. Our results also support previous ﬁndings that users all but universally under-
estimate dot values (Provin 1977).
4.1.2. Graduated dot maps vs. area-proportional circle maps
Because graduated dot maps are visually similar to area-proportional circle maps, the
results of the ﬁve area-proportional circle map estimations were compared to the results
of the 15 graduated dot maps. On average, the area-proportional circles were under-
estimated by 9.3% even though Flannery’s perceptual scaling was applied. The average
underestimation for graduated dot maps was 6.5%. The Kruskal–Wallis one-way ANOVA
test was used again to determine if there are signiﬁcant diﬀerences in estimations of
individual area-proportional circle maps and graduated dot maps. All tests returned
p-values <0.001, demonstrating signiﬁcant diﬀerences in estimations between gradu-
ated dot maps and area-proportional circle maps.
While interpretation of area-proportional circle maps was less error prone than single-
size dot maps, interpretation of area-proportional circle maps was slightly more prone to
errors than graduated dot maps with a moderate number of dots. However, graduated
dot maps with large values (and a large number of dots) seem to result in higher error
than area-proportional circle maps with similar values. For example, the test area in one
graduated dot map had a total of 39 dots (3 dots with a value of 100,000, 31 dots with a
value of 10,000 and 5 dots with a unit value of 1000), a total value of 615,000 and an
average error of −26.3% while an area-proportional circle map (with perceptual scaling
by Flannery) with a value of 405,500 had an average error of −5.3%.
4.1.3. Preference and clarity
The objective for the preference tests was to compare preferences of each map type. For
each map type, users were shown the map and asked to provide a numerical Likert-scale
response for questions of ‘Clarity and Legibility’(1 = Not Clear; 3 = Somewhat Clear;
5 = Very Clear) and ‘Aesthetic Preference’(1 = Not Appealing; 3 = Somewhat Appealing;
5 = Very Appealing) (Likert 1932). The participants were asked to rate two sets of maps;
see Figure 8 for the four maps of one set.
Figure 10 shows the average of the preference and clarity responses. Results indicate
that users found dot maps with random and blue-noise distributions to be least favorable
Figure 10. Mean preference and clarity ratings and standard deviations for two sets of four maps of
the same area.
14 N. D. ARNOLD ET AL.
and approximately equal in clarity and preference. Respondents found graduated dot
maps to be the most preferred maps with the clearest message. Area-proportional circle
maps ranked in-between. Friedman’s test was used to determine if statistically signiﬁcant
diﬀerences were found between each map type in each set (Friedman 1937). Results of the
test show that there is a signiﬁcant diﬀerence between map types (p-values of <0.001) for
each of the two tested sets.
4.1.4. Rank-order preference
Subjects were shown the same maps that were used in the preference and clarity
question in two, 2 ×2 image matrices showing the four types on a single page
(Figure 8). Using the matrix, they were asked to rank-order the maps from 1 to 4
(1 = ‘favorite’and 4 = ‘least favorite’).
Figure 11 shows a histogram of responses for each of the map types. The results show
a clear pattern of ranks. Users ranked graduated dot maps ‘1st’(their favorite) more
often than any other map type. The results also show that blue-noise dot maps were
often ranked ‘3rd’. Dot maps with a random distribution were most commonly ranked
‘last’by respondents. The ranking of area-proportional circle maps does not show a clear
tendency. Friedman’s test was used to determine if statistically signiﬁcant diﬀerences
were found between each map type in each set. p-Values for each test were <0.001,
indicating signiﬁcant diﬀerences between each map type.
4.1.5. Counting vs. estimating
Participants were asked two ﬁnal questions: (1) how often they attempted to count the
dots versus estimate the number of dots; and (2) whether users attempted to count dots
more often for graduated dots or single-size dots. For the ﬁrst question, 78% of
respondents indicated that they sometimes tried to count the dots, 19% of respondents
indicated that they tried to count the dots every time and only 7 respondents estimated
the dots every time. Note that attempting to count the dots for many maps was
impossible due to the large number of dots and the limited viewing time of 10 s.
Responses to the second question were more evenly split. Sixty percent of respondents
stated that they counted the graduated dots more often, 39% stated that they counted
Figure 11. Frequency of map rank responses for the test map sets. Graduated dot maps are ranked
ﬁrst more often than any other map type.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 15
the single-size dots more often and only 3 respondents indicated that they did not
count the dots.
Study participants showed improved accuracy for dot estimation tasks for graduated dot
maps compared to conventional dot maps. Our research shows that conventional dot maps
resulted in a high degree of underestimation, which reaﬃrms the ﬁndings of Olson (1975),
Provin (1977) and Mashoka et al.(1986). Users also underestimated values with graduated
dot maps, but to a much lesser extent. We observe that enumeration areas in graduated dot
maps with many dots did not have an advantage over conventional dot maps. However,
most enumeration areas in graduated dot maps use considerably fewer dots than corre-
sponding enumeration areas on conventional dot maps, which explains the overall advan-
tage. It is by their design that graduated dot maps have smaller numbers of dots per
enumeration area than conventional dot maps. A second reason for their smaller number of
dots is speciﬁc to our study. We designed dot maps following the school that dots should
begin to coalesce in the area with the highest density of dots. This resulted in a considerable
smaller unit value for pseudo-random and blue-noise dot maps (200) than for the smallest
unit value for graduated dot maps (1000). Future studies should compare blue-noise dot
maps and graduated dot maps that use the same (smallest) unit value.
Study participants underestimated the values of area-proportional circles to a slightly
higher degree than graduated dot maps, even though Flannery’sperceptualscalingwas
applied to the circles. Study participants preferred graduated dots to all others in the study.
Respondents indicated that graduated dot maps were clearer, more legible and more visually
appealing than the other maps, and they ranked graduated dot maps as their favorite (ranked
1st) in a rank-order test.
Our user study only focused on identifying absolute values for selected areas of interest.
It did not assess whether spatial patterns can be extracted eﬀectively from the diﬀerent
visualization methods. Additional studies are needed to compare the eﬀectiveness of the
diﬀerent methods for the interpretation of spatial patterns and other visual analysis task.
Although the results are favorable for graduated dot maps, there are some limitations to
the method and remaining questions. The ﬁrst limitation is that the notion of dot density
can possibly be lost due to the reduced number of dots and their placement. Future work
could include relocating small dots between larger ones to reclaim the density and test the
eﬀect on estimation accuracy. Another limitation of the proposed method is that dots are
allowed to move outside of their enumeration area when blue-noise dot patterns are
created. This is problematic when, for example, terrestrial-related dots are moved over
lakes and oceans. Using ancillary data as exclusion areas for geographically based dot
placement is a potential solution; however, future work is necessary to add this additional
constraint to the CCVT blue-noise algorithm. (For analyzing responses in our user study, we
used the number of dots actually placed inside enumeration units.)
Furthermore, the CCVT blue-noise algorithm can move large dots in dense regions
outward from the center of small enumeration areas to prevent overlap. In such cases, it
may appear to the reader that the large dots represent a large enumeration area, which
is misleading. The current implementation also does not guarantee that larger dots
(which replace a cluster of smaller dots) do not overlap with neighboring smaller dots.
16 N. D. ARNOLD ET AL.
Future work is also needed to determine an appropriate number of dot classes for
graduated dot maps. For graduated circle maps, Evans (1977) recommends four or ﬁve
classes for an audience with little experience in reading graphics and notes that experi-
enced readers may appreciate seven or eight classes. We chose three classes of dots to not
confound users’ability to detect diﬀerences between classes; however, we do not attempt
to evaluate the inﬂuence of the number of classes on estimation accuracy.
Another open question relates to the concept of subitizing. Coined by Kaufman et al.
(1949), subitizing refers to the judgment of small numbers of stimuli, a process that is more
accurate, more conﬁdent and more rapid than estimating or counting. Kaufman et al.(1949)
indicated that subitizing occurs when the number of stimuli is less than six. We hypothesize
that graduated dot maps are interpreted with a combination of counting, estimation and
subitizing. Future work could evaluate the processes by which users derive values for
graduated dot maps. Subitizing seems to also be relevant for the selection of appropriate
unit values for graduated dot maps. To minimize estimation errors, the unit values could be
chosen such that map readers subitize rather than estimate or count when extracting values
for an enumeration area. Optimum unit values could be determined that maximize the
number of enumeration areas in a map that only use four or ﬁve dots of each class (the
numbers suitable for subitizing). The unit values, however, need to be numbers that are
simple to sum and multiply and easy to remember, otherwise the advantage of subitizing
would be defeated by error-prone calculations necessary to compute total values.
In addition to the number of dot classes, the size of dots represents an area of potential
research. Given the purpose of the dot map, future work could determine whether area-
proportional dots have an advantage over graduated dots. If the purpose of the map is to
allow map readers to count dots, creating area-proportional dot sizes may not be neces-
sary. Conversely, if the purpose is to show density, then it could be advantageous to use
area-proportional dots (that is, dots with areas proportional to the unit value).
We present a method for creating graduated dot maps that produces visually pleasing
maps with improved estimation accuracy of raw totals. Our method combines blue-
noise dot distributions and a clustering algorithm, and represents the ﬁrst automated
method for producing graduated dot maps.
Graduated dot maps result in more accurate estimation than conventional dot maps.
The reason seems to be that graduated dot maps use fewer dots per enumeration area.
Study participants found graduated dot maps to be clearer, more legible and more
visually appealing than the other maps. Besides aesthetical considerations, the choice
between a graduated dot map and a conventional dot map should be based on the
map-reading task. If the primary goal is to communicate the spatial density pattern, a
conventional dot map is appropriate. If the goal is to communicate raw totals as well as
the density pattern, a graduated dot map should be preferred. The graduated dot map
technique is also recommended when a single dot unit value of a conventional dot map
results in large empty areas or overly dense areas.
INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 17
The authors would like to sincerely thank the reviewers for their valuable comments and suggestions.
We also thank Abby Metzger, Oregon State University, for editing this text, and Jon Kimerling, Oregon
State University, and Guntram H. Herb, Middlebury College, for their help and comments.
No potential conﬂict of interest was reported by the authors.
Bernhard Jenny http://orcid.org/0000-0001-6101-6100
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