|
Comparative Performance of the Pie and the Bar
Prabu David, Hsuan-Huan Hyuang and Stacy Everly
Ohio State University
Address correspondence to:
Prabu David
School of Journalism
Ohio State University
242 West 18th Avenue
Columbus, OH 43210
(614) 292-7438
[log in to unmask]
Comparative Performance of the Pie and the Bar
Abstract
The relative merits of the pie, the divided bar, the oval and the multiple
bar were evaluated in two experiments, one using a comparison task and
the
other using a proportion task. The representations were tested on
three
critical variables: systematic bias, estimation error and ease of
processing. The results support the conventional practice of using the
multiple bar chart for representing comparisons and the pie for
representing
proportions. There is no evidence of systematic bias in visual
perception
associated with any of the four representations. In the future,
researchers
could address how these various representations promote learning and
memory
for the data.
Comparative performance of the pie and the bar
Comparative Performance of the Pie and the Bar
The pictorial representation of data yields graphs that serve as vital
communication tools in the media. Graphs are a part of our everyday life.
They are used increasingly in the mass media and in public discourse.
The
widespread use of visual representations of data is not unique to the
mass
media. In fact, it is part of a societal trend that ranges from
medical
imaging to NASA's renditions of distant celestial objects.
There is a stream of research in the mass communication literature that
specifically addresses the impact of graphics on memory, visual
perception
and cognition. Tankard (1987) addresses cartoon-like quantitative
renderings, which he calls "chartoons," and provides a check list for
producing better graphics. In another study, Tankard (1989) examined
interest and information gain from chartoons. The findings suggest that
readers find chartoons more interesting than simple graphs, although this
did
not translate into any significant information gain. Smith and Hajash
(1988)
provide a content analysis on information graphics in 30 daily
newspapers.
David (1992) examined the accuracy of perception of various mass media
quantitative graphics. Griffin and Stevenson (1992) report that complex
information graphics facilitate memory for news. In a related study,
Griffin
and Stevenson (1994) found that newspaper readers' understanding of
context
of international news improves when the same information is presented
through
text and graphics. Tankard (1994) offers visual crosstabs as an
analytical
tool for exploring relationships.
Despite the growing interest among scholars on the role of visuals and
graphics in the news, Spence & Lewandowsky (1990) indicate that the
theoretical development of the cognitive processing of graphs is still at its
infancy. Hence, the purpose of this paper is to study cognitive
processing
graphs by focusing on the pie and the bar . Three criteria will be
used to
evaluate these graphs: systematic bias, if any, associated with
processing,
estimation error and perceived difficulty while processing.
The intuitive concept of dividing a pie into fractions was introduced by
Playfair (1786) and has been a popular technique ever since. However,
statisticians have regarded the pie as inferior (von Huhn, 1927;
Macdonald-Ross, 1977; Tufte, 1983) to the bar, although there is no
compelling evidence to support this notion. In the earlier studies (Croxton,
1927; Croxton & Stryker, 1927) the pie was compared against the
divided bar.
In the more recent studies, the pie has been compared against both the
divided bar and the multiple bar (Simkin & Hastie, 1987; Spence &
Lewandowsky, 1991). Though not conclusive, recent studies suggest
that the
pie may be particularly suited for representing parts of a whole,
whereas the
bar is better at representing comparisons between various elements in
the
data. The purpose of this study is to pursue this line of research and
to
confirm whether the conventional wisdom of presenting proportions as
parts of
a circle, oval or bar is justified. In other words, when the data to
be
represented are an exhaustive set of parts of a whole, the pie is the
preferred format. On the other hand, for representing comparisons, without
the necessary constraint that the sum of the items add to 100 percent,
the
multiple bar chart is preferred.
Theory
In addition to providing insight into how common data representations are
processed, an integral part of this work is to make some theoretical
and
methodological contributions. Stevens' law from visual psychophysics
will be
used as the theoretical platform to study visual perception and
processing of
graphs. Baird and Noma (1978) suggest that psychophysics is based on
the
premise that human beings act as measuring instruments interpreting
physical
stimuli in the environment. Physical stimuli come in various forms
such as
brightness of light, loudness of sound, sweetness of different drinks,
painfulness of impact and size of objects, to mention only a few. The
overarching goal of psychophysics is to study how sensory and cognitive
mechanisms perceive and interpret the magnitude of the stimuli in the
respective continua. To a large extent, psychophysics investigates the
mapping between the physical magnitude of stimuli and their assessments by
the perceptual systems in people.
Given its relationship to physics, psychophysical theories are often
expressed in the form of mathematical formulae. One formula that has played
a significant role in perception of graphs is Stevens' law (Stevens,
1957;
1975), which was introduced to visual perception of graphs by Cleveland
and
McGill (1984). Stevens' law states that the perceived magnitude of a
stimulus is an exponent function of the physical magnitude of the stimulus.
Therefore, it is sometimes referred to as the exponent law. The
elegance of
the exponent function is its ability to explain experimental results
from a
variety of physical continua, using a simple mathematical expression
with two
variables and two constants.
y = a (x) b
where y is the estimated magnitude of the stimulus, x is the physical
magnitude, a is a scaling factor and b the exponent. See Appendix 1 for a
detailed explanation.
The exponent function has been replicated in hundreds of experiments for a
wide variety of continua. Some examples of continua in which Stevens'
law is
applicable are as follows: electrical shock, color saturation, finger
span,
heaviness, taste, smell, skin vibration and size measurements. The
interesting factor across all these continua is the exponent function,
because it transcends the particular nature of the stimulus and suggests an
underlying order in the perceptual transformation of our environment.
For a
thorough review see Baird and Noma (1978).
For the purposes of this study we are interested in the perceptual
transformation of the size information in graphs to quantitative information.
The value of the exponent in Stevens' law is an index of the systematic bias
if any that is associated with visual perception of any of these
graphs. In
addition to systematic bias, there is also a random error component
that
should be taken into account while evaluating how they graphs are
perceived.
While systematic and random error components are traditional measures
of
accuracy, the linking of systematic error to Stevens' law is the
theoretical
advancement. Further, an additional measure, namely difficulty of
perception, was evaluated in this study.
Hence, the research question is to evaluate accuracy of visual perception of
four common data representations, namely the pie, the oval, the
divided bar
and the multiple bar, on three dependent variables: systematic bias
(indexed
by the Stevens' exponent) during perception, estimation error during
perception and difficulty of perception. However, accuracy of perception of
all four representations cannot be evaluated on one common task
because
representations of parts of a whole should favor proportion judgments,
whereas the multiple bar should favor comparison judgments. Therefore, the
three dependent measures will be evaluated on two estimation tasks,
namely
proportion and comparison.
Experiment 1
The goal of this experiment is to evaluate the pie, the divided bar, the
oval and the multiple bar on a comparison task. Given earlier work on
elementary codes by Cleveland and McGill (1984) and Spence (1990), we
predict
a null model for differences in systematic bias between the four
representations. Since the multiple bar chart is better suited for
representing comparisons and the pie, oval and divided bar are better suited
for representing proportions, the multiple bar will have the least
estimation
error. For difficulty, we predict that the bar would be easiest
because it
appears to be the vehicle of choice for representing comparisons.
__________________________
Figure 1
__________________________
Hypotheses
H1: The systematic bias for the pie, the oval, the divided bar and the
multiple bar are not significantly different from one another.
H2: Since the task involves comparison judgments, the multiple bar is
predicted to be more accurate than the other three representations when
estimation error is used as the dependent measure.
H3: If indeed multiple bars are better suited for representing comparisons,
then performing the comparison task on the multiple bars will be
easier than
the other three.
Method
Magnitude Estimation Technique
In the magnitude estimation task used in this experiment, subjects were
presented with a graph that had seven elements. No two elements of the
graph
were of the same size. The median element was assigned an arbitrary
value
and appeared in the third or fourth position. With the assigned
element as
the standard, subjects were asked to estimate the values associated
with the
other elements. For each subject, the logs of the perceived values
were
plotted against the logs of the physical values and the exponent
estimated.
The Stevens' exponent was used as an index of the systematic bias.
See
Appendix 1 for details on how the exponent was estimated.
Subjects and Design
Twenty-four undergraduate students participated in the experiment as part of
a class requirement. The subjects were run in two groups of 12. A
simple
within-subjects design with four levels was used. Each type of
representation served as a different level.
Stimuli
Subjects were given all four stimuli, shown in Figure 1. The data for the
stimuli were determined by generating a series of seven random numbers
between 10 and 100. Then all four stimuli were graphed to display the same
data. The median element was treated as the standard and assigned a
value of
either 200, 400, 600 or 800. The value of the standard was changed so
that
subjects would not realize that the same data were represented in all
the
graphs. The size of the graphs were controlled to the extent that all
graphs
fit within the dimensions of the multiple bar graph. The stimuli were
labeled from A through G. The value of the standard was stated and
subjects
asked to provide the values of the other elements based on the
standard. The
stimuli were printed on regular 8.5" x 11" white paper.
Four versions of each representation were printed, each with a different
value for the standard, which took one of four values: 200, 400, 600 or
800.
In addition, three backward counting task sheets, with slots for 25 backward
counts were prepared. The stimuli were put together as booklets,
fully
counterbalancing for the order of presentation and the value of the
standard.
Four versions of the booklets were created such that all representations
occurred equally often in all four positions and the four values of the
standard were assigned equally often to the four representations. Once
the
booklets were arranged, the backward counting task sheets were
interspersed
between magnitude estimation tasks.
Procedure
A test booklet was assigned randomly to each subject and they had to work
through the booklet in a classroom setting. Six subjects were run
under each
of the four conditions for a total of 24 subjects. They were
instructed that
the booklet consisted of two types of tasks: size estimation and
backward
counting. The use of a calculator, ruler or other geometric instrument
was
prohibited.
For the size estimation task, they were instructed that the value of one of
the elements is presented, relative to which they had to estimate the
values
of the other elements in the representation. For the backward
counting task,
they were told that they would be given the first five numbers of a
series
and asked to complete the next 20 numbers in the series. Finally,
subjects
were asked to rank order the four graphs on ease of performing size
estimation.
Analysis and Results
Since the Stevens' exponents were used as an index of systematic bias, they
were first estimated by regressing the log of the perceived magnitudes
against the log of the physical magnitudes and estimating the slope
parameter. See Appendix 1 for details on parameter estimation. The mean of
the exponents for the four representations are presented in Table 1
and
Figure 2. As predicted, the exponents for the four representations were
almost equal to one. They ranged between 0.98 and 1.06. The plots for
all
the subjects were examined for goodness of fit of the exponent. The
average
r-squares were also examined, and it was found that about 95 percent
of the
r-squares were greater than 0.9.
Analysis of variance on the exponents was conducted using a simple repeated
measures design with four levels and the main effect for the model was
not
significant. The null effect indicates that there is no difference
between
the exponents for the pie, the oval, the divided bar and the multiple
bar,
which supports our first hypothesis.
Normally, the residuals left over after the extraction of systematic bias
would serve as an indicator of random error. However, since the
systemic
bias was not significant, we decided to use the total estimation error.
Also, from an applied standpoint total error is more interpretable.
The accuracy for the comparison judgments was estimated by taking the total
error defined in the equation below. The error was obtained by taking
the
sum of the absolute differences between the perceived values and the
physical
values and dividing it by the sum of the physical values. This ratio
was
used as the measure of accuracy and submitted to an analysis of
variance. A
simple repeated-measures analysis of variance with four levels
indicated that
the main effect was significant, F(3,69) = 15.780, MSe = 0.082, p <
.001.
Post hoc tests indicated that the multiple bar was the most accurate
and the
oval the least accurate. Except for the difference between the
circular pie
and the divided bar, which was tending toward significance at the p <
0.1
level, all other pairs were significantly different from one another at
p <
0.05. The mean of the estimation errors is presented in Figure 2 and
Table
1.
Estimation Error = f(xleri(Isu(i = 1, 6, (Physical Magnitude)i - (Perceived
Magnitude)i)),Isu(i = 1, 6, (Physical Magnitude)i))
__________________________
Table 1 & Figure 2
__________________________
Next, the ranks for ease of estimation of the four graphs were analyzed.
The mean ranks are presented in Figure 2 and Table 1. The oval was
unanimously ranked as the most difficult, resulting in a mean rank of 4.0.
Hence, the oval was dropped from the analysis and the ranks of the
remaining
three graphs analyzed using a simple repeated-measures design with
three
levels. The main effect of the model was significant, F(2,46) = 17.267,
MSe
= 10.292, p < .001. There was no significant difference between pie
and the
divided bar on ease of estimation. The multiple bar, however, was
significantly different from the pie and the divided bar at p < .001.
Together, these results support the hypotheses. First, there is no
difference in the values of the exponents for the pie, the oval, the divided
bar and the multiple bar. This indicates that there is no
statistically
significant systematic bias associated with any of these representations.
When the estimation errors for the various graphs were compared, the
multiple
bar was more accurate than the other three representations. This
result
supports the second hypothesis that for comparison tasks the multiple bar
is
superior to the pie, the oval and the divided bar. The error
associated with
the divided bar was smaller than the error for the pie and was tending
toward
significance. The oval, however, produced the largest estimation
error.
Further, subjects ranked the oval as the most difficult to estimate.
The
multiple bar was ranked the easiest and the pie and divided bar were
ranked
in between.
Experiment 2
Since the pie, the oval, the divided bar and the multiple bar were evaluated
on comparison judgments in Experiment 1, the goal of this experiment
is to
evaluate these representations on proportion judgments.
In order to study proportion judgments, a constant-sum estimation task was
used in this experiment. According to this method, a pair of elements
were
presented at a time, whose sum equals 100 percent. Hence the name
constant-sum method. One of the elements in the pair is black and the other
element white. On any trial, subjects had to estimate what percentage
of the
whole the black or white portion represented. For the multiple bars,
subjects were presented a pair of bars and instructed that the sum of the
elements in the pair equals 100. In addition to performing the
proportion
task, subjects ranked the four representations on the difficulty of
performing the estimation task.
__________________________
Figure 3
__________________________
Hypotheses
Since there was no evidence of systematic bias in the perception of the four
graph types, we did not pursue it any further. Hence, we evaluated
only two
dependent variables, namely estimation error and perceived difficulty
in
performing magnitude estimation on these representations. Note that in
the
absence of a systematic error, total error serves as a measure of
random
error.
H4: For the proportion task, representations of parts of a whole, namely the
pie, the oval and the divided bar will provide more accurate
estimations
compared to the multiple bar.
H5: Given the complex set of operations that are required to perform constant
sum estimation on multiple bars, it is predicted that the
multiple bar will
be rated most difficult.
Method
Estimation Task
The estimation task in this experiment is different from the one used in the
previous experiment. The stimuli used in this experiment are
presented in
Figure 3. Half the subjects had to estimate the percentage of the
whole the
white portion represented and the other half estimated the percentage
of the
whole the black portion represented. Subjects were instructed that
the sum
of the white and black components always adds to 100 percent. The
absolute
difference between the physical percentage and the perceived
percentage was
used as the key dependent variable.
Subjects and Design
Twenty-four subjects participated in this experiment as part of a course
requirement. A simple within subjects design with four levels was used
in
the experiment. Each of the four types of graphical representations
constituted a level. Subjects were run in three groups of eight in a
classroom setting.
Stimuli
Four sets of random numbers, with seven numbers in each set were generated
using a random number generator. The numbers ranged between 1 and 99.
One
set was randomly assigned to one of the four representations.
Therefore,
seven trials were created for each of the four representations.
However, all
trials were presented on one page. A sample task sheet is presented
in
Figure 4. On half the worksheets subjects were instructed to estimate
the
black portion and on the other half the white portion. For each
representation subjects performed constant-sum estimates on seven
trials.
All seven trials for a representation were presented on one page, which
was
followed by a backward counting task.
__________________________
Figure 4
__________________________
After the task sheets were printed, they were fully counterbalanced such
that the four representations occurred equally often in all four
positions.
Half the subjects estimated the white portions, whereas the other half
estimated the black portions. The backward count worksheets were
interspersed between magnitude estimation worksheets.
Procedure
Subjects were assigned randomly to one of the four counterbalancings within
the black or white condition. In addition to the instructions given
in the
previous experiment, subjects were told that the sum of the black and
white
portions on any representation always equals 100 percent. Since
estimating
proportion in the two-bar representation is not intuitively apparent,
subjects were specifically instructed that the sum of the two bars should
equal 100.
Analysis and Results
The error associated with each representation was estimated by taking the
sum of absolute differences between the physical magnitudes and the
perceived
magnitudes. The errors were analyzed using a repeated-measures
analysis of
variance with four levels. On six out of the total of 672 trials, the
values
of the estimate was transposed. That is, the value of the black
portion was
estimated when the white portion was requested, or vice versa. For
these
trials, mean substitution was used. There was one outlier with a value
greater than 100, which was also substituted with the mean.
The main effect for type of graph was not significant, indicating that there
is no difference in the error associated with proportion judgments for
the
pie, the oval, the divided bar and the multiple bars. Hence Hypothesis
4 did
not hold up.
Subjects ranked the four types of representations on the ease with which
they performed the task. Analysis of variance on these ranks indicate a
significant difference between the various representations, F(3,69) =
16.20,
MSe = 16.53, p < .01. The circle was ranked to be the easiest,
followed by
the divided bar and oval. The multiple bar was ranked the most
difficult.
The difference between the multiple bar and the circle was significant
at p <
.01. Means for ranks and errors are presented in Table 2. Summary
scores on
ease of processing are presented in Figure 5. Hence these results
support
Hypothesis 5.
__________________________
Figure 5 and Table 2
__________________________
Discussion
The finding that there are no differences in terms of systematic bias
between the four data representations confirms related findings by Spence
(1990) and Cleveland & McGill (1984). The differences in the total
error and
the perceived difficulty offer new leads for interesting theoretical
interpretations. Drawing from work on visual routines by Ullman (1984),
Simkin and Hastie (1987) present five elementary operations that they
propose
are sufficient to explain visual processing of simple graphs:
detection,
anchoring, projection, superimposition and scanning. Detection is the
simplest of the five operations and is used to detect differences in size
between two components of an image. The detection operator returns a
simple
dichotomous result such as larger/smaller. Anchoring is the process
by which
a reference point is picked out as a marker or an anchor. Values such
as
50%, 25%, 10% are ideal candidates for an anchor. Projection involves
sending out a ray from one point to another. For example, a horizontal ray
may be projected from one point to another. The superimposition
operation
involves moving elements of an image to a new location so that the
elements
overlap another component of the image. It is used when the simpler
and more
accurate projection operator is not adequate. Finally, scanning is
analogous
to "sweeping" across the distance between two points on the image, or
between
the anchor and another point.
Although the visual system is capable of executing complex visual routines,
the perceived difficulty expressed by the subjects can be explained in
terms
of the number and complexity of the different elementary operations.
For
comparison judgments, the bar chart requires a projection operation,
followed
by anchoring and scanning. But the pie, the oval and the bar char all
require the more challenging superimposition operation, followed by
anchoring
and scanning. Since the projection operation is easier than
superimposition,
the multiple bar is perceived as the easiest. For proportion
judgments, on
the other hand, it is the multiple bar chart that requires the more
difficult
superimposition operator. Hence, the multiple bar chart is ranked
more
difficult than the pie, the oval and the divided bar. In the case of the
oval, its curvature turns into an impediment for both tasks. Future
research
could address the role of such visual operations used in quantitative
graphics.
__________________________
Figure 6
__________________________
Conclusion
The relative merits of the pie, the divided bar, the oval and the multiple
bar were evaluated in two experiments, one using a comparison task and
the
other using a proportion task. The representations were tested on
three
critical variables: systematic bias, estimation error and ease of
processing. The results support the conventional practice of using the
multiple bar chart for representing comparisons and the pie for
representing
proportions. There is no evidence of systematic bias in visual
perception
associated with any of the four representations. In the future,
researchers
could address how these various representations promote learning and
memory
for the data.
References
Baird, J. C. & Noma. (1978). Fundamentals of scaling and psychophysics.
New York: Wiley.
Cleveland, W. S. & McGill, R. (1984). Graphical perception: Theory,
experimentation and application to the development of graphical methods.
Journal of the American Statistical Association, 79, 531-553.
Croxton, F. E. (1927). Further studies in the graphic use of circles and
bars II: Some additional data. Journal of the American Statistical
Association, 22, 36-39.
Croxton, F. E. & Stryker, R. E. (1927). Bar charts versus circle diagrams.
Journal of the American Statistical Association, 22, 473-482.
David, P. (1992). Accuracy of perception of quantitative graphics: An
exploratory study. Journalism Quarterly, 69, 275-292.
Griffin, J. L. & Stevenson, R. L. (1994). The effectiveness of locator maps
in increasing reader understanding of the geography of foreign news.
Journalism Quarterly, 71, 937-946.
Griffin, J. L. & Stevenson, R. L. (1992). Influence of text and graphics in
increasing understanding of foreign news context. Newspaper Research
Journal, 13, 84-99.
Macdonald-Ross, M. (1977). How numbers are shown: A review of research on
the presentation of quantitative data in texts. Audio-Visual
Communication
Review, 25, 359-407.
Playfair, W. (1786). The commercial and political atlas. London: Corry.
Simkin, D. & Hastie, R. (1987). An information-processing analysis of graph
perception. Journal of the American Statistical Association, 82,
454-465.
Smith, E. J. & Hajash, D. J. (1988). Informational graphics in thirty daily
newspapers. Journalism Quarterly, 65, 714-718.
Spence, I. (1990). The visual psychophysics of graphical elements. Journal
of Experimental Psychology: Human Perception and Performance, 16,
683-692.
Spence, I. & Lewandowsky, S. (1991). Displaying proportions and
percentages. Applied Cognitive Psychology , 5, 61-77.
Spence, I. & Lewandowsky, S. (1991). Displaying proportions and
percentages. Applied Cognitive Psychology , 5, 61-77.
Stevens, S. S. (1975). Psychophysics: Introduction to its perceptual,
neural, and social prospects. Wiley: New York, NY.
Stevens, S. S. (1957). On the psychophysical law. Psychological Review, 64
, 153-181.
Tankard, J. W. (1994). Visual crosstabs: A technique for enriching
information graphics. Mass Communication Review, 21, 49-66.
Tankard, J. W. (1989). Effects of chartoons and three-dimensional graphs on
interest and information gain. Newspaper Research Journal, 10,
91-103.
Tankard, J. W. (1987). Quantitative graphics in newspapers. Journalism
Quarterly, 64, 406-415.
Tufte, E. R. (1983). The visual display of quantitative information.
Cheshire, CT: Graphics Press.
von Huhn, R. (1927). Further studies in the graphic use of circles and bars
I: A discussion of Eell's experiment. Journal of American
Statistical
Association, 22, 31-36.
Appendix 1: Magnitude Estimation Technique
In most psychophysical experiments, elements are shown in succession during
magnitude estimation. The subject is asked to provide an initial
value for a
standard element. Relative to this standard, the subject is asked to
assess
the magnitudes of a series of representations.
While this direct magnitude estimation task is theoretically well-grounded,
it is atypical of the task involved in estimating real-world
quantitative
graphics, in which a number of elements are encountered at one time.
Therefore, a slightly modified version of magnitude estimation estimation
will be used in this experiment, in order to provide ecological validity
to
the findings.
In the modified task, subjects were presented with a graph that had seven
elements. No two elements of the graph were of the same size. The
median
element was assigned an arbitrary value -- known as modulus in the
psychometric literature -- of 500 and appeared in the third or fourth
position, which was determined randomly. With the assigned element as the
standard, subjects were asked to estimate the values of the other
elements.
See Figure.
[--- Pict Graphic Goes Here ---]
Although this modification is only minor , it influences the parameter
estimation procedure. Typically, the ratio model outlined by Baird and
Noma
(1978, p. 80) involves a standard stimulus and a series of comparison
stimuli. If Ts and Tc are the physical magnitudes of the standard and
comparison stimuli, and ts and tc are the perceived or estimated magnitudes
of the standard and comparison stimuli, then the equation between the
perceived ratio f(tc,ts) and the physical ratio f(Tc,Ts) can be
expressed
as follows:
tc = a (Tc) b (1)
ts = a (Ts) b (2)
By dividing equations 1 by 2, and taking logarithms we get:
bbc((f(tc,ts)) = bbc((f(Tc,Ts))b (3)
log bbc((f(tc,ts)) = b log bbc((f(Tc,Ts)) (4)
Equation 4 shows how the scaling constant a cancels out and a no-intercept
form of the regression equation is adequate to estimate b, the
underlying
exponent.
However, in the estimation task proposed for this experiment, the standard
is not estimated. It is provided as an arbitrary value, relative to
which
both the physical and estimated magnitudes are scaled. Therefore,
instead of
the two equation form presented in Equation 4, the single equation
form will
be used to estimate the parameters:
log (tc) = log (a) + b log (Tc) (5)
where, tc and Tc are the estimated and physical magnitudes respectively for
the comparison stimuli, both compared against a constant. Here a is
only a
scaling factor and depends on the estimation task used (Kerst &
Howard,
1977), whereas b is the critical parameter, which changes with the
physical
continuum that is estimated.
Equation 5 suggests that the relationship between tc and Tc is a
straight-line function in log-log coordinates. Estimating beta from this
function is straightforward. The logs of the perceived magnitudes are
plotted
against the logs of the physical magnitudes and the value of beta
estimated
using ordinary least squares (OLS) regression. The parameter obtained
from
this procedure pertains to only one graph for one subject. Since four
types
of graphs were tested, four parameters were estimated from each
subject.
These parameters were then averaged over all the subjects.
[--- Pict Graphic Goes Here ---]
Figure 1 The stimulus set for Experiment 1, reduced to 40 percent of the
original size.
Pie
Oval
Divided Bar
Multiple Bar
Exponent
Estimation Error
Rank
0.983
0.134
2.583
1.010
0.187
4.000
1.056
0.097
2.125
0.992
0.048
1.292
Table 1 Means of the exponents, estimation errors and ranks in Experiment 1.
For rank, 1 = easiest and 4 = most difficult.
[--- Pict Graphic Goes Here ---]
[--- Pict Graphic Goes Here ---]
[--- Pict Graphic Go
es Here ---]
Figure 2 Summary data from Experiment 1. Top panel is a dot chart
representing mean exponents for the pie, oval, divided bar and multiple bar,
with 95% confidence intervals. Estimation error for the comparison
task and
the mean rank for ease of processing are presented (1 = easiest and 4
= most
difficult) are presented in the lower panels.
[--- Pict Graphic Goes Here ---]
Figure 3 Stimuli for the constant-sum estimation task used in Experiment 2.
Subjects had to estimate the percentage of the whole occupied by the
black or
white portion, given that the sum of the black and white components
always
equals 100 percent.
[--- Pict Graphic Goes Here ---]
Figure 4 Constant sum estimation task for circles. This figure is presented
at 50% of the original size. Subjects were instructed as follows: If
the
whole circle represents 100%, estimate percent represented by the
shaded
portion in each of the following circles.
Pie
Oval
Divided Bar
Multiple Bar
Estimation Error
Rank
16.66
1.50
18.08
2.88
19.35
2.21
22.64
3.42
Table 2 Estimation error and the rank on the proportion task for Experiment
2. For rank, 1 = easiest and 4 = most difficult.
[--- Pict Graphic Goes Here ---]
Figure 5 Difficulty in estimating proportions for the four representations (1
= easiest and 4 = most difficult) in Experiment 2.
[--- Pict Graphic Goes Here ---]
Figure 6 Figure adapted from Simkin and Hastie (1987) shows three elementary
operations necessary for the comparison judgment for the mulitple bar.
|
