From group diffusion to ratio bias: Effects of denominator and numerator salience on intuitive risk and likelihood judgments

Judgment and Decision Making, Vol. 4, No. 5, October 2009, pp. 436–446
From group diffusion to ratio bias: Effects of denominator and
numerator salience on intuitive risk and likelihood judgments
Paul C. Price? and Teri V. Matthews
California State University, Fresno
Abstract
The group-diffusion effect is the tendency for people to judge themselves to be less likely to experience a negative
outcome as the total number of people exposed to the threat increases — even when the probability of the outcome is
explicitly presented (Yamaguchi, 1998). In Experiment 1 we replicated this effect for two health threat scenarios using a
variant of Yamaguchi’s original experimental paradigm. In Experiment 2, we showed that people also judge themselves
to be less likely to be selected in a lottery as the number of people playing the lottery increases. In Experiment 3, we
showed that explicitly presenting the number of people expected to be selected eliminates the group-diffusion effect,
and in Experiment 4 we showed that presenting the number expected to be affected by a health threat without presenting
the total number exposed to the threat produces a reverse effect. We propose, therefore, that the group-diffusion effect
is related to the ratio bias. Both effects occur when people make risk or likelihood judgments based on information
presented as a ratio. The difference is that the group-diffusion effect occurs when the denominator of the relevant ratio
is more salient than the numerator, while the ratio bias occurs when the numerator is more salient than the denominator.
Keywords: risk judgment, probability judgment, group-diffusion effect, ratio bias.
1 Introduction
outcome. One of his scenarios, for example, read as fol-
lows.
In many situations, there is “safety in numbers.” A person
An infectious disease is prevalent in a foreign city. The
in a group is less susceptible to a wide variety of threats
disease comprises a fever and temperatures of over 39
than a person who is alone. Consider, for example, a trav-
degrees for more than a week together with severe diar-
eler who is walking in a strange city at night. If this per-
rhea. Although the death rate is not high, the disease has
son is in a group, he or she is probably less likely to get
after-effects such as total hair loss. The city authorities
lost, less likely to be mugged, and more likely to receive
are afraid of losing tourists from abroad and have kept
help in the event that he or she twists an ankle. However,
the matter con?dential. A group of 10 Japanese tourists
there are many other kinds of threats for which being part
including yourself has arrived in the city and plan to stay
of a group makes no difference. Imagine, for example, a
for one week. How likely do you think it is that you will
traveler who eats at a local restaurant. If the food in the
catch the disease if you stay as planned for one week?
restaurant were found to be contaminated with E. coli,
For each scenario, there was an alone condition in
then the person would be just as likely to get sick after
which the participant was exposed to the threat alone
dining in a group as after dining alone. There is evidence,
(“You have arrived in the city . . . ”), a small-group con-
however, that people tend to confuse the latter type of sit-
dition in which the participant was one of 10 people ex-
uation with the former. That is, people sometimes per-
posed, and a large-group condition in which anywhere
ceive an illusory safety in numbers.
from 100 to 1 million people were exposed, depending
In the ?rst demonstration of this effect, Yamaguchi
on the scenario.
(1998) presented college students with one of six scenar-
Across all the scenarios, there was a strong tendency
ios in which they were exposed to a threat (e.g., a carcino-
for participants to give lower probability judgments as the
gen, a ?nancial risk) and asked them to judge the prob-
number of people exposed to the threat increased. Fur-
ability that they would experience an associated negative
thermore, the function relating group size to perceived
?We would like to thank Andrew Smith and the members of the
risk was roughly logarithmic. There was a large drop
Judgment and Reasoning Lab at California State University, Fresno for
in perceived risk from the alone to the small-group con-
their help in conceptualizing and carrying out this research. Address:
dition, with a smaller drop from the small-group to the
Paul C. Price, Department of Psychology, California State University,
Fresno, 2576 East San Ramon Drive, Fresno, CA, 93711. Email:
large-group condition. Yamaguchi (1998) referred to this
paulpri@csufresno.edu.
as the group-diffusion effect and he proposed that it oc-
436

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
437
curs because people use an interdependence heuristic.
ory. People do not attend to and weight the denominator
They apply the general rule that “I am safer in a larger
more than the numerator because it makes them feel safe.
group” even when group size is irrelevant. According
They attend to and weight it more because it is more at-
to Yamaguchi, the interdependence heuristic could have
tentionally salient.
evolved as the cognitive concomitant of the motivational
A potential problem with this idea is that there is con-
mechanisms that lead humans — and many other animals
siderable research on a phenomenon called the ratio bias
— to form and maintain social groups. Of course, these
that seems to show just the opposite. When reasoning
mechanisms evolved because there often is safety in num-
about likelihoods based on ratios, people attend to and
bers. Individuals in groups are better able to fend off at-
weight numerators more than denominators. For exam-
tacks, ?nd mates, ?nd or create shelter, and forage suc-
ple, Denes-Raj and Epstein (1994, following Piaget & In-
cessfully. But the group-diffusion effect is the result of
helder, 1951/1975) asked people to choose between two
over-applying this generally useful rule.
gambles. In one, they would win if they selected a red
Our purpose in conducting the present research on the
jelly bean from a bin containing 10 jelly beans, where
group-diffusion effect was twofold. First, we wanted to
one of them was red. In the other, they would win if they
replicate it, because there have been only two published
selected a red jelly bean from a bin containing 100 jelly
studies on it since Yamaguchi’s (1998), which was con-
beans, where between 5 and 9 of them were red. Sur-
ducted in Japan. One replicated the basic effect in Hong
prisingly, many people preferred to select from the sec-
Kong (Ho & Leung, 1998). The other was only par-
ond bin, implying that they were focusing on the greater
tially successful in replicating the effect in the United
number of red jelly beans in that bin (for additional ex-
States (Chua, Yamaguchi, & Yates, 2001, described in
amples, see also Dale, Rudski, Schwarz, & Smith, 2007;
Yamaguchi et al., 2008), with the group-diffusion ef-
Denes-Raj, Epstein, & Cole, 1995; Reyna & Brainerd,
fect being observed for only two of the six scenarios
2008). Yamagishi (1997) has shown something similar in
tested. These results leave open the possibility — as Ya-
the domain of risk perception. His participants judged the
maguchi himself pointed out — that culture might play
riskiness of various causes of death when the death rates
a role in the group-diffusion effect. Because both Japan
were presented as ratios, and they appeared to attend to
and Hong Kong are culturally more collectivist than the
and weight the numerators more than the denominators.
United States (e.g., Triandis & Tramifow, 2001), partic-
For example, they judged the risk of dying of cancer to be
ipants in those countries might be more attentive to the
greater when told that it kills 1,286 people out of 10,000
number of people exposed to a threat or be more likely to
than when told that it kills 24.14 people out of 100. The
apply the interdependence heuristic, perhaps because of
ratio bias has also been shown to in?uence the elicitation
their greater sense of collective control as opposed to in-
of health-state utilities (Pinto-Prades, Martinez-Perez, &
dividual control (Yamaguchi, Gelfand, Ohashi, & Zemba,
2005; Yamaguchi et al., 2008). It is important, therefore,
Abellán-Perpiñán, 2006) and the perceived guilt of a de-
to replicate the group-diffusion effect in an individualistic
fendant based on DNA evidence (Koehler & Macchi,
culture such as that of the United States.
2004).
Our second purpose in conducting this research was to
Our proposal, however, is that, although people attend
consider an alternative explanation of the group-diffusion
to and weight the numerator more than the denomina-
effect. Speci?cally, we thought it might be the result of
tor in many situations, they do the opposite in others.
people’s attending to and weighting the number of people
Furthermore, this is mainly a consequence of the rela-
exposed to the risk more heavily than other information
tive salience of the numerator and denominator. In terms
in making their likelihood judgments. Consider two sce-
of the classic jelly bean scenario that has been used to
narios. In one, 1 person is expected to be taken ill out of
demonstrate the ratio bias, people might exhibit a group-
10 people exposed to a virus. In the other, 10 people are
diffusion effect — judging the likelihood of selecting a
expected to be taken ill out of 100 people exposed. The
red jelly bean to be lower when there are 100 jelly beans
probability that any individual will be taken ill is given by
in the bin than when there are 10 — if their attention is
the ratio of the number of people expected to be affected
drawn primarily to the total number of jelly beans rather
to the number of people exposed, and of course the two
than the number of red ones. One way to do this might be
probabilities in this example are the same (1/10 = 10/100
to present the number of red jelly beans implicitly rather
= 10%). However, if people attend to and weight the
than explicitly by giving the total number of jelly beans
number of people exposed (the denominator of the frac-
along with the probability of selecting a red one (10%).
tion) more than the number expected to be affected (the
Consider the analogous situation described in the fol-
numerator), then they would perceive lower risk when
lowing letter to a popular media columnist who answers
the number of people exposed is 100 than when it is 10.
people’s mathematical, scienti?c, and technical questions
Note that there is no motivational component to our the-
(vos Savant, 2006).

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
438
Family and friends have ganged up on me, but we agree
in which the number of deaths (the numerator) was made
to believe what you say. I say the odds of winning a six-
more salient (e.g., “100 people in Australia die every day
number lottery (in which you choose the numbers) are the
from cancer.”) and a condition in which the time frame
same whether 100 or 100,000 tickets are sold. They say
(the denominator) was made more salient (“Every day in
the chances are better if only 100 are sold. Who’s right?
Australia 100 people die from cancer.”). Consistent with
Although the chances of winning are the same regard-
research on the ratio bias, they found that most people
less of how many tickets are sold, the letter writer’s fam-
rated the causes of death riskier in the per-year condition
ily and friends appear to be in?uenced by that number
than in the per-day condition — with a minority showing
— perhaps because it is the most salient one presented
the opposite effect. The salience manipulation, however,
explicitly in the scenario.
had no effect.
The scenarios used by Yamaguchi (1998) are simi-
We began our research by replicating the group-
lar to our hypothetical jelly bean example and the lot-
diffusion effect on college students in the United States
tery example above in that they seem to draw attention
with two health scenarios to be sure that it occurs with
to the denominator of the relevant ratio — the number
participants from a more individualistic culture. We con-
of people exposed to the threat. While all six of Yam-
tinued by replicating it again in Experiment 2, but in a
aguchi’s scenarios prominently featured the number of
new context. Instead of health-threat scenarios, we used
people exposed to the threat, none of them explicitly pre-
lottery scenarios in which participants had a chance to
sented the number of people expected to be affected. In
lose or win money. Our rationale was that if participants
one of his scenarios (and also in the scenarios of Ho &
exhibited a group-diffusion effect for a positive outcome
Leung, 1998), the probability of the negative outcome
(i.e., winning money), then it is unlikely that the effect
was presented (e.g., a 15% chance of developing can-
is the result of an interdependence heuristic that is ap-
cer), but, given the dif?culty that people have in under-
plied in response to a threat. Such a result would be con-
standing single-event probabilities (Gigerenzer, 1994), it
sistent, however, with the idea that people simply attend
seems likely that the number exposed to the threat re-
to and weight the denominator of the relevant ratio more
mained a highly salient piece of information — certainly
than the numerator. In Experiment 3, we show that the
more salient than the non-presented or implicitly pre-
group-diffusion effect is eliminated when we change the
sented number of people expected to be affected.
scenarios slightly to include an explicit presentation of
Similar ideas have been explored by other researchers,
the numerator of the relevant ratio as well as the denom-
although not in connection with the group-diffusion ef-
inator. Finally, in Experiment 4, we show that explicitly
fect. For example, Stone et al. (2003) speculated that
presenting the numerator but not the denominator pro-
the greater impact of certain graphical risk communi-
duces an effect in the opposite direction — a ratio bias.
cation methods occurs because these graphical methods
All of these results are consistent with our proposal that
emphasize the number of people expected to be affected
the group-diffusion effect is a result of people’s attend-
more than the number exposed. For example, a bar graph
ing to and weighting the denominator of the relevant ratio
that compared the gum-disease risk associated with two
more than the numerator. They also demonstrate a theo-
brands of toothpaste in terms of the number of people ex-
retical connection between the group-diffusion effect and
pected to be affected produced relatively large differences
the much better known ratio bias.
in what people were willing to pay for the two products
(Stone, Yates, & Parker, 1997). However, Stone et al.
(2003) found that a stacked bar graph that shows both the
2 Experiment 1
number expected to be affected and the total number who
use each toothpaste produced much smaller differences
We decided to replicate the group-diffusion effect using
on par with the differences produced by presenting the
a variant of Yamaguchi’s (1998) paradigm. The primary
risks in terms of probabilities.
differences are that we used a within-subjects design in
Bonner and Newell (2008), however, found no effect
which participants responded to several scenarios that
of a conceptually similar manipulation. They presented
varied systematically in terms of the health threat un-
people with information about various causes of death
der consideration, the objective probability of the neg-
in terms of the number of people who die per day from
ative outcome, and the number of people exposed to
that cause or the number who die per year. For exam-
the threat. Second, we asked participants to make non-
ple, the frequency of death from cancer in Australia was
numeric likelihood judgments on a 12-point scale rather
presented as 100 deaths per day or as 36,500 deaths per
than numeric judgments on a standard percentage scale.
year. In essence, these are ratios in which the number
The reason is that some non-normative likelihood judg-
of deaths is the numerator and the time period is the de-
ment phenomena — such as the alternative-outcomes and
nominator. These researchers also included a condition
dud-alternative effects — are observed only when peo-

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
439
ple make non-numeric judgments (Windschitl & Krizan,
12
1%
2005). This is probably because numeric response scales
20%
11
prompt people to treat their task as a mathematical one
ed
10
with a precise answer to be calculated, which they duti-
f
ect
fully calculate. Because we are more interested in peo-
9
ple’s intuitions about risk and likelihood, we decided to
8
use the non-numeric response format.
7
6
2.1 Method
5
elihood of being af
2.1.1 Participants
4
3
The participants were 31 students at California State Uni-
Judged lik
2
versity, Fresno, who participated in return for partial
credit in an introductory psychology course. There were
1
20 women, 6 men, and 5 participants whose sex was not
1
10
100
1000
recorded.
Number exposed to the health threat
2.1.2 Design and procedure
Figure 1: The mean intuitive likelihood judgment as a
Participants completed a questionnaire with 16 short
function of the number of people exposed to the health
items loosely based on the scenarios used by Yamaguchi
threat and the probability of being affected in Experiment
(1998). Eight of the items concerned a bacteria scenario
1. The results are collapsed across the bacteria and car-
and had the following general form.
cinogen scenarios.
Imagine that you are one of N people eating at a
restaurant. Afterward, you ?nd out that you were exposed
tremely Poor Chance on the left-hand side and Extremely
to a certain kind of bacteria in the food. Furthermore,
Good Chance on the right-hand side. Participants com-
medical experts say that people exposed to these bacteria
pleted the questionnaire in small, non-interacting groups.
have a P probability of becoming seriously ill as a result.
They read a short set of instructions that described the use
The other eight items concerned a carcinogen scenario
of the response scale and then completed the 16 items at
and had the following general form.
their own pace.
Imagine that you are one of N people in your neigh-
borhood whose drinking water is found to be contami-
nated with a cancer-causing chemical. Scientists say that
2.2 Results and discussion
people exposed to this chemical have a P probability of
developing liver cancer.
Each response was coded as an integer from 1 to 12, with
The number of people exposed to the health threat, N,
lower numbers indicating a lower chance of experiencing
was varied systematically across the items. It was either
the negative outcome. Despite the instructions, two par-
1, 10, 100, or 1000. When N was 1, the wording of the
ticipants circled an anchor label or a cluster of asterisks
scenario was changed slightly to seem more natural (e.g.,
— rather than a single asterisk — for each item. These
“Imagine that you are eating alone in a restaurant.”). The
participants were dropped from the analyses. Two oth-
objective probability of experiencing the threat, P, was
ers circled an anchor label rather than an asterisk for just
also varied systematically across the items; it was either
a few items. On these items, we coded the response as
1% or 20%. Thus, the items represented all 16 possible
the most extreme response at that end of the scale (i.e., 1
combinations of the number of people exposed (1, 10,
or 12). The ?nal analyses, therefore, were based on the
100, or 1000), the objective probability (1% or 20%), and
responses of 29 participants.
the threat (bacteria or carcinogen). The items were ar-
Figure 1 presents the mean gut-feeling likelihood judg-
ranged on the questionnaire in a randomized order, and
ment as a function of the number exposed to the threat,
a second form was created by reversing the order of the
separately for the two objective probabilities. As the
items.
?gure shows, there was a clear group-diffusion effect,
The question that participants responded to for each
with participants judging their chances of experiencing
item was of the form, What is your “gut feeling” about
the negative outcome to be lower as the number of people
your chance of [becoming seriously ill / developing liver
exposed to the threat increased.
cancer]? They responded by circling one of 12 hori-
To con?rm this interpretation, we conducted a
zontally arrayed asterisks, anchored with the labels Ex-
repeated-measures analysis of variance (ANOVA) with

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
440
the number exposed, objective probability, and health
to cue the interdependence heuristic at all so that there
threat as within-subjects factors, and item order as a
would be no effect of the number exposed. A second pos-
between-subjects factor. There was no main effect of
sibility is that people would continue to feel safer in larger
health threat, F(1,27) = 0.49, p = .49, or item order,
groups and that this positive feeling would cause them to
F(1,27) = 0.01, p = .92, which is why the data are col-
judge themselves to be more likely to experience positive
lapsed across these factors in Figure 1. Not surprisingly,
events — the opposite of a group-diffusion effect. On the
there was a linear effect of objective probability, F(1,27)
other hand, if the group-diffusion effect is the result of
= 93.53, p < .001. Most importantly, for our purposes, is
a salience-based over-weighting of the number exposed,
that there was a linear effect of the log of the group size,
then people should judge themselves to be less likely to
F(1,27) = 35.80, p < .001 — a group-diffusion effect. The
experience positive events, in addition to negative events,
only signi?cant interaction was that among group size,
as the number exposed increases.
threat, and item order, F(1,27) = 9.85, p = .004. Interac-
In Experiment 2, therefore, we used items that featured
tions involving item order are likely to be caused by par-
a lottery scenario. Participants imagined that they were in
ticipants’ responding differently to particular items when
a room with N people and that each person had a P prob-
they happen to be near the beginning versus the end of
ability of being selected. For some items, the outcome
the questionnaire. Because such effects are not relevant
of being selected was that they lost $50. For other items,
to our concerns here — and because they were inconsis-
the outcome was that they won $50. Again, our atten-
tent across the present studies — we do not discuss them
tional explanation predicts that people should judge their
further.
likelihood of both losing and winning to decrease as the
Recall that Bonner and Newell (2008) found evidence
number of people participating in the lottery increases.
of individual differences in their study of ratio bias.
While most of their participants judged causes of death to
be riskier when they were presented in terms of the num-
3.1 Method
ber dying per year, a signi?cant minority showed the op-
posite effect. To look for such differences in the present
3.1.1 Participants
experiment, we computed an effect for each participant
The participants were 38 students at California State Uni-
by taking the simple correlation between the log of the
versity, Fresno, who participated in return for partial
group size and his or her likelihood judgments. These
credit in an introductory psychology course. There were
correlations ranged from –.70 to +.07, with a median of
21 women, 8 men, and 2 additional participants whose
–.30. Only one participant had a positive correlation.
sex was not recorded.
Thus, we found no evidence of individual differences in
the direction of the effect.
In summary, we convincingly replicated the group-
3.1.2 Design and procedure
diffusion effect using an American sample in Experi-
ment 1. Again, our procedure differed from Yamaguchi’s
Again, participants completed a questionnaire with 16
(1998) in that we used a within-subjects design and a non-
short items, all of the following general form.
numeric response format. Of course it would be interest-
Picture yourself as one of N people in a room. Ev-
ing to explore the extent to which these differences affect
eryone in the room has a P chance of being randomly
the results, but we chose to explore whether these results
selected to [win / lose] $50.
are best explained by the idea that people use an interde-
The 16 items represented the 16 different combinations
pendence heuristic or by the idea that they are exhibiting
of the number of people in the lottery (1, 10, 100, or
a salience-based tendency to attend to and weight the de-
1000), the objective probability of being selected (1% or
nominator of the relevant ratio more than the numerator.
20%), and the outcome (winning $50 or losing $50). The
items were arranged on the questionnaire in a random-
ized order, and a second form was created by reversing
3 Experiment 2
the order of the items. For each item, participants re-
sponded to the question, What is your gut feeling” about
If the group-diffusion effect is the result of people’s us-
your chance of being selected? They responded by cir-
ing an interdependence heuristic, then it seems reason-
cling one of 11 horizontally arrayed asterisks, anchored
able that they would exhibit the effect when judging their
with the labels Extremely Poor Chance on the left-hand
risk of experiencing negative outcomes as in Experiment
side and Extremely Good Chance on the right-hand side.
1. But what if people were to judge the likelihood that
The change from a 12-point scale in Experiment 1 to an
they would experience positive outcomes? We can think
11-point scale in the rest of the experiments was inciden-
of two possibilities. One is that such situations would fail
tal.

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
441
11
Win 1%
1
.0
Lose $50
Win 20%
Win $50
10
Lose 1%
ed
Lose 20%
9
0.8
8
7
0.6
obability
6
v
ed pr
5
0.4
elihood of being select
4
Obser
3
0.2
Judged lik
2
1
0.0
1
10
100
1000
0.0
0.2
0.4
0.6
0.8
1.0
Number in the lottery
Expected probability
Figure 2: The mean intuitive likelihood judgment as a
Figure 3: The proportion of individual participants’ p val-
function of the number of people in the lottery, the prob-
ues that are less than or equal to the expected proportion
ability of being selected, and the outcome of the lottery
for both the lose and win conditions in Experiment 2.
in Experiment 2.
The evidence for individual differences in the direction
3.2 Results and discussion
of the effect in this experiment is merely suggestive. Un-
Each response was coded as an integer from 1 to 11, with
der the lose condition, the simple correlations between
lower numbers indicating a lower chance of experiencing
the log of the group size and each participant’s likelihood
the outcome. Again, two participants circled an anchor
judgments ranged from –.95 to +.27, with a median of
label for a small number of items and their responses for
–.24. Only ?ve of those correlations were positive and
these items were coded as the most extreme response at
only one exceeded .15. Thus, the effects were fairly con-
that end of the scale (i.e., 1 or 11).
sistent in the lose condition. Under the win condition,
Figure 2 presents the mean likelihood judgment as a
however, the correlations ranged from –.96 to +.64, with
function of the number of people in the lottery, separately
a median of –.02. To examine the possibility of individ-
for the two objective probabilities and two outcomes. As
ual differences more closely — especially in the win con-
the ?gure shows, there appears to have been a group-
dition — we plotted individual participants’ p values in
diffusion effect, with participants tending to judge them-
Figure 3. Each p value is shown in terms of both the pro-
selves less likely to be selected as the number of people
portion of p values expected to be less than or equal to
in the lottery increased. An ANOVA analogous to that
it and the proportion that was actually less than or equal
described previously con?rmed that there was no main
to it. For example, we would expect 5% of the p values
effect of outcome, F(1,33) = 0.15, p = .70, or item order,
to be less than or equal to .05. Here a p value below .50
F(1,33) = 0.09, p = .77. There was a linear effect of the
indicates a negative correlation (a group-diffusion effect)
objective probability, F(1,33) = 35.41, p < .001, and also
and a p value above .50 indicates a positive correlation
a linear effect of the log of the number of people in the
(a ratio bias). Effects in both directions would be indi-
lottery, F(1,33) = 40.27, p < .001, con?rming that there
cated by a greater proportion of low p values than ex-
was an overall group-diffusion effect.
pected (points appearing above the 1:1 diagonal on the
There were also signi?cant interactions between ob-
left) and a greater proportion of high p values than ex-
jective probability and outcome, F(1,33) = 6.15, p < .02,
pected (points appearing below the 1:1 diagonal on the
and among the number of people, objective probability,
right). The actual pattern, however, shows no evidence
and outcome, F(1,33) = 4.85, p = .003. These seem to
of an effect in the opposite direction for either the lose or
indicate that participants were more sensitive to the dif-
win conditions.
ference between the two probabilities for winning than
An alternative way of looking at individual differences
for losing — and especially when there were 100 people
involves examining the relationship between the size of
in the lottery — although it is not immediately clear why
the effect under the lose and win conditions across partic-
this should be.
ipants. To the extent that individual participants show a

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
442
4.1 Method
1
.0
4.1.1 Participants
0.5
The participants were 43 students at California State Uni-
or winning $50
y f
versity, Fresno, who participated in return for partial
er
tt
credit in an introductory psychology course. There were
33 women, 8 men, and 2 participants whose sex was not
0.0
he lo
recorded.
?0.5
4.1.2 Design and procedure
Participants completed a questionnaire with 12 items of
f
ect of number in t
.0
the following general form.
Ef
?1
Picture yourself as one of N people in a room. n of the
?1.0
?0.5
0.0
0.5
1.0
N people in the room will be randomly selected to [win /
Effect of number in the lottery for losing $50
lose] $50.
In this experiment, N was either 10, 100, or 1000. (The
Figure 4: The relationship between the effect size under
structure of the task made it impossible to have a condi-
the lose and win conditions across participants in Experi-
tion in which N was 1.) The value of n was chosen so that
ment 2. The effect size is the simple correlation between
the probability of being selected was either 10% (e.g., n
the number of people in the lottery and the participant’s
= 1 when N = 10) or 30% (e.g., n = 30 when N = 100).
intuitive likelihood judgment.
Note that we had to increase our lower probability from
1% to 10% in this experiment to avoid ratios of 0.10 out
of 10. The 12 items, then, represented the 12 different
consistent group-diffusion effect across conditions, these
combinations of the number of people in the lottery (10,
variables should correlate highly. In fact, the correlation
100, or 1000), the objective probability of being selected
is surprisingly weak, r(36) = .12, p = .45. Figure 4 sug-
(10% or 30%), and the outcome (winning $50 or losing
gests that this might be because there was a distinct clus-
$50). The key is that participants were given the numer-
ter of participants who had a substantial negative corre-
ator (n) explicitly as opposed to having to infer it from N
lation under the lose condition but a substantial positive
and P as in Experiments 1 and 2.
correlation under the win condition. This could represent
Again, the items were arranged on the questionnaire in
a true “safety-in-numbers” effect for these participants.
a randomized order, and a second form was created by re-
Being in a larger group caused them to give more opti-
versing the order of the items. For each item, participants
mistic judgments regardless of whether optimism meant
responded to the same likelihood judgment question as in
a lower chance of losing or a higher chance of winning.
Experiment 2.
4 Experiment 3
4.2 Results and discussion
Despite the suggestion of individual differences, the re-
Each response was coded as an integer from 1 to 11, with
sults of Experiment 2 are generally consistent with our
lower numbers indicating a lower chance of experiencing
salience explanation of the group-diffusion effect but
the outcome. Five participants made unusable responses
inconsistent with Yamaguchi’s (1998) interdependence
and were dropped from the analyses.
heuristic explanation. Experiment 3 was designed to test
Figure 5 presents the mean likelihood judgment as
another straightforward implication of our explanation.
a function of the number of people in the room, sep-
If the way a scenario is presented calls as much attention
arately for the two objective probabilities and two out-
to the number of people expected to be affected (the nu-
comes. Again, there was no effect of winning versus los-
merator of the relevant ratio) as to the number of people
ing, F(1,36) = 1.30, p = .26, or item order, F(1,36) = 0.05,
exposed (the denominator), then the group-diffusion ef-
p = .83, but there was a linear effect of the objective prob-
fect should be eliminated. In Experiments 1 and 2, the
ability, F(1,36) = 35.83, p < .001. Most importantly —
numerator was not explicitly presented; it had to be in-
and unlike in Experiments 1 and 2 — there was no linear
ferred from the number exposed and the stated probabil-
effect of the log of the group size, F(1,36) = 0.22, p =
ity of being affected. In Experiment 3, we presented the
.64. Thus, explicitly including the numerator of the rel-
numerator explicitly.
evant ratio in addition to the denominator eliminated the

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
443
11
Win 10%
1
.0
Lose $50
Win 30%
Win $50
10
Lose 10%
ed
Lose 30%
9
0.8
8
7
0.6
obability
6
v
ed pr
5
0.4
elihood of being select
4
Obser
3
0.2
Judged lik
2
1
0.0
10
100
1000
0.0
0.2
0.4
0.6
0.8
1.0
Number in the lottery
Expected probability
Figure 5: The mean intuitive likelihood judgment as a
Figure 6: The proportion of individual participants’ p val-
function of the number of people in the lottery, the prob-
ues that are less than or equal to the expected proportion
ability of being selected, and the outcome of the lottery
for both the lose and win conditions in Experiment 3.
(winning vs. losing $50) in Experiment 3.
group-diffusion effect, which is what we predicted based
a subset of participants who were affected differently by
on our attentional explanation.
the number of people in the room under the lose and win
As in Experiment 2, there were also signi?cant interac-
conditions.
tions between objective probability and outcome, F(1,36)
= 7.02, p = .01, and among group size, objective proba-
bility, and outcome, F(1,36) = 6.16, p = .02. And again
5 Experiment 4
these seem to indicate that participants were more sen-
sitive to the difference between the two probabilities for
Experiments 1 through 3 have replicated the group-
winning than for losing — especially for the larger group
diffusion effect, shown that it occurs when people judge
sizes.
the risk of both negative and positive outcomes, and
Given the null effect of group size in Experiment 3,
shown that explicitly presenting the numerator of the rel-
it is especially important to consider the possibility that
evant ratio in addition to the denominator eliminates the
there are individual differences in the direction of the ef-
effect. In Experiment 4, we addressed two issues. One
fect that cancel each other out. Again, we computed the
is that we have not yet shown that the presence versus
simple correlation between the log of the group size and
absence of information about the number of people ex-
each participant’s likelihood judgments separately for the
pected to be affected determines whether or not we ob-
lose and win conditions. In the lose condition, the corre-
serve the group-diffusion effect in a single experiment
lations ranged from –.76 to +.82, with a median of –.12.
in which all other factors are controlled. Experiment
In the win condition, the correlations ranged from –.92
2 demonstrated a group-diffusion effect when numera-
to +.82, with a median of .00. Figure 6 shows, for in-
tor information was not explicitly presented, while Ex-
dividual participants’ p values in both the lose and win
periment 3 failed to demonstrate an effect when both
conditions, the proportion that would be expected to be
numerator and denominator information were explicitly
less than or equal to that p value if there were no effect in
presented. Although both experiments concerned losing
either direction and the proportion that was actually less
and winning lotteries, the Experiment 2 scenarios explic-
than or equal to it. Note that the observed values track
itly presented the probability of being selected but the
the expected values quite closely, meaning that this ap-
Experiment 3 scenarios did not. Experiment 4, there-
pears to be a true null effect. The correlation between the
fore, included the denominator-only and numerator-plus-
effect size for the win and lose conditions across partic-
denominator conditions in the same study, while explic-
ipants was somewhat higher than in Experiment 2, r(36)
itly presenting the probability of being selected in both
= .30, p = .07. And this time there was no indication of
conditions. The second issue addressed by Experiment

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
444
4 is whether we can observe a reversal of the group-
11
Denominator only
diffusion effect — a ratio bias — in a third condition in
Numerator plus denominator
10
which we explicitly present only the numerator so that the
Numerator only
ed
denominator must be inferred.
9
f
ect
In all three conditions, participants were given the
8
probability of experiencing a negative health outcome. In
7
the denominator-only condition, they were also given the
number of people exposed to the threat but not the num-
6
ber expected to be affected, as in Experiments 1 and 2.
5
elihood of being af
In the numerator-plus-denominator condition, they were
4
given both the number of people exposed and the number
3
expected to be affected, similar to Experiment 3. In the
numerator-only condition, they were given the number of
Judged lik
2
people expected to be affected but not the number of peo-
1
ple exposed. Our prediction was that we would observe
a group-diffusion effect in the denominator-only condi-
10
100
1000
tion, no effect in the numerator-plus-denominator condi-
Number exposed to the health threat
tion, and a ratio bias in the numerator-only condition.
Figure 7: The mean intuitive likelihood judgment as a
5.1 Method
function of the number of people exposed to the health
threat, separately for the three information conditions.
5.1.1 Participants
The results are collapsed across the two probabilities and
the two health threat scenarios.
The participants were 88 students at California State Uni-
versity, Fresno, who participated in return for partial
credit in a health psychology course or an introductory
psychology course. There were 68 women, 16 men, and
versing the order of the items. For each item, participants
4 participants whose sex was not recorded.
responded to the same likelihood judgment question as in
Experiments 2 and 3.
5.1.2 Design and procedure
Participants completed a questionnaire with 12 items sim-
ilar to those in Experiment 1. Participants were randomly
assigned to one of three information conditions. In the
5.2 Results
denominator-only condition, the items were of the same
general form as in Experiment 1. In the numerator-plus-
denominator condition, the last sentence continued, “so
Figure 7 presents the mean likelihood judgment as a func-
[the medical experts] expect about n of the people to [be-
tion of the number of people exposed, separately for each
come seriously ill / develop cancer],” where n was simply
information condition. Consistent with the previous stud-
the speci?ed percentage (P) of the N people exposed. In
ies, there was no main effect of health threat, F(1,82) =
the numerator-only condition, the items still indicated the
0.83, p = .36, or of item order, F(1,82) = 0.93, p = .76,
number of people expected to be affected, but they only
but there was a main effect of the objective probability,
indicated that “several” people were exposed instead of
F(1,82) = 80.33, p < .001. Most strikingly, there was a
the precise number. In all three conditions, the number
signi?cant interaction between the number exposed and
of people exposed to the health threat (N) was either 10,
the information condition, F(2,82) = 36.95, p < .001.
100, or 1000, and the probability of being affected (P)
There was a group-diffusion effect in the denominator
was either 10% or 30%. These values, of course, deter-
only condition, F(1,28) = 21.37, p < .01, and no effect
mined the corresponding numbers of people expected to
in the numerator-plus-denominator condition, F(1,28) =
be affected (n). The 12 items, then, represented the 12
0.41, p = .53 — replicating the previous results in a single
different combinations of the number of people exposed
experiment. But there was also a ratio bias in the numer-
(N = 1, 10, 100, or 1000), the objective probability of
ator only condition, F (1,26) = 36.71, p < .001. Partic-
being affected (P = 10% or 30%), and the health threat
ipants presented with explicit numerator information but
(bacteria or carcinogen).
no explicit denominator information judged their risk to
Again, the items were arranged on the questionnaire in
be greater as the number of people exposed (and therefore
a randomized order, and a second form was created by re-
the number expected to be affected) increased.

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Judgment and Decision Making, Vol. 4, No. 6, October 2009
From group diffusion to ratio bias
445
6 General discussion
sented both the numerator and the denominator produced
ratio biases (Bonner & Newell, 2008; Yamagishi, 1997).
In the present studies, we replicated Yamaguchi’s (1998)
One reason for this might be that there were additional
group-diffusion effect on people’s intuitive likelihood
factors that caused people in those previous studies to at-
judgments for health threat scenarios and extended it to
tend to and weight the numerator more than the denomi-
lottery scenarios. People judged the likelihood of the var-
nator. In the case of Yamagishi’s study, this could be the
ious outcomes to be lower as the number of people ex-
fact that his questionnaire listed the number of people ex-
posed to the threat or involved in the lottery increased. In
pected to die from each cause separately, but it presented
addition, we showed that explicitly presenting the num-
the size of the reference class (e.g., 100 vs. 10,000) only
ber of people expected to be affected or selected (the nu-
once in the instructions at the beginning. By the time par-
merator) in addition to the number exposed to the threat
ticipants started in on their task, they may not have been
or playing the lottery (the denominator) eliminates the ef-
thinking about the size of the reference class anymore. In
fect, and that explicitly presenting the numerator but not
the case of Bonner and Newell’s study, it could be that
the denominator reverses the effect. This entire pattern of
the verbal expressions every day versus ever year do not
results seems inconsistent with Yamaguchi’s (1998) sug-
adequately communicate the quantitative fact that the lat-
gestion that the group-diffusion effect is the result of peo-
ter is 365 times greater than the former, which prevents
ple’s using an interdependence heuristic and perceiving
that manipulation from having much impact on people’s
an illusory safety in numbers.
risk judgments.
Instead, it is consistent with the idea that information
Other factors are likely to in?uence the relative
can be presented so that participants attend to and weight
salience of the denominator and numerator as well. For
either the denominator or the numerator more heavily in
example, in much of the research on ratio bias, partic-
their likelihood judgments. Thus, this research dovetails
ipants choose between two lotteries described in terms
nicely with research on the ratio bias (e.g., Denes-Raj &
of numerators and denominators. Under such conditions,
Epstein, 1994; Yamagishi, 1997). Our contention is that
numerators might have an especially large effect on peo-
the primary difference between situations in which the
ple’s choices because they are generally easier to compare
group-diffusion effect is observed and situations in which
than denominators (Denes-Raj & Epstein, 1994). When
the ratio bias is observed is the amount of attention drawn
people make intuitive likelihood judgments about indi-
to the denominator versus the numerator of the relevant
vidual lotteries, group-diffusion effects might be much
ratio. The group-diffusion effect is more likely to be ob-
easier to observe. Another factor is imaginability, which
served when the denominator is more salient and the ratio
has been implicated as a cause of the ratio bias and simi-
bias is more likely to be observed when the numerator is
lar effects (Koehler & Macchi, 2004; Newell, Mitchell, &
more salient. This is consistent with the ideas of Reyna
Hayes, 2008). For example, Slovic, Monahan, and Mac-
and Brainerd (2008), that the ratio bias occurs in part be-
Gregor (2000) found that a psychiatric patient was per-
cause reasoning about situations in which the members of
ceived as more dangerous when it was reported that 20 in
a smaller category (e.g., people expected to be affected by
100 similar patients will commit a violent act than when
the threat) are included in a larger category (e.g., people
it was reported that there is a 20% chance that the patient
exposed to the threat) is inherently dif?cult. This inher-
will commit a violent act. They argued that the “20 in
ent dif?culty, combined with easily remembered or pro-
100” phrasing is more likely to bring to mind images of
cessed numerator information, is what produces the ratio
violent acts being committed. But perhaps denominators
bias. In essence, we are adding the idea that problems
can be made more imaginable too. In fact, this might be
can also be structured so that the denominator is more
part of the reason that we observed the group-diffusion
easily remembered or processed, in which case there is a
effect here. We began each item by asking people explic-
group-diffusion effect.
itly to imagine the number of people exposed: “Imagine
It is clear, though, that we still need a comprehensive
that you are one of N people. . . .”
theory that speci?es all the conditions under which peo-
The development of a more comprehensive theory is
ple are more sensitive to the denominator than the numer-
important for the practical domain of risk communica-
ator of the relevant ratio and therefore allows us to predict
tion. It is likely that there are many factors that effect
when people will exhibit a group-diffusion effect, a ratio
the relative salience of the number of people exposed to a
bias, or even a null effect. Consider, for example, that our
risk relative to the number expected to be affected. Cre-
Experiment 3 produced neither a ratio bias nor a group-
ating the most effective methods of risk communication
diffusion effect. But previous research that explicitly pre-
will require that we understand what they are.

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