The Effect of Using Large versus Small Units in Quantitative Estimates of Length,
Weight, and Volume
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of Arts in the
Graduate School of The Ohio State University
By
Jonghun Sun
Graduate Program in Psychology
The Ohio State University
2012
Thesis Committee:
Michael L. DeKay (Advisor), Ellen Peters, Thomas Nygren
Copyrighted by
Jonghun Sun
2012
Abstract
Previous research has demonstrated that changes in units can affect judgments
and decisions. For example, people interpret quantitative information that is expressed as
a larger number of small units as being larger than the same quantity expressed as a
smaller number of large units. In related work, estimates of length or distance or the
monetary value of goods are sometimes larger when participants use large units rather
than small units. In all of this research, however, the focus has been on other phenomena
(e.g., anchoring, attribute weighting, consumer behavior) rather than on the underlying
relationship between numerical estimates and units. In two studies, we anticipated that
participants’ estimates of physical quantities (e.g., the weight of a brick) would be larger
when participants used larger units (e.g., pounds) than when they used smaller units (e.g.,
ounces), because their numerical answers would not be adjusted enough to compensate
for differences in units. In Study 1, estimates of items’ length, weight, and volume were
larger when made in larger units than when made in smaller units. This “unit effect”
remained significant when we adjusted for participants’ incorrect knowledge of unit
ratios and when we considered only those participants who knew the correct ratios. In
addition, the unit effect was larger when participants were less familiar with the units
used to make the estimates. In Study 2, participants estimated weight and volume using
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either familiar or unfamiliar (fictional) units. The unit effect was significant for
unfamiliar units but not for familiar units. Possible mechanisms for these results are
discussed and further research hypotheses are presented.
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Acknowledgments
I would like to express my gratitude to my advisor, Dr. Michael L. DeKay, for his
patience, support, and guidance. I also would like to thank my committee members, Dr.
Ellen Peters and Dr. Thomas Nygren for their helpful advice. My colleagues in the
department, Hyebin Rim and Louise Meilleur, have consistently motivated me and
provided emotional support. I dedicate this work to my parents, who have always shown
endless trust in my potential and talent.
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Vita
2001 ...............................................................Daewon Foreign Language High School,
Seoul Korea
2006 ..............................................................B.A. Business Administration, Korea
University, Seoul Korea
2010................................................................B.A. Psychology, The Ohio State University
Fields of Study
Major Field: Psychology
v
Table of Contents
Abstract .............................................................................................................................. ii
Acknowledgments.............................................................................................................. iv
Vita...................................................................................................................................... v
List of Tables ................................................................................................................... viii
List of Figures .................................................................................................................... ix
Chapter 1: Introduction ....................................................................................................... 1
Effects of Scale on the Interpretation of Numeric Information ...................................... 1
Reversing the Effect of Unit Size ................................................................................... 5
The Role of Units in Magnitude Estimation ................................................................... 6
The Potential Role of Familiarity in the Unit Effect..................................................... 11
The Goals of the Current Research ............................................................................... 13
Chapter 2: Study 1 ............................................................................................................ 15
Method .......................................................................................................................... 15
Results and Discussion ................................................................................................. 17
Chapter 3: Study 2 ............................................................................................................ 33
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Method .......................................................................................................................... 33
Results and Discussion ................................................................................................. 35
Chapter 4: General Discussion and Conclusions .............................................................. 48
Inferences from conversational Norms ......................................................................... 51
Conversational Norms in Estimates of Physical Quantities ......................................... 54
Implications of the Unit Effect ..................................................................................... 59
References ......................................................................................................................... 61
Appendix A ....................................................................................................................... 64
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List of Tables
Table 1. The Units Used in Three Conditions (Study1) ................................................... 17
Table 2. Participants’ answers to the conversion ratio and familiarity questions (Study1)
........................................................................................................................................... 24
Table 3. Self Reported Familiarity with Units on a 5-Point Scale (Study 1).................... 28
Table 4. Units Used in the Six Conditions (2 (Unit Size) x 3 (Familiarity) ) for Each of
the Two Items (Study2) .................................................................................................... 35
Table 5. Mean of logged estimates converted to the smallest unit for the target (N=159)
(Study2)............................................................................................................................. 37
viii
List of Figures
Figure 1. Average estimates of length, weight, and volume of six items in the three unit
conditions, with ± standard error bars............................................................................... 18
Figure 2. Objective estimates of length, weight, and volume of six items, plotted as
functions of objective units ............................................................................................... 23
Figure 3. Subjective estimates of length, weight, and volume of six objects, plotted as
functions of subjective units ............................................................................................. 27
Figure 4. Objective estimates of weight and volume of two items, plotted as functions of
objective units by familiarity condition ............................................................................ 40
Figure 5. Subjective estimates of weight and volume of two items, plotted as functions of
subjective units by familiarity condition........................................................................... 44
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Chapter 1: Introduction
Quantitative information can be expressed in various units. Depending on the size of the
chosen unit, the corresponding numeric value can be easily adjusted to communicate the desired
quantity. For instance, we can express one hour of time in minutes or seconds, and adjust the
numeric value to 60 or 3600 to make all three expressions equivalent (1 hour = 60 minutes =
3600 seconds). Recent research shows that judgment and decision making is often affected by
the way in which quantitative information is expressed, or “framed,” even though there is no
difference in the actual magnitude or quantity.
Effects of Scale on the Interpretation of Numeric Information
Burson and colleagues demonstrated that people’s preference for one option over
another can be reversed by manipulating the scales used to describe the attributes of those
options (Burson, Larrick, & Lynch, 2009). In their first study, participants evaluated two cellphone plans. One plan was superior in quality (fewer dropped calls), while the other was superior
in price. These two attributes , quality and price, were presented using either an expanded scale
(e.g., dropped calls per 1000 calls) or a contracted scale (e.g., dropped calls per 100 calls). Price
was presented using an expanded scale (price per year) when the number of dropped calls was
presented on the contracted scale, or vice versa, with price on a contracted scale (price per month)
when the number of dropped calls was on the expanded scale. If the price was expressed on the
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expanded scale ($384 per year) so that number was larger than on the contracted scale ($32 per
month), and the other attribute was described on the contracted scale (4.2 calls dropped per 100
calls) so that number was smaller than on the expanded scale (42 calls dropped per 1000 calls),
participants preferred the option with superior price. On the other hand, when price was
described on the contracted scale and the number of dropped calls was described on the
expanded scale, participants preferred the option with superior number of dropped calls.
In Burson et al.’s (2009) second study, participants indicated their willingness to pay for
the movie-rental services. For the option that was labeled as the “above average plan,”
participants increased their willingness to pay for the service when the plan was described as 468
movies per year (expanded scale) rather than 9 movies per week (contracted scale). The authors
suggest that the expanded scale, which used a larger number to describe the same deal, was
responsible for which attribute participants appreciated more in making a choice.
In four similar experiments, Pandelaere, Briers, and Lembregts (2011) presented
participants with pairs of consumer goods or services to compare. One characteristic of each
option was framed using either large or small units. For example, the duration of warranty
service was framed either in “months and weeks” or in “days.” Participants reported that the
perceived difference between the options was larger when the units were smaller. This larger
perceived difference between the options resulted in more participants choosing the option with
superior score on that attribute. For example, the caloric content of chocolate candy and an apple
was described in either large units (kilocalories) or small units (calories). The difference was
perceived as larger when it was framed in small units, leading more participants to choose the
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low-calorie apple. This effect, however, was eliminated when participants were manipulated to
pay more attention to the unit describing the quantitative information.
A similar framing effect is found in other studies of consumer behavior, where the same
amount of money in different forms affects spending decisions and self control (Raghubir &
Srivastava, 2009). The “denomination effect” refers to the tendency of people to refrain from
spending money in the form of a large denomination (e.g., one $5 bill) rather than in a small
denomination (e.g., five $1 bills). In their second study, participants who needed to control their
spending preferred to receive a monetary prize in a larger denomination, such as a single $100
bill rather than five $20 bills. Moreover, this preference was followed by a lower likelihood of
spending the money. The authors argued that the large denomination makes participants hesitant
to spend the money, and participants opted for the larger bill as a strategy to help themselves
avoid painful spending. However, once they begin to spend from the large-denomination budget,
they tend to spend more than if they had the same amount of money in a small denomination.
Even though the main purpose of Raghubir and Srivastava’s research was to establish the
denomination of money as a self-control device, it provides clear evidence that the same budget
plays a different role in our behavior depending on how it is framed.
In a study of budget constraints with foreign currency (Wertenbroch, Soman, &
Chattopadhyay, 2007), participants forecasted their future spending in different categories when
given a budget in a specified currency. Importantly, the total amount that they intended to spend
differed depending on the currency. They indicated that they would spend more when the
currency of the budget had a smaller (less valuable) unit and therefore a higher number of units
compared to the base currency of U.S. dollars (e.g., 1100 Korean won = 1 U.S. dollar).
3
Somewhat confusingly, the authors referred to the numerical part of an amount as the “face value”
of the currency. The authors’ interpretation was that the budget given to participants looked
greater when expressed as a large number of small units, so participants estimated that they
would spend more. On the other hand, when the foreign currency was more valuable than the
base currency (e.g., pounds rather than dollars), participants indicated that they would spend less
because the allowed budget looked tighter to them. This result suggests a similar tendency in
people’s perceptions of the quantity as that in the previous research example. The same monetary
value is perceived differently depending on how it is framed by the currency, with larger
numbers of small units being perceived differently than smaller numbers of large units.
The four examples presented so far all point in the same direction. The general tendency
is that the high numerosity associated with small units results in greater magnitude perceptions
for the given information, while the low numerosity associated with large units results in the
opposite. In other words, expanded scales (i.e., those with small units and large numbers) might
be responsible for inflated perceptions of the quantity in question. A possible explanation for this
effect is people’s relatively high sensitivity to the numeric information compared to their
sensitivity to the size of the unit. Heavy reliance to the numeric portion of the information in
interpreting quantities is evident from past research. Pelham (1994) explains, for example, that 8
slices of 1 pizza looks like more than 1 whole pizza, because the numerosity that “8” has in
former case draws more attention than the true amount of pizza. In a series of studies,
participants estimated the total area of visual stimuli, the sum of multiple numbers, and the total
value of multiple coins as larger when the quantities were divided into multiple pieces. This
tendency was exacerbated when the task was difficult and when the cognitive load or the time
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pressure was high. Pelham argued that people make inferences based on numerosity rather than
others aspects of the information when making judgments.
Reversing the Effect of Unit Size
More recent research shows that the effect of unit size can be reversed by manipulating
presence or absence of information. Monga and Bagchi (2011) show that the attention paid to
either the number or the unit is determined by the relative salience of the two types of
information and that this attention affects related estimates. In four separate experiments,
participants estimated the length of time or an amount of a resource that was presented with
different sizes of units. In the first study, the researchers manipulated the relative salience of
numbers and units in a graphical presentation showing the progress of a construction project.
When the numbers were made salient, estimated completion times were longer when small units
(feet) and large numbers were used than when large units (floors) and small numbers were used.
This “numerosity” result is consistent with the other research reviewed above. However, when
the unit was made salient to participants (by removing the numbers), the result was reversed,
with the remaining work estimated as larger when the unit was larger. The authors called this
tendency “unitosity” because participants depended more on the unit than the number. In later
experiments, Monga and Bagchi obtained similar results by manipulating participants’ construal
level (Trope & Liberman, 2000; Trope & Liberman, 2003). Participants in the low-construallevel condition focused more on the number than the units, resulting in larger estimates when the
information was presented in small units (and large numbers). On the contrary, participants in the
high-construal-level condition focused more on the units than the numbers, resulting in larger
estimates when information was presented in large units (and small numbers). The results of this
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research indicate two opposite estimation patterns, with the tendency called “unitosity” being the
opposite of the standard or default result. The important message from Monga and Bagchi’s
research is that estimates for the same target are systematically influenced by the size of the units
chosen for the scale. Moreover, unless we manipulate attention as the authors did, people tend to
be more sensitive to the number than to the unit.
The Role of Units in Magnitude Estimation
The effect of scale or frame found in the above literature seems robust. The examples
given so far, however, mainly deal with comparisons of number and unit information.
Researchers presented both the number and the unit to participants so that they could interpret
the given information. For example, participants in Pandelaere et al.’s (2011) third experiment
read the calorie information about the food, which included both number and unit. The
participants’ task was to compare the caloric content of two snacks and decide what to have. In
such cases, the task is not a direct estimation of quantity, but is rather an interpretation and
comparison.
Our primary research question is whether the same tendency will appear in magnitude
estimation, where participants have to come up with the number as their own answer. Compared
to the amount of research on comparing information in different units (particularly in tasks
involving consumer choice), there has been little research the straightforward estimation of
physical quantities in different units. Therefore, it is too early to conclude that the same
psychological processes will lead to an effect of unit size in magnitude estimation.
The research in this thesis tests for an effect of unit size when participants estimate the
physical properties of familiar objects. We expect that using larger units will result in larger
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magnitude estimates. Specifically, we expect that participants will adjust their numerical
responses in the correct direction (i.e., larger numbers for smaller units and smaller numbers for
larger units), but to an insufficient degree. Thus, the ratio of the numerical responses will be less
than that required to compensate for the difference in units (i.e., numerical responses in inches
will be less than 12 times those in feet). When all estimates are converted to the same unit (e.g.,
inches), estimates originally made using the large unit will be larger than those originally made
with the small unit. In the current research, we will use the term “unit effect” to refer this
directional tendency in physical estimation. We have found some evidence regarding this
hypothesis in past research, even though that research was not directly concerned with the unit
effect.
For example, Wong and Kwong (2000) demonstrated that incidental numerical anchors
can have different effects in an estimation task depending on whether the units used in that task
implied that the anchor was either high or low. Although their primary results on anchoring are
tangential to our hypothesis, the control condition of Wong and Kwong’s second study does
provide some relevant information. Specifically, participants who had not been exposed to an
anchor estimated that a familiar river in Hong Kong was longer if they made the estimate in
kilometers than if they made the estimate in meters. Converted to meters, the mean estimates in
the two conditions were 4,267 m and 3,908 m, respectively. Unfortunately, it is impossible to
assess the statistical significance of this difference between means based on the information
provided. The main purpose of the current research is to replicate and investigate the significance
of differences like this one.
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Other evidence related to our current hypothesis comes from LeBoeuf and Shafir’s
(2009) research regarding time and length estimation. In their Study 2, for example, participants
estimated the time to a particular event in the future or past, using either a small unit (days) or a
large unit (weeks). The authors found that estimates made using the large unit were significantly
longer those made using the small unit, at least for estimates of the time until a personal event in
the future. More closely related and similar to the current research, participants also estimated
the geographical distance between well-known locations (buildings on campus). In one task,
participants estimated the distance using either “unspecified” units (familiar units of the
participants’ choosing, such as yards or miles) or unusual “specified” units, such as the number
of soda cans, cars, or steps. The estimates made by two groups were systematically different,
showing that the measurement scale influences distance estimation. More important for our
interest in the size of unit, we noted that the same target distance was estimated with the number
of cars in one condition and with the number of soda cans in another condition. After the authors
converted both estimates to feet using conversion factors provided by the participants, the
distance appeared somewhat greater when the larger unit was used (cars, M=557.9 ft, SD = 470.1,
N=123) than when the smaller unit was used (soda cans, M=500.6 ft, SD = 360.1, N=38).
According to our test, this difference is not statistically significant (t(37)=.79, p > .1), though the
direction of the effect does correspond to our hypothesis.
A larger, significant effect is reported by Raghubir and Srivastava (2002) in their
research on people’s willingness to spend foreign currency. In four experiments, participants
imagined situations in which they had to buy merchandise from six different foreign markets,
each with a different currency. They indicated the maximum amount of money that they would
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be willing to spend for each product in the different markets. The results showed that participants
were willing to pay less for the same product when the foreign currency had a lower value
(smaller units) than the base currency (F(5,480)=5.86, p<.001). For instance, they spent less with
Turkish lira (with an exchange rate of 685,000 lira to one U.S. dollar) than they spent with
Norwegian krone (with an exchange rate of 9.5 krone to one U.S. dollar). The authors claim that
goods seemed more expensive in Turkish lira than in Norwegian krone due to the much higher
numbers required, resulting in less willingness to spend.
At first glance, this result seems to contradict the results of Wertenbroch et al.’s (2007)
study, which we reported earlier. However, it is important to note that the two articles focus on
different factors related to spending behavior. Wertenbroch et al.’s study measures how
participants perceive “what they have in the pocket” while Raghubir and Srivastava’s (2002)
study focuses on “what they will take out from the pocket.” Wertenbroch et al. present the total
budget allocated to participants before they decide how much to spend from it. Therefore, the
expanded scale, with large numbers of small currency units, is followed by participants’ feeling
that they have enough money to spend and hence greater spending. Raghubir and Srivastava’s
(2002) study, on the other hand, measures the amount of money that participants want to pay for
the certain goods, so the price of the product is the key. As participants imagine spending money
on an expanded scale, with large numbers of low-value currency units, the number is perceived
as greater and they are less likely to pay that amount out of pocket. In short, the two studies tell
similar stories: People think that an amount of money in small units is larger than the same
amount of money in large units, because the number of units is larger.
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Notable effects of unit size have also been reported in the domain of energy
consumption (Frederick, Meyer, & Mochon, 2011). In one of their studies, participants estimated
the energy used by eight different electric appliances, either in watts or kilowatts. Interestingly,
participants underestimated energy use by a factor of 6 when making estimates in watts, but
overestimated by a factor of 51 when making estimates in kilowatts. Thus, the estimates were
over 300 times larger with the large unit (kW) than with the small unit (W). Because a kilowatt is
equal to 1,000 watts, participants’ numerical responses should be 1,000 times greater with watts
than with kilowatts. The numbers given in watts, however, were on average only 3 times larger
than those given in kilowatts. The magnitude of the unit effect in this study is remarkably large,
compared to the other examples stated above. We suspect that participants’ lack of familiarity
with the energy units in the task played a role in the magnitude of the effect. Such unfamiliarity
may exacerbate the discrepancy in estimates. We will return to this issue again in the next
section.
Findings so far indicate that magnitude estimates are influenced by unit size, at least in
some cases. Smaller unit size is logically associated with higher numerical responses, and larger
unit size is logically associated with lower numerical responses, but participants’ numerical
responses appear to differ by less than they should. The result is that estimates are larger with
larger units than with smaller units. The generality of a unit effect in estimates of physical
quantities, however, has not yet been proven. For example, the effect found in Raghubir and
Srivastava’s (2002) study is large and significant, but the tasks given to participants were more
complicated than simple estimation. Participants had to imagine the market situation in foreign
countries and consider their preferences for various products, possibly complicating the process
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of estimation using different currency units. For such reasons, it seems worthwhile to conduct a
simple and straightforward experiment to investigate the unit effect in physical estimation.
The Potential Role of Familiarity in the Unit Effect
We hypothesize that the unit effect is moderated by one’s familiarity with the units in
question. More specifically, we suspect that individuals who are very familiar with the units in
question will be less susceptible to the unit effect than are those who are less familiar with the
units. Some evidence supporting this hypothesis is found in the literature.
In one example, Poulton stated in his 1979 review that the “stimulus equalizing bias” is
not found when participants use familiar units (Poulton, 1979; Harvey & Campbell, 1963). The
stimulus equalizing bias is one of several biases that affect magnitude judgment. In a typical
demonstration, participants estimate the weight of two target objects, which are among either a
wide range of other objects or a narrow range of other objects. The difference in the range of
stimuli, according to the authors, biases participants’ perceptions of the difference between the
objects (a fixed difference is perceived as larger if in a narrow range and smaller if in a wide
range). Importantly, the bias was found when they used 5-point scale to estimate the weights of
the objects, but not when they used a conventional and more familiar unit, the ounce. It is evident
that the familiarity with the unit is responsible for the different results. Of course, the bias in this
example is different from the unit effect, and the difference between a 5-point scale and a
conventional unit (ounce) may be larger than the difference between two conventional units that
differ in familiarity (e.g., gram and ounce). Nonetheless, the magnitude of the unit effect may
vary depending on one’s familiarity with the units.
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Recent research by Shen and Urminsky (2012) includes the manipulation of participants’
familiarity with the units in an estimation task. We want to clarify up front that Shen and
Urminsky’s research is not directly comparable to the current research on the unit effect.
Although different units are used in the experiment, the size of the unit was not an independent
variable in any of their studies, and the research was not designed to test the effect of unit size.
Instead, it shows the possibility that participants’ familiarity with the unit results in different
estimates. They define the “unit salience effect” as the tendency to pay less attention to the
numeric information when an unfamiliar unit is made salient, resulting in reduced sensitivity to
the number. They manipulated the salience of the unit by increasing the font size or making the
text bold. Participants rated the price of foreign hotels, with the price described in either familiar
U.S. currency or unfamiliar Brazilian currency, and with the unit either made salient or not
(Study 3). When the unit was familiar, participants were somewhat sensitive to the number
regardless of unit salience. When the unit was unfamiliar, participants were sensitive to the
number when the unit was not salient, but were less sensitive to the number when the unit was
made salient. Apparently, making an unfamiliar unit more salient made participants more
hesitant to infer anything from the numeric information, because they realized that they did not
know how to interpret the numbers. In other words, the salience of the unit cued people to their
lack knowledge regarding the unfamiliar unit. Our question is how such tendencies will affect
magnitude estimation. Based on Shen and Urminsky’s results and other examples regarding the
unit effect, we predict that unfamiliarity with the units will lead people to make smaller
adjustments to compensate for the size differences between units, thus yielding a larger unit
effect.
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Frederick et al.’s (2001) study of energy consumption, mentioned earlier, supports this
hypothesis. The estimates for the energy consumed by the appliances varied greatly depending
on the size of the unit participants were asked to use. The gap between the estimates from those
using different units was remarkably large. We believe that participants’ unfamiliarity with
energy units (watts and kilowatts) was responsible for the insufficient adjustment in numerical
responses. When people are not familiar with a unit, they have little or no information on which
to base their numerical estimate. A reasonable strategy in such instances is to avoid giving
extremely large or small numerical responses, in order to avoid providing an extremely wrong
answer. However, when the estimates are converted to a common unit, this insufficient
adjustment for the units will result in a larger unit effect.
To summarize, we expect that familiarity with the unit moderates the unit effect in
quantitative estimation, such that a greater unit effect will be found when participants are less
familiar with the unit.
The Goals of the Current Research
The primary goal of the current research is to assess the effect of unit size in quantitative
estimation. More specifically, we expect that estimates made in larger units (e.g., pounds) will be
bigger than estimates made in smaller units (e.g., ounces), because numeric answers will not be
adjusted enough to compensate the difference in the size of units. We expect this effect because
participants are likely to avoid extreme values in estimating uncertain answers, and instead focus
more on generating a “reasonable” number so that it doesn’t seem too small when expressed in
larger units or too large when expressed in smaller units.
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Our second goal is to assess the moderation of the unit effect by familiarity. We predict
that unfamiliarity with the unit will magnify the unit effect. In order to test this hypothesis in the
first study, we asked participants to report their familiarity with the units on a 5-point scale. In
the second study, we manipulated participants’ familiarity with the unit by creating fictitious
units. Further details will be provided later in this thesis.
This research uses estimation tasks that are direct and straightforward. Participants
estimate the length, weight, and volume of objects that are either present or are otherwise well
known to the participants. Although the psychophysics literature has long history of exploring
perceptions of visual or aural stimuli (Poulton, 1979; Harvey & Campbell, 1963; Engen & Levy,
1958; Frederiksen, 1975), we could not find an example in which participants directly estimated
the physical dimensions of objects using a variety of everyday units, perhaps because it has been
argued that biases are unlikely in such situations (Poulton, 1979). A second explanation for this
gap in the literature is based on the desire to avoid the “stimulus error” (Boring, 1921), in which
participants’ knowledge regarding objects and units is said to interfere with the study of internal
perception processes. Nevertheless, we believe that the current research is meaningful and of
practical importance. Convincing results from the current research will help to solidify the
presence of the unit effect and its relevance in a wider variety of situations, and contribute to a
better understanding the psychological process behind the effect.
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Chapter 2: Study 1
The objective of the first study was to examine whether estimates of the length, weight,
and volume of items differ depending on the units used. Specifically, we predicted that people do
not adjust their numerical estimates enough to compensate for a change in units (e.g., from feet
to inches). In addition, we assessed whether this effect was moderated by participants’ selfreported familiarity with the units. We predicted that the effect would be stronger for less
familiar units than for more familiar units.
Method
Participants. One hundred two undergraduate students from The Ohio State University
participated in the experiment for course credit.
Materials and procedures. Participants estimated the length, weight, or volume of six
items while sitting at separate computers. They estimated the length of the table at which they
were sitting, the length of a very familiar lake on campus, the weight of a brick, the weight of a
one-gallon jug of milk, the volume of a trashcan, and the volume of a bathtub. The table, brick,
and trashcan were in the room where the experiment was conducted (there was a brick on each
participant’s table, but only one trashcan), while the other three items were not. Participants were
randomly assigned to one of three conditions that varied in the size of the units (small, medium,
or large) used for each item (see Table 1). Note that the designation of a particular unit as small,
15
medium, or large differs by item. For example, foot is the large unit for the length of the table but
the small unit for the length of the lake.
In each condition, participants estimated the length, weight, or volume of the items in
the specified units. For example, participant saw a question as “Imagine Mirror Lake at Ohio
State University. How long is Mirror Lake (straight across, in its longest dimension) in feet?
_____feet” on the computer screen (see Appendix A for complete materials). After making the
six estimates, participants rated their familiarity with every unit in the experiment on a 5-point
scale, with 1 = Not at all familiar and 5 = Extremely familiar. They also estimated the conversion
ratios for all pairs of units on each dimension (e.g., length). Before providing the numerical ratio,
they first indicated which of two units was larger. For example, if a participant answered
(correctly) that an inch is larger than a centimeter, the ratio question was formatted as “1 inch =
____ centimeters.” If a participant answered (incorrectly) that a centimeter is larger than an inch,
the ratio question was formatted as “1 centimeter = ____ inches.” Participants provided
demographic information and were debriefed before being dismissed.
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Dimension
Length
Weight
Volume
Target Item
Small Unit
Medium Unit
Large Unit
Table*
Mirror Lake
Brick*
Gallon jug of milk
Trashcan*
Standard bathtub in average house
Centimeter
Foot
Gram
Ounce
Fluid Ounce
Quart
Inch
Yard
Ounce
Pound
Liter
Liter
Foot
Meter
Pound
Kilogram
Gallon
Gallon
Note: The objects with asterisk were physically presented in the sight of experiment.
Table 1. The Units Used in Three Conditions (Study1)
Results and Discussion
All quantitative estimates were converted to the smallest unit used for the given item. For
example, all estimates for the length of the table were converted to centimeters, while all the
estimates for the length of the lake were converted to feet. Estimates were then logged to reduce
positive skew.1
Comparison by condition. Figure 1 shows the means of logged estimates in each
condition. If there were no effect of units on estimates, all lines in Figure 1 would be flat.
However, some lines show clear increases, with larger estimates as the units get larger. We
conducted further analyses to determine whether these increases are significant.
A repeated-measures ANOVA with unit condition (between-subjects) and object (withinsubjects) as predictors yielded significant differences of estimates across the three unit conditions
1
All analyses for Study 1 were replicated with estimates converted to the largest unit for each
object and to the medium unit for each object. Key results were essentially unchanged. We
therefore report results only for conversions to the smallest unit.
17
(χ2 (2, 102) = 29.42, p <.0001) and the six items (χ2 (5, 102) = 87.21, p <.0001).2 The main effect
of unit condition indicates that the overall increase evident in Figure 1 was significant. The main
effect of item is not psychologically interesting, because there were real differences in the
physical magnitudes of the items. However, there was also a significant interaction between unit
condition and item, implying that the size of the unit effect depends on the item being assessed
(χ2 (10, 102) = 28.08, p <.005). This tendency is also evident from the different slopes of the
lines in Figure 1. Specifically, the lines for the length of the table and the length of the lake seem
less steep than the other lines. Therefore, each item was analyzed separately in next step.
`
Figure 1. Average estimates of length, weight, and volume of six items in the three unit
conditions, with ± standard error bars
2
All repeated-measures analyses were conducted using SAS PROC GENMOD.
18
Differences between the three unit conditions were significant for the weight of the brick
(χ2 (2, 102 = 16.22), p < .001), the weight of a one-gallon jug of milk (χ2 (2, 102) = 17.64), p <
.001), the volume of the trashcan (χ2(2, 102) = 12.72, p <.01), and the volume of a bathtub (χ2 (2,
102) = 10.83, p< .01). The effect of unit condition was not significant for the length of the table
(χ2 (2, 102) = 4.20, p =.12) or the length of the lake (χ2(2, 102) =2.95, p =.23).
The above analysis supports the initial hypothesis of the current research by showing that
participants in the large-unit condition made larger estimates of the same item than did
participants in the small-unit condition. However, other possible causes for these results still
exist. In order to clarify the unit as a main factor causing the increasing estimates, these other
possibilities will be considered and ruled out through further analysis.
Objective Unit Effect. One concern about the above analysis is that it does not reflect the
true differences between the sizes of the units. The units used by the participants in each
condition are not equally distant to each other in terms of true size. For example, the difference
between a quart and a liter, the units used in the small- and medium-unit conditions for the
bathtub, is not identical to the difference between a liter and a gallon, the units used in the
medium- and large-unit conditions. The relatively steep line between the medium and large
conditions for the bathtub in Figure 1 compared to the relatively flat line between the small and
medium conditions might be caused by this discrepancy.
To address this issue, we treated unit size as a continuous variable. For each item, the
smallest unit used was assigned a size of 1.00. Then, the true ratios of the medium and large
units to the small unit were used as the sizes for the medium and large units. For the bathtub
example, a quart, liter, and gallon had sizes of 1.00, 1.06, and 4.00, respectively. These values
19
were then logged. The resulting values are called “objective units” in the current research. It is
important point to note that this procedure affected only the values of predictors in the repeatedmeasures regression model below, not the predicted values. As in the initial analysis, the
predicted values were participants’ numerical estimates, which were converted into the smallest
unit for each item and logged.
Again, the repeated-measures regression revealed a positive and significant main effect of
objective unit size (b=.24, χ2 (1, 102) = 17.40, p<.0001) and a significant interaction between
unit size and item (χ2 (5, 102) = 21.94, p<.001). Regarding the interaction between unit size and
item, we again investigated the unit effect separately for each item. For four items, regression
analysis showed that estimates increased significantly as the objective unit size increased (weight
of brick (b= .23, t(101)=5.34, p<.0001), weight of a jug of milk (b= .44, t(100)=4.57, p<.0001),
volume of the trash can (b= .25, t(101)=4.12, p <.0001), volume of a bathtub (b = .49,
t(101)=3.33, p<.005)), but this effect was not significant for the other two items (length of the
table (b=.04, t(101)=.90, p=.37), length of the lake (b= -.01, t(101)=-.04, p=.97)).
Figure 2 presents these results using common scales for the six different items. The
positive slopes in the bottom four panels of the figure indicate that participants’ numerical
responses were less sensitive to units than they should have been. For example, in the bottom
right panel, responses for the volume of a bathtub in quarts were less than four times as large as
responses for the volume of a bathtub in gallons. So when the estimates are converted to quarts
(and logged), estimates that had been made in gallons (the rightmost points) are larger than those
that had been made in quarts (the leftmost points). This is the unit effect.
20
Subjective Unit Effect. An additional concern with the above analyses is that we used the
correct ratios between the sizes of units (rather than participants’ stated ratios) to convert
numerical estimates of length, weight, and volume to the smallest unit for each item. If
participants did not know the sizes of the units in which they were asked to respond, then they
may have provided numerical estimates that were either too high or too low. For example, a
participant who thought that a liter is smaller than it really is may have indicated that the
trashcan would hold more liters of water than it really would. Our use of the participant’s
numerical response together with the correct size of a liter would have misrepresented the
participants’ volume estimate as being larger than it really was. In fact, if participants generally
underestimate the ratios between the relevant units (e.g., if they think that there are fewer fluid
ounces in a liter than there really are), then our using the correct conversion ratios could lead to
positive slopes like those seen in Figures 1 and 2.
Continuing the example, there are 33.81 fluid ounces in one liter, but the geometric
mean of participants’ stated ratios was only 17.39 (whether this error stems from misperceptions
of the size of a fluid ounce, a liter, or both is not particularly important here). For those
participants who responded in liters, we multiplied their answers by 33.81 to convert them to
fluid ounces. If this factor was too big (in the sense it is larger than participants’ imagined ratios)
then the middle set of points for trashcan in Figures 2 is too high relative to the leftmost set of
points for trashcan.
Indeed, there is some evidence that participants underestimated many of the relevant
ratios, and that they were more likely to do so for weight and volume than for length. Table 2 is
provided in order to show how aware participants are of the units’ true ratios. See the first three
21
columns of Table 2. Column 1 gives the true ratios, whereas Column 2 gives the geometric
means of participants’ stated ratios. To illustrate, first two columns in the first row shows that
there are 1.09 meters in one yard actually, but on average participants thought that there are only
.78 meters in a yard. Column 3 gives the mean of ln(true ratio/stated ratio), which is positive if
participants overestimated the true ratio and negative if participants underestimated the ratio.
Many of these means are negative, particularly for weight and volume.
Table 2 implies that not all participants have the right idea regarding the size of units.
Especially, the answers for the weight and the volume units far deviate from the true answers.
Such incorrect perceptions of the units can be responsible for the positive slopes we see in Figure
2.
22
Figure 2. Objective estimates of length, weight, and volume of six items, plotted as functions of
objective units
23
Units
1)True
Ratio
meter/yard
1.09
2)Geometric
Mean of
Participants'
answer
0.78
3)Mean of
Log (P's
answer/True
answer)
-0.33
4)Mean of
|Log(P’s
answer/True
answer)|
0.79
5)Average
familiarity
cm/inch
2.54
4.34
0.30
0.56
3.81
foot/yard
3.00
4.19
-0.10
0.13
4.28
foot/meter
3.28
4.45
0.33
0.39
3.86
inch/foot
12.00
10.81
0.54
0.67
3.99
cm/foot
30.48
42.62
0.34
0.61
3.99
pound/kg
2.20
0.97
-0.82
2.77
3.41
ounce/pound
16.00
14.57
-0.09
0.61
3.37
gram/ounce
28.35
3.09
-1.43
2.28
2.91
ounce/kg
35.27
9.87
-1.27
2.91
2.79
gram/pound
453.59
50.32
-2.20
2.44
3.53
quart/liter
1.06
1.59
0.41
1.17
2.85
liter/gallon
3.79
2.77
-0.31
0.81
3.21
quart/gallon
4.00
4.00
0.00
0.48
3.04
f.ounce/liter
33.81
17.39
-0.66
1.22
2.59
f.ounce/liter
128.00
34.98
-1.30
1.50
2.70
3.39
Table 2. Participants’ answers to the conversion ratio and familiarity questions (Study1)
To see whether the apparent unit effect in our initial analysis resulted from participants’
inaccurate perceptions of unit sizes, we redid the analysis using subjective rather than objective
unit ratios. In this analysis, we used each participant’s stated conversion rates to convert
magnitude estimates in the medium- and large-unit conditions to the smallest unit. For example,
imagine that a participant who estimated the length of the lake in yards (the medium unit) also
indicated that there are 5 feet in one yard. Her “subjective estimate” for the length of the lake in
24
feet (the small unit) would be calculated by multiplying her original numerical estimate by 5
rather than 3. The aim of this process was to eliminate the misrepresentation of the estimates
resulting from participants’ misunderstanding the sizes of units. In addition to such changes in Y
values, we also changed participants’ X values to their subjective ratios, for consistency. As
before, both the X values (subjective unit sizes) and Y values (magnitude estimates for the items,
in subjective units) were logged for the regression analysis.
The overall effect of subjective unit (b=.61, χ2 (1, 102) = 35.05, p<.0001) and the
interaction with item (χ2 (1, 102) = 32.51, p <.0001) were again significant. This result shows
that the effect of the unit remains even when we account for incorrect knowledge of the units.
The effect of the unit was positive and significant for the same four items as in the previous
analysis (the weight of the brick (b=.50, t(100)=6.89, p<.0001), weight of a one-gallon jug of
milk (b= .76, t(100)=10.34, p<.0001), volume of the trash can (b= .40, t(101)=4.58, p <.0001),
and the volume of a bathtub (b = .86, t(101)=10.34, p<.0001)). Interestingly, the effect of the unit
was also significant for the two other items ((length of the table (b= .28, t (101)=5.74, p<.0001)
and the length of the lake (b= .83, t(101)=1.08, p<.0001)). Figure 3 presents the subjective
estimates as a function of subjective unit for each item.
Extreme Cases. Looking at Figure 3, some of the observations are unusually off from the
rest of observation. These outliers may have amplified the positive regression lines, which could
weaken the validity of our finding. Outliers on the lower left of each panel, which is far left
from the point in X axis, result from participants not knowing which of two units is larger. To
check if such outliers are distorting the effect of unit, we calculated Cook’s distance for each
point (Cook, 1977). Then the same regressions were conducted without observations where
25
Cook’s distance was larger than .04, which is 4 divided by number of observations for each item
(Bollen & Jackman, 1985). In this process, 4 observations for the table, 7 for the lake, 3 for the
brick, 4 for the milk, 5 for the trashcan, and 7 for the bathtub estimates were excluded. Even after
excluding the outliers, the effect of subjective unit on the subjective estimates remained
significant for all six items (length of the table (b = .22, t(97)=4.74, p<.0001), length of the lake
(b = .60, t(94)=6.38, p<.0001), weight of the brick (b = .47, t(97)=6.63, p<.0001), weight of a
one-gallon jug of milk (b = .61, t(96)=7.66, p<.0001), volume of the trashcan (b = .38,
t(96)=4.74, p<.0001), and volume of a bathtub (b = .83, t(94)=1.15, p<.0001)).
26
Figure 3. Subjective estimates of length, weight, and volume of six objects, plotted as functions
of subjective units
27
Familiarity and the Unit Effect. Taking into account the finding that participants’
knowledge of the relative sizes of the units varies, we also expected that participants would have
different familiarity with the various units. Table 3 presents the means of participants’ selfreported familiarity with each unit. It is notable that participants tended to be more familiar with
length units than with units for other dimensions. Regarding the fact that conversion ratios for
the length units were relatively accurate than for other dimension units (see column 3 of Table
2), this result is consistent with other results presented thus far.
Length
Weight
Volume
Unit
Mean
Standard
Deviation
Meter
3.34
1.21
Yard
3.43
1.29
Centimeter
3.70
1.09
Feet
4.28
0.97
Inch
4.28
0.88
Ounce
2.75
1.22
Kilogram
2.83
1.19
Gram
3.07
1.23
Pound
3.99
1.09
Fluid Ounce
2.17
1.07
Quart
2.68
1.17
Liter
3.02
1.08
Gallon
3.40
1.18
Note: 1 = Not at all familiar, 5 = Extremely familiar
Table 3. Self Reported Familiarity with Units on a 5-Point Scale (Study 1)
28
To establish the validity of self-reported familiarity, we assessed the relationship
between the accuracy of participants’ conversion ratios and their familiarity with the units
involved in each comparison. One measure of the accuracy of participants’ conversion ratios
appears in column 4 of Table 2. That column gives the means of |ln(actual ratio/stated ratio)|,
which is smaller when participants report more accurate ratios. The average of the familiarity
ratings for the two units in each comparison is given in column 5 of Table 2 (for example,
familiarity for meter/yard is the average of the familiarity values for meters and yards). Although
Table 2 presents averages, our analysis used values from the individual participants. As
expected, participants who were more accurate about the conversion ratio tended to be more
familiar with those units (b= -.50, F(1,1521)=174.05, p<.0001). This result was obtained from
predicting the average familiarity (column 5 in Table 2) from inaccuracy in the conversion ratio
(column 4 in Table 2). Note that the relationship is negative, as it indicates that less inaccuracy is
associated with more familiarity.
The more important question is whether the familiarity with the unit affects the
participants in making quantitative estimation. Therefore, we investigated the possibility of the
effect caused by the familiarity with the unit. Controlling the knowledge on the size of units did
not eliminate the trend of increasing estimates as the unit increases. Accordingly, we expected
that the familiarity should show the similar outcome, assuming that familiarity correlates with
the knowledge.
We asked whether participants’ self-reported familiarity with the units in which they
were making judgments affected the magnitude of the unit effect. Specifically, we predicted
subjective estimates on the basis of subjective unit size, as before, but with familiarity and the
29
interaction between familiarity and subjective unit size added as predictors in the model.
Familiarity and subjective unit size were both centered for this analysis. The effect of unit size
remained significant (b=.60, χ2 (1, 102) = 38.18, p<.0001). The effect of familiarity was not
significant (b=.12, χ2 (1, 102) = 2.20, p=.14), but the interaction between familiarity and unit size
was significant and negative (b=-.10, χ2 (1, 102) = 11.11, p<.0001). In other words, familiarity
moderated the unit effect, such that the unit effect was smaller when participants’ familiarity
with the unit was higher.
It is notable, however, that the unit effect was significant even when participants reported
that they were most familiar with the unit. Examined separately by the level of familiarity, the
unit effect was significant at all five levels of familiarity. The unit effect was the smallest (b=.28,
t(152)=5.12, p<.0001) among observations with the highest familiarity score (5), while it was the
largest with the lowest familiarity (1) (b=.79, t(46)=7.92, p<.0001).
Participants who knew the exact conversion ratios. Lastly, we examined the unit effect
using only observations from participants who correctly reported the exact ratios between
relevant units. Because conversion ratios between metric and nonmetric units are not integers,
we restricted this analysis to estimates made in United States customary units. Next, estimates
from participants who entered the incorrect conversion ratio between the given unit and the
smallest unit for the item question were dropped. Through this procedure, 384 observations were
deleted from the initial 612, leaving 228 observations for analysis. Note that this procedure
ensures that the subjective units are the same as the objective units, because only observations
from participants who correctly reported the true ratios were left. The estimates were converted
in to the smaller of the two remaining units for each item, for example ounce rather than gram.
30
The same analysis as before was repeated with these 228 observations. First, the overall
effect of unit was measured. As before, there were significant effects of unit size (b=.21, χ2 (1,
93) = 5.71, p<.05) and item (χ2 (5, 93) = 34.12, p<.0001), but the interaction between unit size
and item was no longer significant (χ2 (5, 93) = 6.00, p=.31). We suspect that the unequal
number of observations per item is responsible for the nonsignificant interaction. For instance,
there were only 5 observations left among estimates for the trashcan, since only 5 participants
knew exact conversion ratio between fluid ounce and gallon (1 gallon=128 fluid ounces.)
Next, self-reported familiarity with the units was added to the model, along with the
interaction between familiarity and unit size. The effect of unit size remained significant (b=.23,
χ2 (1, 93) = 8.11, p<.005), as did the effect of item (χ2 (5, 93) = 15.34, p<.01). No other effect
was found. Especially, the interaction between familiarity and unit size was not significant (b=.08, χ2 (1, 93) = 2.51, p=.11). It is noticeable, however, that the direction of the coefficient is still
negative, indicating the tendency toward a larger unit effect as familiarity gets smaller.
The current finding shows that effect of unit still remains even after eliminating the
possibility that incorrect knowledge about the units is responsible for the observed tendency.
Even participants who knew the exact size of units compared to the other units estimated the
same item as larger when they used larger units of measurement. Unlike when all the
observations were used, the interaction of unit and familiarity was reduced to the insignificant
level with observations made by participants who know the conversion ratio. One possible
interpretation is that participants who knew the correct conversion ratio were all familiar with the
units to the relatively similar degree (i.e., the range of familiarity is restricted), so that familiarity
does not make a substantial difference in the size of the unit effect. The mean familiarity with the
31
units used for estimates was indeed higher (M=3.8, SD=1.17, N=228) than before excluding the
participants who didn’t know the exact conversion ratio (M=3.3, SD=1.36, N=612). The
significant main effect of unit, however, still shows that participants are affected by the unit
when they make quantitative estimates.
Study 1 demonstrates the hypothesized effect of unit size on quantitative magnitude
estimates for physical objects. This effect remained significant when subjective rather than
objective unit conversion ratios were used and when participants who did not know the relevant
conversion ratios were excluded. The study also found a significant interaction, indicating that
the unit effect was smaller when participants reported greater familiarity with the units. Because
this result is correlational, however, it would be premature to conclude that familiarity with the
relevant units is responsible for the magnitude of the unit effect based on the current finding.
Therefore, next study is designed to manipulate participants’ familiarity with units, in order to
determine the role of familiarity in the unit effect.
32
Chapter 3: Study 2
The purpose of Study 2 was to examine the role of familiarity in the unit effect. In Study
1, we found a significant correlation between self-reported familiarity with the units and the
magnitude of the unit effect (i.e., moderation of the effect by familiarity). Correlational
significance, however, does not prove the casual impact of familiarity. In addition, one can still
question the credibility of participants’ self-reported familiarity with units. The following study
is intended to address these issues.
In order to manipulate participants’ familiarity with the units, we introduced fictitious
units that were defined in terms of familiar units. It was predicted that a stronger unit effect
would be observed for fictitious (and hence unfamiliar) units than for familiar units. As in the
previous study, subjective unit size was considered in comparisons of magnitude estimates.
Method
One hundred sixty participants from The Ohio State University estimated the volume of
the trashcan and the weight of the brick (the same brick and trashcan as in Study 1). In a 2 (unit
size) x 3 (familiarity) x 2 (item) mixed design, participants were assigned to either “small” or
“large” unit size, and to either “familiar,” “easy unfamiliar,” or “difficult unfamiliar” units.
Participants in the familiar-unit condition used conventional units such as gallons, liters, pounds,
and ounces. In both unfamiliar-unit conditions, participants read a scenario about foreign units
33
that are used in a fictitious country. The scenario explained the size of these unfamiliar units
using conversion ratios to United States customary units. All participants used their assigned
units to estimate the weight of the brick and the volume of the trashcan.
In easy unfamiliar-unit condition, participants were told that the foreign units were 1.33
times larger than the units in the corresponding familiar condition. For example, the small
familiar unit for the brick was ounce. Participants in the easy unfamiliar-unit condition were told
that the foreign unit “sio” was equal to 1.33 ounces. The purpose of setting imaginary units as
1.33 times the conventional unit instead of 1 or some other natural number was to reduce the
possibility that the new units would be perceived as equally familiar as the conventional units. In
the difficult unfamiliar-unit condition, the unit was the same size, but it was explained with a
more difficult conversion ratio to a different conventional unit. For example, participants were
told that foreign unit “sio” was equal to .083 pounds. Table 4 shows the design of Study 2 and
the units used in each condition.
As in Study 1, participants reported the conversion ratio for all possible pairs of units
used in the study, including the imaginary units used in the unfamiliar condition. Participants in
all conditions also indicated their familiarity with the units. The questions and response scales
were identical to those used in Study 1. Participants were dismissed after demographic questions
and debriefing.
In Study 2, we made more effort to encourage participants to pick up the brick and
examine the trashcan. We noticed that some of participants in Study 1 did not actually pick up
the brick or look back to examine the trashcan. In order to promote participants getting more
involved and physically experiencing with the items, we appended additional instructions to the
34
previous questionnaire (e.g., “We encourage you to take a closer look at the trash can to improve
your estimates”).
Target Item
Brick
(Weight)
Trashcan
(Volume)
Unit Size
Familiar Unit
Easy Unfamiliar Unit
Difficult
Unfamiliar Unit
Small
Ounce
Sio (=1.33 oz)
Sio (=0.083 lb)
Large
Pound
Sio (=1.33 lb)
Sio (=21.28 oz)
Small
Liter
Tou (=1.33 l)
Tou (=0.35 gal )
Large
Gallon
Tou (=1.33 gal)
Tou (=5.04 l)
Table 4. Units Used in the Six Conditions (2 (Unit Size) x 3 (Familiarity) ) for Each of the Two
Items (Study2)
Results and Discussion
Manipulation Check. We first checked whether the manipulation of familiarity was effective.
Across all conditions, participants reported that they were less familiar with the fictitious units
tou (M = 1.37, SD = .71) and sio (M =1.40, SD =.76) than with the conventional units liter (M =
3.49, SD =1.07), gallon (M = 3.74, SD =1.11), ounce (M = 3.37, SD =1.20), and pound (M =
4.04, SD =1.14). These means are from all participants regardless of what they used in their own
tasks. In order to see if participants’ familiarity in each condition differed as we intended, selfreported familiarity scores for the units used in each condition were extracted and observed.
Then, self-reported familiarity scores were predicted by three familiarity conditions. This
analysis was conducted separately by dimension, weight units for the brick and volume units for
35
the trashcan. Familiarity with weight units used in Study 2 was significantly different by
condition (X2(2, 159) = 61.52, p < .0001). The same was found for volume units (X2(2, 159) =
58.87, p < .0001). There was no significant difference in familiarity between the easy and
difficult unfamiliar-unit conditions for either dimension.
Data Preparation. As in Study 1, estimates were converted to the smallest unit used for
the item. Estimates for the brick in all conditions were converted into ounces and estimates for
the trashcan in all conditions were converted to liters.3 All estimates were then logged before
analysis.
Replication of Unit Effect. The means of logged estimates in each cell of the design
appear in Table 5. For both items in all familiarity conditions, estimates were larger when
participants used the larger unit, though some of these differences (e.g., 5.35 vs. 3.21) were much
larger than others (e.g., 4.69 vs. 4.63).
The first analysis of these data focused on the effect of “objective unit.” As in the
previous study, the smallest unit was set as 1. Hence, ounce was set as 1 for the brick and liter
was set as 1 for the trashcan. The true sizes of other units compared to these smallest units were
considered as the “objective units” in each condition. For the fictitious units, we used the
conversion ratio that we presented to the participants in the scenario. For example, the objective
unit sizes for participants in the small, easy unfamiliar-unit condition would be 1.33 ounces or
liters. All of these unit-size numbers were then logged.
3
Analyses were repeated with estimates converted to the largest unit as well. Results did not
differ, so only the results based on the smallest unit are presented in the main text.
36
Target
Item
Unit Size
Familiar Unit
Easy Unfamiliar Unit
Difficult Unfamiliar Unit
Brick
(Weight)
Small
4.63
4.32
3.21
Large
4.69
4.67
5.35
Trashcan
(Volume)
Small
2.26
1.74
2.34
Large
2.68
2.68
3.06
Table 5. Mean of logged estimates converted to the smallest unit for the target (N=159) (Study2)
As in the objective unit size analysis of Study 1, we regressed participants (logged)
estimates onto the (logged) unit sizes. The main effect of the objective unit size was again found
to be significant (b=.41, X2(1, 160) = 26.18, p <.0001), as was the effect of item (X2(1, 160)
=42.67, p <.0001). However, there was no interaction between unit size and item (X2(1, 160) =
1.68, p =.20). The uninteresting effect of item will be ignored as before. The effect of unit size
was examined by item even though there was no interaction. Estimates for both items were larger
when the size of the unit was larger (weight of the brick (b=.32, X2(1, 160) = 18.87, p <.0001),
volume of the trashcan (b=.49, X2(1, 160) = 15.55, p <.0001)).
The effect of the unit remained significant when subjective units were used. In this case,
the estimates were converted to the smallest unit for each dimension, based on participant’s
stated conversion ratio. Such change was also done for unit size (the predictor variable), as in
Study 1. Again, significant effects of unit size (b=.71, X2(1, 160) = 17.01, p <.0001) and item
(X2(1, 160) = 57.45, p =<.0001) were found. The interaction of the item and the subjective size
of the unit was nearly significant (X2(1, 160) = 3.68, p = .05), and we again examined the effect
of unit size for each item. As predicted, the significant effect of subjective unit size was found
37
from both items (weight of the brick (b=.56, X2(1, 160) = 17.52, p <.0001), volume of the
trashcan (b=.86, X2(1, 160) = 8.33, p <.01)).
We repeated this analysis after excluding outliers with larger Cook’s distances (Cook,
1977.) Among estimates for the brick, 12 observations with Cook’s distance larger than
.03(=4/160) were excluded. The effect of subjective unit size remained significant (b=.36, X2(1,
148) = 16.15, p <.0001). Similarly, 11 observations were excluded for the trashcan and results
still showed a significant effect of subjective unit size (b=.74, X2(1, 145) = 24.34, p <.0001).
Self-Reported Familiarity and Unit. The interaction between self-reported familiarity
and the size of the unit was also found, as in the previous study. When self-reported familiarity
with the unit and the interaction between familiarity and subjective unit size were added to the
model for predicting subjective estimates (with appropriate centering), there were significant
main effects of subjective unit size (b= .58, X2(1, 158) = 26.22, p < .0001), familiarity (b=.22,
X2(1, 158) = 9.34, p <.005), and item (X2(1, 158) = 5.77, p <.0001). More importantly, the
interaction of familiarity and unit size was significant (b=-.21, X2(1, 158) = 15.00, p < .0005),
where the negative value of the coefficient indicates that the unit effect was smaller when
familiarity was higher. So far, these findings show that the unit effect and the interaction with
familiarity are replicated successfully.
Objective Unit Effect Moderated by Familiarity Condition. The most important goal of
this second study was to assess the difference in magnitude of the unit effect depending on
familiarity, as manipulated by experimental conditions. We therefore conducted an ANOVA to
predict objective estimates on the basis of unit size, manipulated familiarity condition, and item.
All three independent variables had significant effects on the objective estimates (objective unit
38
(b=.39, X2(1, 159) = 27.27, p <.0001), familiarity condition (X2(2, 159) = 1.48, p = <.01), and
item (X2(1, 159) = 48.66, p < .0001)), but the main effect of unit size is of primary interest. This
unit effect differed in magnitude depending on the familiarity condition, as indicated by a
significant two-way interaction between unit size and familiarity condition (X2(2, 159) = 6.63, p
< .05). This result supports the hypothesis of Study 2, which was that familiarity with the unit of
measurement moderates the effect of unit size on quantitative estimates.
However, the interpretation of this hypothesized two-way interaction is qualified by a
significant three-way interaction between unit size, familiarity condition, and item (X2(2, 160) =
8.14, p < .05). To find the details, the two-way interaction between unit size and familiarity
condition was assessed separately for each item. In estimating the weight of the brick, main
effect of objective unit size (b= .31, X2(1, 160) = 21.05, p < .0001), familiarity (X2(2, 160) =
18.39, p < .0005), and the interaction with familiarity (X2(2, 160) =22.20, p < .0001) were all
significant. The estimates for the volume of the trashcan showed significant main effects of unit
size (b= .50, X2(1, 160) = 17.22, p < .0001) and familiarity (X2(2, 160) =7.42, p < .05). The
interaction, however, was not significant (X2(2, 160) = 2.43, p=.30). See Table 5 for the relevant
means.
The effect of objective unit was once again measured separately in the three different
familiarity conditions. In the familiar-unit condition where the participants used only
conventional units, the unit effect was not significant (b= .17, X2(1, 46) = 2.19, p= .14). In
comparison, this effect was significant in two other conditions where participants used fictitious
units (easy unfamiliar (b=.42, X2(1, 55) = 11.59, p < .001), difficult unfamiliar (b= .63, X2(1, 59)
= 15.52, p < .0001)).
39
Brick
Trashcan
Brick
Trashcan
Brick
Trashcan
Figure 4. Objective estimates of weight and volume of two items, plotted as functions of
objective units by familiarity condition
40
The same procedure was once again repeated by item. The three left panels in Figure 4
show the estimates for the weight of the brick predicted by objective unit size in each familiarity
condition. Simple regression results indicated a significant unit effect in the difficult unfamiliarunit condition (b= .78, t (58) = 6.11, p < .0001), but not in the familiar-unit condition (b= .02, t
(45) = .27, p =.79) or the easy unfamiliar-unit condition (b= .13, t (54) = 1.0, p =.32). Estimates
for the trashcan were also examined separately, with results presented in the three right panels of
Figure 4. A significant unit effect was found in the easy unfamiliar-unit condition (b= .70, t (54)
= 4.33, p < .0001) and the difficult unfamiliar-unit condition (b= .48, t (58) = 2.11, p < .05), but
not in the familiar-unit condition (b= .31, t (45) = 1.5, p=.14).
Subjective Unit Effect Moderated by Familiarity Condition. These analyses were repeated
using subjective units, with the estimates and units converted to subjective values for each
participant using the same procedure as in Study 1. Again, the purpose of this procedure was to
clarify the fact that incorrect knowledge of the units is not responsible for the observed unit
effect in the previous findings. Subjective estimates were predicted by subjective unit size,
familiarity condition, and item. The unit effect remained significant (b= .53, X2(1,160) = 27.08, p
< .0001), as did the two-way interaction between unit size and familiarity condition (X2(2,160) =
20.07, p < .0001). These effects were again examined by item. Looking only at estimates for the
weight of the brick, the main effect of unit size (b= .39, X2(1,160) = 23.36, p < .0001) and the
interaction between unit size and familiarity (X2(2,160) = 28.09, p < .0001) were significant.
Similar results were found for estimates of the volume of the trashcan (main effect of unit size
(b= .67, X2(1,160) = 17.84, p < .0001), interaction between unit size and familiarity condition
(X2(2,160) = 9.68, p < .01)).
41
Figure 5 presents subjective estimates for the brick and the trashcan in each familiarity
condition. Seen individually, estimates for the brick showed a significant effect of unit size only
in the difficult unfamiliar-unit condition (b= .87, t(58) = 9.80, p < .0001), but not in the familiarunit condition (b= .00, t(45) = .05, p =.96) or the easy unfamiliar-unit condition (b= .30, t(52) =
1.85, p =.07). For the trashcan, the unit effect was found in both unfamiliar-unit conditions (easy
(b=1.01, t(52) = 11.87, p <.0001), difficult (b= .81, t(58) = 8.52, p <.0001)), but not in the
familiar-unit condition (b= .18, t(45) = .79, p =.45).
Figure 5 also shows how participants’ answers to the conversion ratio questions deviate
from the true answers depending on the familiarity condition. In top two graphs where familiar
units were used, the distribution of the values on the X axis is much narrower than in the other
four graphs. Especially in the difficult unfamiliar-unit condition, where participants were
expected to have less accurate conversion ratios, the subjective unit sizes are widely distributed
in horizontal direction, indicating that many participants were unable to provide the correct ratio.
Extreme Cases. Figure 5 contains extreme outliers, which are easily noticed in multiple
panels. To ensure that these outliers are not responsible for the significant effect of subjective
unit size, we repeated the same regression analyses without the outliers. We again calculated
Cook’s distance for each point and excluded observations with Cook’s distance bigger than .03
(=4/159) from the regressions. Twelve estimates were excluded for the brick and thirteen
estimates were excluded for the trashcan. Results were essentially unchanged. The unit effect in
estimates for the brick was found in the difficult unfamiliar condition (b=.62, t(51)=6.32,
p<.0001), but not in the familiar unit (b=.04, t(44)=.50, p<.62) or the easy unfamiliar- unit
conditions (b=.25, t(48)=1.84, p=.07). In estimates for the trashcan, a significant unit effect was
42
found in both unfamiliar-unit conditions (easy (b=.78, t(49)=5.98, p<.0001), difficult(b=.79,
t(50)=8.81, p<.0001)), but not in the familiar- unit condition (b=.33, t(43)=1.72, p=.09). As in
Study 1, significant unit effects remained even after excluding outliers.
43
Brick
Trashcan
Brick
Trashcan
Brick
Trashcan
Figure 5. Subjective estimates of weight and volume of two items, plotted as functions of
subjective units by familiarity condition
44
Difficult vs. Easy conversion. One may argue that the way subjective estimates were
calculated in Study 2 is partially responsible for larger effect of unit size in the unfamiliar-unit
conditions. Some participants in those conditions were told the conversion ratio of fictitious units
to large familiar units (e.g., pound and gallon), but their subjective unit sizes were calculated
based on their reported conversion ratios to the small unit (e.g., ounce and liter). For example, a
participant might be told that one sio was equal to 1.33 pounds, but her subjective unit was based
on her stated conversion ratio for sio to ounces. Such participants had to perform additional
calculations to convert the fictitious unit to the smallest unit. This discrepancy in the ease of
converting the unit may be partly responsible for the effect of subjective unit size in our
regressions. To guard against this possibility, we repeated the same analysis with subjective unit
sizes and estimates calculated differently. This time, all fictitious units were first converted into
the conventional units that appeared in participants’ instructions, using participants’ stated ratio
for that conversion. Next, those values were converted to the smallest unit based on participants’
subjective conversion ratio between the relevant conventional units. For instance, if a participant
learned that one sio is equal to 1.33 pounds from the scenario, her estimate made in sio was
converted to pounds first. Next, her answer for the conversion ratio from pounds to ounces was
used to convert her estimate into ounces. Following this procedure, the overall effect of
subjective unit size was still significant (b=.62, X2(1,160) = 27.61, p < .0001), as was the
interaction between subjective unit size and familiarity condition (X2(2,160) = 22.21, p < .0001).
Results for separate analyses by familiarity condition or item were also similar to those from our
original analyses. The only difference found was for the subjective unit effect in estimates for the
45
brick in the easy unfamiliar condition (b=.52, t(52) = 3.59, p < .001), which was not significant
in the previous analysis (b= .30, t(52) = 1.85, p =.07 ). Remember, this result is from the
individual regression analysis by familiarity and item condition.
Participants who didn’t pick up the brick or look at the trashcan. To ensure that
estimates are based on physical experience, we asked participants in Study 2 if they actually
picked up the brick and looked at the trashcan. Seven of the 159 participants reported that they
did not actually pick up the brick, and 19 reported that they did not look back to see the trashcan
in the room. When the estimates from these participants were excluded from the data, the results
of our primary analyses were essentially unchanged.
Discrepancy between Study 1 and Study 2. Even though the effect of unit size was
significant overall, separate regressions by familiarity condition showed that unit size did not
have a significant effect on estimates in the familiar-unit condition. Since this result contrasts
with the result from Study 1, where all participants used conventional units throughout the entire
experiment, we investigated the difference between these two results. First, only estimates for the
same items and the same units used in Study 2 were extracted from the data of Study 1. That is,
only the estimates for the brick made in ounces and pounds and the estimates for the trashcan
made in liters and gallons were retained in both studies. As a result, estimates made by 71
participants in Study 1 were compared to the estimates made by 46 participants from familiarunit condition of Study 2. The effect of objective unit size was significant in Study 1 (b=.25,
X2(1,71) = 7.26, p < .01), but not in Study 2 (b=.17, X2(1,46) = 2.19, p = .14). The effect of
subjective unit was also significant in Study 1 (b=.51, X2(1,71) = 10.21, p < .005) but not in
Study 2 (b=.09, X2(1,46) = .79, p =.37). Seen separately by item, the objective unit effect was
46
significant for the trashcan (b=.39, X2(1,71) = 5.08, p < .05) but not for the brick (b=.10, X2(1,71)
= 2.54, p = .11) in Study 1, while neither effect was significant in Study 2. The subjective unit
effect was significant for the both items in Study 1 (brick (b=.23, X2(1,71) = .4.46, p < .01),
trashcan (b=.80, X2(1,71) = 7.85, p < .001)), but for neither item in Study 2. A possible
interpretation for this discrepancy between the two studies will be discussed in general
discussion section.
47
Chapter 4: General Discussion and Conclusions
The current research found significant evidence for a unit effect in quantitative
estimation. Again, we use the term “unit effect” to refer people’s tendency to make greater
quantitative estimates when larger units are used than when smaller units are used. We found
positive and significant effects of unit size in participants’ estimates for the weight and volume
of four items (Study 1). The overall effect remained significant after we eliminated the possible
influence of participants’ incorrect knowledge about the units. The results of Study 1 also
indicate that the unit effect is moderated by participants’ self-reported familiarity with the units.
As participants reported lower familiarity with units, the effect was stronger. This finding was
replicated and extended in Study 2, where we manipulated familiarity by providing fictional
units. As we predicted, participants in the two unfamiliar-unit conditions were affected by the
size of unit to a greater degree than were participants in the familiar-unit condition.
The existence of a unit effect in quantitative estimation is consistent with findings from
the literature reviewed in the introduction of this thesis (Burson et al., 2009; LeBoeuf & Shafir,
2009; Pandelaere et al., 2011; Raghubir & Srivastava, 2009; Wertenbroch et al., 2007; Wong &
Kwong, 2000). That literature indicates that the unit in which quantitative information is
expressed affects evaluations and comparisons because people naturally focus more on the
number than on the unit, even though both are necessary. The current research shows that in
48
quantitative magnitude estimation, people’s adjustments to their numerical responses are
insufficient to compensate for differences in units.
Although Monga and colleagues (Monga & Bagchi, 2011) reported that unit effects
could be reversed when units were made salient, we found the standard unit effect in estimation
tasks in which the unit is very salient. Those authors argued that the direction of the effect
differs depending on the relative salience of the number and the unit. They showed that the unit
effect can be reversed if the unit is made salient to participants so that the unit draws more
attention than the number does. The results from our two experiments, however, show that the
unit effect still remains even when the unit is very salient. We explicitly specify the unit for
participants to use in our primary estimation tasks. Participants have to fill the blank with their
own numeric response corresponding to the presented unit. In this manner, the units in our
experiments are even more salient than those in the comparative estimation tasks in Monga’s
and others’ research, where both units and numbers are presented. Despite this unit salience, the
unit effect remains consistent, with quantitative estimates made in large units being larger than
quantitative estimates made in small units.
The current research supports the existence of a unit effect in quantitative estimation and
the moderation of this effect by participants’ familiarity with the relevant units. At this time,
however, we do not have sufficient evidence to explain the psychological processes behind the
unit effect. The next step in our research, therefore, is to investigate the possible mechanisms
for the unit effect.
Three plausible mechanisms come to mind. The first is that people avoid using
extremely large or small numbers in their estimates, regardless of the units. Although
49
participants might have the general intention to provide a correct estimate, they might assume
that an extreme number is not likely to be correct. Such a tendency would lead participants to
avoid very small numbers in situations where large units are used and to avoid very large
numbers in situations where small units are used, even if more extreme answers would be more
accurate. This would lead to magnitude estimates being larger for large units than for small units,
as observed.
A second mechanism is based on anchoring and insufficient adjustment (Tversky &
Kehneman, 1974; Epley & Gilovich, 2001). It is possible that, when asked to estimate the
magnitude of some item, participants’ first response is to estimate the magnitude in whatever
unit seems most natural and appropriate for the question, and then convert that initial answer into
the unit requested. For example, when asked to estimate the weight of the brick in ounces, a
participant might spontaneously think that the brick probably weighs about 5 pounds. If the
participant knows that there are 16 ounces in a pound, and is able and willing to do the math, he
can convert pounds to ounces and answer the question. However, if he knows only that an ounce
is a good bit smaller than a pound, he may simply adjust his numerical answer upward to
compensate for the smaller unit. If, as in other tasks, this adjustment is insufficient, his final
answer will be too small. Of course, this would work in the other direction if the most natural
unit were small and the question asked for a response in a larger unit. Either way, insufficient
adjustment would lead to a unit effect in the observed direction.
A third mechanism involves the role of conversational norms in magnitude estimation.
We discuss this mechanism in much greater detail below, because we think that it is particularly
plausible.
50
Inferences from Conversational Norms
We propose conversational norms as a likely factor driving the unit effect. Grice’s
(1975) maxims for conversation have long been discussed in linguistics and psychology. People
follow implicit rules when engaging in smooth, meaningful conversation, even though these
rules sometimes violate formal implications of rational logic. Grice summarized several
maxims that we follow when answering questions. The ultimate goal of following maxims is to
be as informative and appropriate as possible, given the context provided by the question. The
“maxim of relation,” for example, commands us to provide the relevant answer to the speaker’s
question. In order to obey this principle, we make inferences based on the context of the
question, in addition to the surface content of the question. Grice called such inferences
“implicature.” The reason we make inferences from questions when forming answers is because
we believe that other people will also follow the maxims. Especially when we are not sure
about the answer, we believe that the speaker selected the particular wording of the question for
good reasons. In our studies, it is likely that linguistic information from the unit, in addition to
the objective request for quantitative information, had an influence on the judgments of
participants who had the intention to follow maxim of relation.
Sher and McKenzie (2006) illustrate that people make inferences based on the framing
of options, even though such framing is logically irrelevant to people’s choices. In their Studies
1, 2, and 3, normatively equivalent sentences describing the same object led to different
behavior by participants. All participants had one cup filled with liquid and one empty cup in
the beginning. Half of them were requested to make “a half (or 1/p) full cup” while the other
51
half were told to make “a half (or 1-1/p) empty cup.” The dependent variable was which of two
cups participants choose to use when furnishing the requested amount of liquid. The authors
predicted that more participants would select previously full cup when the instruction stated
“empty” rather than “full,” and the results supported this hypothesis. The explanation for this
result is that information about the correct selection (even though there is no correct or incorrect
choice) “leaks” from the language used in the instruction and is reflected in participants’
behavior. Participants, therefore, infer from the word “empty” that the original status of the
requested cup was “full.” They therefore choose the originally “full” cup to set required amount
of liquid in it.
Similar tendencies were observed in subsequent studies where participants selected a
phrase to describe the outcome of a rolled die or a flipped coin (Study 4), or the probability of
project success (Study 5). The choice of language to describe the event reflected the reference
point that participants set at the outset. For instance, a participant who rolls a die with 5 white
sides and 1 black side sets her reference point as the white side coming up frequently. When the
participant rolls the die multiple times, having the black side come up more often is above the
initial reference point. This valence is reflected in the choice of words to describe the outcome.
Therefore, the participant chooses to say that the die came up “black p times” out of total,
although she could just as easily use “white (1-p) times” to convey the same result. The
meaningful point for our current research is that participants often rely on the nuances of
language when communicating quantitative information. Just as Sher and McKenzie’s (2006)
participants went behind the explicit meaning of the numeric information to find the relevant
52
answer with a consistent pattern, there is a similar possibility that such tendencies influenced
the reported quantitative estimates in our research.
A further example is even more specific to the use of units. Recent research illustrates
the possible role of conversational norms in communicating quantitative information in different
units (Zhang & Schwarz, 2011). In that research, participants received the schedule of a
construction project and forecasted the actual completion time. The schedule was communicated
either in a large unit (e.g., years) or a small unit (e.g., weeks). The predicted completion date was
closer to the initially presented schedule when the schedule was expressed in a smaller unit
(Study 1). However, this tendency was found only when the schedule information was
communicated by a credible source (Studies 2 and 3). Zhang and Schwarz argue that participants
interpret the use of a smaller unit as conveying a more accurate and confident estimate of the
completion time. Such interpretations carried over to choice of consumer goods when attributes
were framed in different units (Study 4). What these results imply to us is that participants make
inferences based on the unit in order to evaluate the credibility of a number. The fact that
participants relied on unit size to estimate other attributes of information suggests the possibility
that conversational norms play a role in the interpretation of quantitative information.
On the other hand, results from studies regarding foreign currency (Wertenbroch et al.,
2007; Raghubir & Srivastava, 2009) may provide an example in which the unit effect is not
driven by conversational norms. In their studies, participants read a scenario involving the
purchase of goods in a foreign market. Participants had to provide their willingness to pay for a
product in, for instance, Korea, using Korean won as the currency (Raghubir & Srivastava,
2009). Because the Korean won is the currency typically used in Korea, the scenario provided a
53
clear reason for using a unit of that size. Hence, it is unlikely that participants would use
conversational norms to infer the value of the product based on the size of the chosen unit.
However, the unit effect was still observed. At least for studies regarding currency, inferences
based on conversational norms may not be an essential part of the unit effect.
Conversational Norms in Estimates of Physical Quantities
Although it may not be the only driving factor behind the unit effect, it would be
reasonable for participants to use conversational norms when estimating physical quantities. For
future research, we can make predictions regarding the effects of conversational norms on the
unit effect. Conversational norms can involve inferences either from the unit to the item or from
the item to the unit. We hypothesize that the direction of inferences is mostly influenced by the
participant’s relative familiarity with the unit and the target item being estimated. However, it is
important that the influence of the conversational norm should result in a unit effect in the same
direction as that observed, regardless of the direction of the inference. What follows is a detailed
explanation of this hypothesis.
Participants who are more familiar with the unit than with the item are predicted to rely
on the size of unit to make an appropriate inference regarding the item. According to Gricean
implicature (Grice, 1975), they assume that the question is reasonable and intended to elicit
relevant answers. In particular, they presume that a unit of appropriate size has been selected for
the question regarding the given item. As a result, they approximate the estimate for the item
based on what they know about the unit. For example, if a participant is asked to estimate the
weight of a brick in ounces, and she knows that an ounce is a small unit of weight, she might
54
infer that the brick in question is relatively light (particularly if she has not picked up the brick).
A person asked to estimate the weight of brick in pounds might infer that the brick is heavier (or
at least that it’s not particularly light). Participants in these two conditions make different
inferences regarding the weight of the brick and make quantitative estimates that differ in the
same direction as their inferences.
On the other hand, inferences from the item to the unit are likely to occur when
participants are more familiar with the item than with the unit. For example, a participant might
be relatively unfamiliar with a gram. If he knows that bricks are usually heavy, he might infer
that a gram isn’t very small (otherwise it would be an inappropriate unit for assessing bricks).
And if he thinks that a gram is bigger than it really is, then he will provide a numerical response
that is too low. As before, such tendencies would lead to a unit effect in the observed direction.
Thus, the effect of conversational norms would be in the same direction, regardless of whether
the inferences are from units to items or from items to units. Moreover, a greater reliance on
conversational norms would result in a larger unit effect.
Although we have no strong evidence regarding inferences from conversational norms in
our studies, we see two ways of assessing their potential effects. The first assessment involves
inferences from units to items. Presumably, inferences form units to items would be greatly
lessened if participants became more familiar with the items through experience. Once you’ve
picked up the brick, there is much less need to infer anything about its weight from the unit used
in the question. In order to check this hypothesis, we compared the unit effects observed in Study
1 and Study 2. In Study 1, we noticed that the majority of participants did not pick up the brick to
assess its weight or look back the trashcan to estimate its volume. Therefore, we strongly
55
encouraged participants in Study 2 to actually pick up the brick and take a close look at the
trashcan before making their estimates. We even placed the brick in front of the keyboard so that
a participant would have to move it in order to type. Such physical experience should reduce the
influence from conversational norms in Study 2, relative to Study 1. We extracted the
observations made for the same items in the same units from the two studies (See Chapter 2,
Discrepancy between Study 1 and Study 2). The effect of subjective unit was significant in Study
1(b=.51, X2 (1,71) = 10.21, p < .005) but not in Study 2 (b=.09, X2(1,46) = .79, p =.37). To test
the difference more formally, we analyzed the combined data using a new variable for study, a
subjective unit-size variable (as before), and the interaction between these two predictors. The
main effect of subjective unit (b=.30, X2(1,117) = 11.96, p < .001) and interaction with study
(b=.21, X2(1,117) = 8.16, p < .005) were both significant. We interpret this result as indicating
that physical experience with items overrides the conversational norms implied by the unit in the
questions. Thus, participants in Study 1 relied more on what they could infer from the
questionnaire, but such inference became unnecessary when participants in Study 2 physically
sensed the weight and volume of items. This interpretation remains speculative, of course,
because we did not randomly assign participants to the different situations.
The second assessment involves inferences from items to units. Recall that after the
primary estimation task, we assessed participants’ knowledge of the ratios between units. If
participants made inferences from an item to a unit in the primary task (e.g., that a gram isn’t
very small if it is an appropriate unit for a brick), it is possible that this inference would be
carried over to the later ratio task. Inferences about the size of a gram would be reflected not
only in their answer to the primary estimation task for the brick, but also in their conversion ratio
56
between gram and other units (e.g., 1 pound = ____ grams). More generally, if participants used
an unfamiliar unit to make an estimate for items that they knew relatively well, they would have
made an inference regarding the size of unit. There is a chance that this inference would still be
active when they answered our conversion-ratio questions. Indeed, not many participants in
Study 1 reported accurate conversion ratios or high familiarity with the gram. A pound is
equivalent to 454 grams, but the geometric mean for the conversion ratio was only 50.32 (See
Table 2). The average familiarity rating was 3.07 for gram (SD=1.23, N=102), compared to 3.99
for pound (SD=1.09, N=102) on the 5-point scale (See Table 3). On average, the low conversion
ratio is consistent with many participants thinking that a gram is larger than it really is. However,
it is important to note that a conversational-norm explanation for the low ratio applies only to
those who were asked to estimate the weight of the brick in grams, not to those who were asked
to estimate the weight of the brick in pounds, because the latter group would have no reason to
make brick-related inferences regarding the size of a gram. If this is correct, then the average
grams/pound ratio should be smaller in the gram condition than in the pound condition.
Following this rationale, we compared the grams/pound conversion ratios reported by
participants from small unit condition (g) and large unit condition (lb) in Study 1. According to
our prediction for inferences from the item to the unit, we expected smaller ratios in small-unit
condition. However, the mean of logged responses for the conversion ratio for grams/pound in
the small-unit condition (M=4.52, SD=1.96) was higher than that in the large-unit condition
(M=3.76, SD=2.31). Although the difference was not statistically significant (t(64)=1.41, p=.16),
it is inconsistent with our prediction. We also tested a separate but similar pair of units (gallons
and fluid ounces, both used for estimating the volume of the trash can) in the same manner.
57
Participants in Study 1 were relatively familiar with the gallon (M=3.40, SD=1.18, N=102), but
not the fluid ounce (M=2.17, SD=1.07, N=102). If participants made inferences regarding the
size of a fluid ounce on the basis of the trash can, and if these inferences carried over to the
conversion-ratio task, we would predict that the reported number of fluid ounces per gallon
would be smaller in the small-unit condition (the reasoning is exactly the same as that for grams
and pounds). However, the conversion ratio (1 gallon = ________ fluid ounces) was again higher
in the small-unit condition (M=4.16, SD=1.17) than in large-unit condition (M=3.23, SD=1.31).
This difference was significant, but in the opposite direction of our prediction (t(66)=3.07,
p=.0031). Thus, this assessment provides no evidence that participants made inferences from
items to units. This is a fairly weak test, however, because it relies on questions that followed the
primary estimation task. It remains possible that participants used conversational norms to make
inferences from items to units in the primary task, but that such inferences had no effect on
responses in the later conversion-ratio task. To summarize, we cannot yet draw strong
conclusions regarding the influence of conversational norms in magnitude estimation, because
the evidence is mixed and weak. Collecting more direct evidence is an important task for future
research.
Recently, we have designed an experiment to test the possibility of eliminating
conversational norms from our magnitude estimation task. Participants will estimate the target
item in different units, as in our previous experiments. The difference is that we will tell half of
the participants that the units have been chosen randomly by the computer. In previous research,
mentioning to participants that the given information is selected randomly influenced judgment
and decision making. For example, participants relied more on base-rate information than
58
irrelevant personal information when the description was sampled randomly by a computer
rather than when it was based on an expert’s statement (Schwarz, Strack, Hilton, & Naderer,
1991). In this study, random selection by a computer was used as a tool to eliminate inferences
from the source of the information (human) to the type of information (personal information).
Other researchers also suggest that participants make fewer inferences from how information is
presented when they are aware of randomness in the selection of frames (Sher & McKenzie,
2011). Therefore, we expect such instruction to eliminate or at least reduce the role of
conversational norms because, in the absence of an experimenter choosing an “appropriate” unit,
participants would have no reason to infer anything about the item from the unit and no reason to
infer anything about the unit from the item. If the unit effect is significantly smaller in the
random condition than in standard condition, the results will indicate the responsibility of
conversational norms in magnitude estimation, as well as the size of their effect. This in turn will
help us to understand the mechanisms underlying the unit effect and to plan further research.
Implications of the Unit Effect
The current studies establish the existence of a unit effect in magnitude estimation.
Findings suggest that we make larger estimates for length, weight, and volume when we use
units of larger size than when we use units of smaller size. In addition, this unit effect is
moderated by familiarity, with the effect being larger for unfamiliar units and smaller for
familiar units. Speculation regarding psychological mechanisms suggests the possibility that
conversational norms play an important role in the unit effect. The evidence collected so far,
however, is not consistent or robust enough to draw strong conclusions regarding the effects of
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conversational norms. Hence, more effort to clarify the influence of conversational norms on the
unit effect, including evidence on the direction and magnitude of such influence, is critical to
understand our general tendencies in quantitative estimation. Depending on the results regarding
conversational norms, it may also be fruitful to investigate other possible mechanisms for the
unit effect.
Even if it turns out that conversational norms are a main driving factor in the unit effect,
the findings from the current research still make an important contribution. The findings
underline the importance of unit choice in marketing strategy, risk communication, financial
decision making, and many other domains of everyday life. For instance, it might be beneficial
for a delicatessen or grocery store to use larger units when customers can specify the amount of a
good that they wish to purchase, since customers might overestimate the amount of food they
want when using larger units. For example, they might request a larger amount of cheese when
they using pounds rather than ounces. Similarly, when planning a project like finishing one’s
thesis, the unit of time used to forecast the final completion date can affect one’s estimate. A
student who is asked to plan in months is more likely to perceive a longer time until the end (and
possibly take longer to finish) than is a student who is asked to plan in weeks.
60
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Appendix A: Example Questionnaire in Study 1 and 2
Question in Study1
Estimations
The purpose of this research study is to improve our understanding of how people make
estimates of physical quantities. The questions in this computer-based survey involve estimating
length, weight, and volume.
There are no anticipated risks as a result of your participation in this study. We will not ask
for your name as part of the study, so your responses will not be linked with your name in any
way. The only benefit to you and others as a result of your participation is a greater
understanding of decision processes, as explained in the debriefing materials that will be
provided at the end of the session. The experiment will last about 30 minutes and you will
receive half of one credit hour towards your REP requirement. If you have questions about the
research, or in the extremely unlikely event of a research-related injury, please contact Dr. Mike
DeKay, 224 Lazenby Hall, phone 292-1837. For questions about your rights as a participant in
this study or to discuss other study-related concerns or complaints with someone who is not part
of the research team, you may contact Ms. Sandra Meadows in the Office of Responsible
Research Practices at 1-800-678-6251.
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Your participation in this study is voluntary. Refusal to participate will involve no penalty or
loss of benefits to which you are otherwise entitled. You may also discontinue participation at
any time without penalty or loss of benefits to which you are otherwise entitled.
If you wish to participate in this study, please click the Next button.
In this part of the study, we will ask you to estimate the length, weight, or volume of objects in
the room. We don’t expect you to know the exact answers exactly, but please answer each
question with your best estimate. You may use decimals in your answers if you would like. You
may begin when you are ready.
[1]4 Please take a look at the table you are sitting at right now. How wide is the table from left to
right in centimeters(inches/feet)?
______________ centimeters(inches/feet)
[2] Please pick up the brick that is on the table in front of you. How much does the brick weigh
in grams(ounces/pounds)?
______________ grams(ounces/pounds)
[3] There is an empty trash can on the table by the door. How many fluid ounces(liters/gallons)
of water would it take to fill the trash can up to the brim? You are welcome to take a closer look
at the trash can to improve your estimate.
______________ fluid ounces(liters/gallons)
4
Words in the bracket did not appear to the screen for participants.
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In this part of the study, we will ask you to estimate the length, weight, or volume of objects not
in this room. We don’t expect you to know the answers exactly, but please answer each question
with your best estimate. You may use decimals in your answers if you would like. You may
begin when you are ready.
[4] Imagine Mirror Lake at Ohio State University. How long is Mirror Lake (straight across, in
its longest dimension) in feet(yards/meters)?
______________ feet(yards/meters)
[5] Imagine an unopened one-gallon plastic jug of milk. How much does the jug of milk weigh in
ounces(pounds/kilograms)?
_____________ounces(pounds/kilograms)
[6] Imagine a typical bathtub in an average American house. How many quarts(liters/gallons) of
water would it take to fill the bathtub to a depth of eight inches?
______________ quarts(liters/gallons)
In this part of the study, we will ask you to compare different units of measurement. Please
answer to the best of your ability.
[1] Which of the following two unit is larger? [These are listed in alphabetical order.]
_____ centimeter
_____ inch
[If centimeter] Please indicate how many inches it would take to equal one centimeter.
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1 centimeter = _____ inches
[If inch] Please indicate how many centimeters it would take to equal one inch.
1 inch = _____ centimeters
[Same question would compare each of the following unit pairs.]
(centimeter & foot), (inch &foot), (foot & yard), (foot & meter), (yard & meter), (gram &
ounce), (gram & pound), (ounce & pound), (ounce & kilogram), (pound & kilogram),
(fluid ounce & liter), (fluid ounce & gallon), (liter & gallon), (quart & liter), (quart & gallon)
In this part of the study, we would like to know how familiar you are with a number of different
units. Just rate your familiarity on the scale provided.
[Appear in random order]
1. How much are you familiar with “centimeters”?
2. How much are you familiar with “inches”?
3. How much are you familiar with “feet”?
4. How much are you familiar with “yards”?
5. How much are you familiar with “grams”?
6. How much are you familiar with “ounce”?
7. How much are you familiar with “pounds”?
8. How much are you familiar with “fluid ounce”?
9. How much are you familiar with “liters”?
10. How much are you familiar with “quarts”?
11. How much are you familiar with “meters”?
12. How much are you familiar with “kilograms”?
13. How much are you familiar with “gallons”?
[The following scale (in horizontal form) will appear for each question above.]
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1) Not at all familiar
2)
3) Moderately familiar
4)
5) Extremely familiar
What system of measurement was most commonly used in the country you were raised in, before
you began attending Ohio State University? Please check one.
1) The United States Customary System (inch, foot, ounce, pound, quart, etc.)
2) The metric system (centimeter, meter, gram, kilogram, liter, etc.)
3) Some other system (please specify: __________)
Finally, we would like you to answer a few questions about yourself. This information will be
very useful in helping us describe the types of people who participated in our study.
Do you have any guesses about the specific goal of this study or about the specific
hypothesis that we are testing? If yes, please describe your guess(es) in the box below. If no, just
type "no"
Answer_______________________________________________________________
What is your sex? Please check one.
 Male
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 Female
What is your age in years? _______________
Are you Hispanic or Latino? Please check one.
 Yes
 No
How would you describe your race? Please check all that apply.






American Indian or Alaska Native
Asian
Black or African American
Native Hawaiian or Other Pacific Islander
White
Other (Please specify)
Is English your first language? Please check one.
 Yes
 No
Thank you for participating in this study!
[The end of the questionnaire]
Question in Study2
Estimations
69
[…]5In this part of the study, we will ask you to estimate the weight or volume of the object in
the room. We don’t expect you to know the exact answers, but please answer each question with
your best estimate. You may use decimals in your answers if you would like. You may begin
when you are ready.
[Familiar Unit Condition only]
[1] There is an empty trash can on the table by the door. How many gallons(liters) of water
would it take to fill the trash can up to the brim? We encourage you to take a closer look at the
trash can to improve your estimate. Please use decimals as needed.
______________ gallons(liters)
[2] Please pick up the brick that is on the table in front of you. How much does the brick weigh
in pounds(ounces)? Please use decimals as needed.
______________ pounds(ounces)
[Easy Unfamiliar Unit Condition only]
In an island named Woodhaki, the most prevalently used unit for volume and weight are “Tou”
and “Sio”. “Tou” is equivalent to 1.33 Gallons(1.33 Liters), and “Sio” is equivalent to 1.33
Pounds(1.33 Ounces). Please answer to the following questions using the “Tou” and “Sio” units.
5
First instruction was identical to what was shown in Study 1.
70
[1] There is an empty trash can on the table by the door. How many “Tou”s of water would it
take to fill the trash can up to the brim? We encourage you to take a closer look at the trash can
to improve your estimate. Please use decimals as needed.
______________ Tous
[2] Please pick up the brick that is on the table in front of you. How much does the brick weigh
in “Sio”? Please use decimals as needed.
______________ Sios
[Difficult Unfamiliar Unit Condition only]
In an island named Woodhaki, the most prevalently used unit for volume and weight are “Tou”
and “Sio”. “Tou” is equivalent to 0.35 gallons(5.04 liters), while “Sio” is equivalent to 0.083
pounds(21.28 ounces). Please answer to the following questions using the “Tou” and “Sio” units.
[1] There is an empty trash can on the table by the door. How many “Tou”s of water would it
take to fill the trash can up to the brim? We encourage you to take a closer look at the trash can
to improve your estimate. Please use decimals as needed.
______________ Tous
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[2] Please pick up the brick that is on the table in front of you. How much does the brick weigh
in “Sio”? Please use decimals as needed.
______________ Sios
[Familiarity Questions from Study 1 in all conditions]
[…]
1. How much are you familiar with “liters”?
[Similar questions forTou, Gallon, Ounce, Sio, Pound were repeated]
In this part of the study, we will ask you to compare different units of measurement. Please
answer to the best of your ability.
Which of the following two unit is larger?
_____ Liter
_____ Tou
[If liter] Please indicate how many tous it would take to equal one liter.
1 Liter= _____ Tou
[If tou] Please indicate how many liters it would take to equal one tou.
1 Tou = _____ Liter
[Same question would compare each of the following unit pairs.]
( Tou &Gallon),(Ounce & Sio), (Sio &Pound), (Liter & Gallon), (Ounce & Pound)
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Now, we would like to check how you did in the previous questions. Your answers to these
questions have no impact on your REP credit.
Did you actually pick up the brick to estimate the weight?
Did you actually approach the trashcan to estimate the volume?
Thank you for participating.
[The same demographic questions from Study 1 were asked]
[The end of the questionnaire]
73