Judgment and Decision Making, Vol. 3, No. 6, August 2008, pp. 435–448
Dissecting the risky-choice framing effect: Numeracy as an
individual-difference factor in weighting risky and riskless options
Ellen Peters∗
Decision Research
Eugene, OR
Irwin P. Levin
Department of Psychology
University of Iowa
Abstract
Using five variants of the Asian Disease Problem, we dissected the risky-choice framing effect by requiring each
participant to provide preference ratings for the full decision problem and also to provide attractiveness ratings for
each of the component parts, i.e., the sure-thing option and the risky option. Consistent with previous research, more
risky choices were made by respondents receiving negatively framed versions of the decision problems than by those
receiving positively framed versions. However, different processes were evident for those scoring high and low on
numeracy. Whereas the choices of the less numerate showed a large effect of frame above and beyond any influence of
their evaluations of the separate options, the choices of the highly numerate were almost completely accounted for by
their attractiveness ratings of the separate options. These results are consistent with an increased tendency of the highly
numerate to integrate complex numeric information in the construction of their preferences and a tendency for the less
numerate to respond more superficially to non-numeric sources of information.
Keywords: numeracy, framing, individual differences, risky choices, attribute framing.
1 Introduction
Tversky and Kahneman’s (1981) introduction of the
Asian Disease Problem was among the earliest exam-
ples of the malleability of human decision making. At
the heart of this problem is the choice between a risk-
less option and a risky option of equal expected value.
Because the current study will dissect the components of
the prototypical risky-choice paradigm as exemplified by
the Asian Disease Problem, we now describe these com-
ponents. The Sure-Thing option offers a fixed (riskless)
outcome. In the Positive framing condition it is “save
200 (out of 600) lives” whereas in the Negative condition
it is “400 will die.” The Risky option offers a “one-third
chance of saving all the lives and a two-thirds chance of
saving no lives” in the Positive condition and a “one-third
chance that no one will die and a two-thirds chance that
all will die” in the Negative condition.
In response to this choice problem, the majority of de-
cision makers choose the riskless or “Sure-Thing” op-
tion over the Risky option when potential outcomes are
framed as gains (lives saved) but choose the Risky option
over the Sure-Thing option when the exact same objective
∗We thank Joshua Weller, Paul Windschitl, Paul Slovic, two anony-
mous reviewers, and Jon Baron for their helpful comments. This
work was supported by grants from the National Science Foundation
(0517770 and 0350984) to the first and second authors, respectively.
Address: Ellen Peters, Decision Research, 1201 Oak Street, Suite 200,
Eugene, Oregon, 97401. Email: empeters@decisionresearch.org
outcomes are framed as losses (deaths). Later attempts to
replicate this phenomenon and extend it to other domains
such as money gained or lost rather than lives did not al-
ways duplicate the literal preference reversal, but a gen-
eral preference shift of more risky choices to avoid losses
than to achieve gains is one of the most solid findings
in judgment and decision making research (see reviews
by Kühberger, 1998; Levin et al., 1998). Later research
uncovered task characteristics and individual difference
factors that moderated the reliability and magnitude of
the risky-choice framing effect (Fagley & Miller, 1997;
Highhouse & Paese, 1996: Levin et al., 2002; Wang,
1996). The present study focuses on one such individual-
difference factor.
The aim of the current study is to dissect the risky-
choice framing effect into its component parts and to ex-
amine the moderating effect of an important individual-
difference variable, numeracy, defined as the ability to
understand probabilistic and mathematical concepts. We
asked participants in each framing condition to judge the
full scenario and also to separately judge both the Sure-
Thing component and the Risky component. In that way,
we can assess the extent to which the Full Scenario fram-
ing effect is driven by framing of the separate compo-
nents, and we can compare this for individuals differing
on a variable known to be associated with more superfi-
cial vs. more complex processing of numeric information
in decisions.
435

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 436
1.1 Numeracy moderates framing effects
Numeracy refers to the ability to understand and use
mathematical and probabilistic concepts. Based on the
National Adult Literacy Survey, almost half of the gen-
eral U.S. population has difficulty with relatively simple
numeric tasks such as calculating the difference between
a regular price and sales price using a calculator or esti-
mating the cost per ounce of a grocery item. These in-
dividuals do not necessarily perceive themselves as “at
risk” in their lives due to limited skills; however, the re-
search reviewed below demonstrates that having inade-
quate numeric skills is associated with lower comprehen-
sion and use of numeric information in health and finan-
cial domains.
Not surprisingly, greater ability with numbers leads
to more comprehension of numeric information in im-
portant decisions (e.g., mammograms; Schwartz et al.,
1997). Numeracy relates in somewhat less intuitive ways
to a variety of cognitive and affective biases in deci-
sion making (Peters et al., 2006). For example, Dehaene
(1997) suggests that, while children spend a lot of time
learning the mechanics of math, they may not really un-
derstand how to apply those mechanics even in adult-
hood. We propose that those high in numeracy will be
more likely to do so. As a result, they should, for ex-
ample, find alternative frames of the same number more
accessible and more influential in decisions.
Peters et al. (2006) examined numeracy’s effect on
framing of a single attribute by presenting participants
with the exam scores of five psychology students and
asking them to rate the performance of each student on
a 7-point scale from –3 (very poor) to +3 (very good).
The framing of the exam scores was manipulated as ei-
ther percent correct or percent incorrect so that “Emily,”
for example, was described as having received either 74%
correct on her exam or 26% incorrect. In a repeated-
measures analysis of variance of the rated performance,
the usual framing effect was shown such that the more
positive frame elicited more positive ratings. Further-
more, the interaction of numeracy with the frame was
also significant, with the less numerate participants show-
ing a stronger framing effect. These findings are con-
sistent with high-numerate participants being more likely
to retrieve and use appropriate numerical principles and
transform numbers presented in one frame into a differ-
ent frame, and the less numerate responding more to the
affect communicated by the single given frame of the in-
formation. We believe that less numerate decision makers
are left with information that is less complete and lacks
the complexity and richness available to the more numer-
ate. Controlling for a proxy measure of intelligence (self-
reported SAT scores) did not alter the results. Actual
number ability appears to matter to judgments and de-
cisions in important ways not captured by other measures
of achievement or ability.
Although Peters et al. (2006) did not examine risky-
choice frames, an unpublished Master’s Thesis by Gar-
cia (2006, supervised by Peters), using a risky-choice
paradigm, found no effect of numeracy on risky-choice
framing problems. We were curious about this lack
of finding given the robust nature of numeracy’s influ-
ence on attribute framing, and our speculation that risky
choices in such problems were based on evaluations of
the two options comprising the choice: the Sure-Thing
option and the Risky option. In prior studies of numeracy,
highly numerate individuals have demonstrated deeper
processing of numeric information by showing smaller
framing effects (presumably caused by transforming the
given numeric frame to its normative equivalent) and by
being more likely to draw meaning from number com-
parisons in judgments (Peters et al., 2006). The highly
numerate appeared to integrate more sources of informa-
tion than the less numerate. In a separate study, the highly
numerate were more likely to be sensitive to numeric in-
formation in judgments of the attractiveness of a hospital
whereas the less numerate were insensitive to provided
numeric information and appeared to misattribute their
current mood to the judgment instead (Peters et al., un-
der review). Thus, it was curious in Garcia (2006) that
numeracy did not influence risky-choice framing effects
in a similar manner with greater effects of the provided
frame on the less numerate. However, as pointed out by
Levin et al. (1998), the risky-choice framing paradigm is
more complex than the attribute-framing paradigm which
has been the source of previous work on the influence
of numeracy. In attribute framing, a single attribute of
an object is alternatively labeled in positive or negative
terms (e.g., success rate versus failure rate of a medical
treatment) and its effect on the evaluation of that object is
assessed. No manipulation of risk is involved. In risky-
choice framing, the labeling of outcomes is manipulated
and the element of risk is added by creating choice op-
tions of varying risk level.
We developed the following hypotheses:
1. In order to replicate the basic Risky-Choice Framing
effect, we expect that the Risky option will be pre-
ferred more than the Sure-Thing option in the neg-
ative framing condition than in the positive framing
condition.
2. Based on Garcia’s (2006) findings, individuals high
and low in numeracy will demonstrate similar fram-
ing effects in risky choices.
3. Because the Sure-Thing option is similar to an
attribute-framing problem (i.e., a single attribute is
manipulated such as “400 of 600 lives will be lost”),

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 437
we expect the frame to influence evaluations of the
Sure-Thing option more for the less numerate than
the highly numerate.
4. Because the less numerate appear to integrate fewer
pieces of information and to respond more than the
highly numerate to non numeric sources of informa-
tion such as mood states, they would be expected
to focus on a single favorable statement such as the
sure gain provided by the Sure-Thing option in the
Positive frame or the possibility of no loss provided
by the Risky option in the Negative frame. By con-
trast, the highly numerate who use numeric informa-
tion more completely are expected to be more capa-
ble of integrating all the information from both op-
tions in their choices. Thus, choices of the highly
numerate should be more influenced by their evalu-
ations of the separate options.
2 Method
2.1 Participants
Participants were 108 students (42% female) fulfilling a
research participation component of an introductory mar-
keting course at the University of Iowa.
2.2 Design
Participants were randomly assigned to a Positive frame
group (N = 53) or a Negative frame group (N = 55).1
Within each group, participants rated their degree of pref-
erence between the options in the Full Scenario task and
then provided separate ratings of the attractiveness of the
Sure-Thing and Risky options. They did this for each
of five scenarios. Further procedural variations are de-
scribed below.
2.3 Materials
Five scenarios were constructed, each patterned after the
Asian Disease Problem but different in content domain
and in the expected value of the options. The Positive
and Negative frame versions of the scenarios are repro-
duced in Appendix A. Briefly, one is an exact replication
of the Asian Disease Problem except that it was simply
called an “unusual disease” from Sweden, one involves
animals endangered by wildfires, one involves crop de-
struction from a severe drought in another country, one
involves loss of medical benefits in another country, and
one involves investment losses. The introduction to the
1The positive group is Versions 1–4 in the accompanying data file.
full scenario was repeated each time a response was re-
quired for either the full risky-choice problem or one of
its components.
2.4 Procedure
Participants rated the five Full Scenarios, the five Sure-
Thing options in two formats, and the five Risky options
in two formats, each presented in separate blocks in their
booklet to ensure that the separate ratings for each part of
the same scenario were spaced far enough apart to reduce
memory effects. Participants made four other responses
(the other four scenarios) before revisiting the same sce-
nario. Each response took about one minute. Further-
more, the response scales were varied between the full
risky-choice problem and the components.
Each participant received the same frame throughout
the experiment. The Full Scenarios were always pre-
sented first. Participants were not allowed to look back
at earlier responses. Each participant then responded to
both the Sure-Thing and Risky options in two separate
formats in different blocks of trials. The Sure-Thing op-
tion was presented in one block as a numerical count
(e.g., 200 people will be saved) and in another block as
a fraction (1/3 of the 600 people will be saved) in or-
der to examine whether different effects of frame were
produced; the Risky option was presented once with the
better outcome (e.g., 1/3 chance of saving all lives) first
and again with the worse option (2/3 chance of saving no
lives) presented first. Four different orders of presenta-
tion of these four blocks were constructed and counter-
balanced across participants.
In the Full Scenario, participants were asked to check
one of seven boxes labeled from “Much prefer A” (the
Sure-Thing option) to “Much prefer B” (the Risky op-
tion) with a midpoint of “A and B are equal.” This expan-
sion of the usual dichotomous choice was done in order
to provide continuous numerical data for the statistical
(regression) analyses (see Levin et al., 2002). Responses
were scored such that higher numbers represent greater
preferences for the Risky option.
To evaluate the Sure-Thing option and the Risky option
separately, participants were asked to circle a number be-
tween –3 (Very bad) and +3 (Very good) with a midpoint
of 0 (Neither bad nor good) to indicate their evaluation
of that particular option.
2.5 Individual difference measures
After completing the ratings tasks, participants were
asked to complete the following: a demographic informa-
tion sheet including age, gender, GPA, and ACT scores;
the 18-item Need for Cognition scale (Cacioppo et al.,

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 438
1984); and the 11-item Numeracy scale shown in Ap-
pendix B (Lipkus et al., 2001).2
All inferential statistics used mean-deviated continu-
ous scores (Irwin & McClelland, 2001). A median split
on numeracy was used for descriptive statistics and to
identify low and high scorers so that inferential analyses
could be conducted separately within each group.
3 Results
In contrast to some previous studies, men and women
scored about the same on numeracy (scores = 9.5 and 9.2,
respectively, t(106) = 1.0, p = .32). Higher numeracy was
associated with higher self-reported GPA and higher ACT
scores (r = .16, p < .10 and .28, p < .01). Numeracy and
Need for Cognition were not significantly related (r = .10,
ns).
3.1 Analysis of the dual formats for the
sure-thing and risky options
We first examined whether the two formats of the
Sure-Thing options (counts versus proportions) and,
separately, of the Risky option (the two orders)
produced different framing effects on evaluations.
A repeated-measures analysis of variance (repeated-
measures ANOVA) was conducted of the attractiveness
ratings for the Sure-Thing options in the five scenar-
ios with format, frame, numeracy (continuous, mean-
deviated), and their interactions as predictors. A sim-
ilar analysis was conducted of attractiveness ratings of
the Risky options. Format did not significantly influence
the attractiveness ratings as a main effect or in interac-
tion with frame or numeracy for either the Sure-Thing or
Risky options.
Correlations between responses to the two formats
were similar for the low and high numerate (average r =
.58 and .60 between the two Sure-Thing formats, respec-
tively, for individuals low and high in numeracy across
the five scenarios and average r=.41 and .53, respectively,
between the two Risky formats). This consistency might
be considered puzzling from the standpoint that individ-
uals lower in numeracy presumably have more difficulty
using numbers in judgments and decisions and therefore
should perhaps be less consistent. The consistency is not
puzzling, however, from the standpoint that the less nu-
merate may process different information than the highly
numerate, with the less numerate processing numeric in-
formation more superficially. Our expectation is that the
2An example of an easy item is “Which of the following numbers
represents the biggest risk of getting a disease? 1 in 100, 1 in 1000, 1
in 10.” An example of a hard item is “In the Acme Publishing Sweep-
stakes, the chance of winning a car is 1 in 1,000. What percent of tickets
of Acme Publishing Sweepstakes wins a car?”
less numerate will respond more to the given frame of in-
formation (which was the same across the formats for a
given participant) rather than the numbers. We retained
only the usual formats (the count format for the Sure-
Thing option and the better outcome first for the Risky
option) in further analyses.
3.2 Separate analyses of Full Scenarios and
components
We next examined the Full Scenarios to test for the usual
risky-choice framing effect and to verify that numeracy
again did not moderate the effects of framing at this level.
A repeated-measures ANOVA of choice preferences was
conducted with the five problems as the repeated mea-
sures and frame (–1, 1), numeracy (continuous and mean-
deviated), and their interactions as the predictors. Con-
sistent with Hypothesis 1, an overall effect of frame was
found, F(1, 104) = 18.9, p < .001, with individuals pre-
ferring the sure-thing option in the domain of gains —
the positive frame — and the risky option in the domain
of losses — the negative frame (average choice prefer-
ences = 3.51 and 4.31, respectively, where a response
of 4 indicates no preference between the two options
and lower numbers indicate a preference for the sure op-
tion). Numeracy and its interaction with frame were non-
significant (F(1, 104) = .04, p = .85 and F(1, 104) = 1.8,
p = .28, respectively). The effects of frame differed by
scenario, F(4, 416) = 2.9, p < .05, with framing effects
being nonsignificant in the scenarios in which the risky
option had a higher expected value than the sure-thing
option (the Spanish drought and Delta’s medical-benefit
crisis). In the positive frame of both scenarios (where de-
cision makers are generally risk-averse), preferences for
the risky option were noticeably stronger for both high-
and low-numerate participants. See Table 1 for prefer-
ence means by frame and numeracy.
As a result of this initial analysis, we dropped the
two non-significant framing problems and focused fur-
ther data analysis on the three scenarios that showed sig-
nificant effects of frame on risky choices (but see the
tables for results with the two dropped scenarios). A
repeated-measures ANOVA of those three scenarios re-
vealed a stronger overall effect of frame, F(1, 104) =
30.5, p < .001, with average choice preferences of 3.0
and 4.2, in the positive and negative frames, respectively.
In this analysis, the main effect of numeracy remained
nonsignificant, but the highly numerate demonstrated a
marginal tendency towards smaller framing effects than
the less numerate, F(1, 104) = 3.3, p = .07.
As suggested in Hypothesis 2, the effects of frame
by numeracy were not conventionally significant in the
risky-choice frame (the Full-Scenario decision), and the
effects of frame were significant in separate analyses of

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 439
Table 1: Preference means by frame and numeracy.
Overall Less numerate (5–9) Higher numerate (10–11)
Full Scenario (1–7 scale) Neg Pos
Main
effects of
frame in
ANOVA
F (1,106)
Neg Pos
Main
effects of
frame in
ANOVA
Neg Pos
Main
effects of
frame in
ANOVA
1. Sweden - disease 4.45 3.32 15.4,
p < .001
4.65 3.00 p < .001 4.28 3.63 ns
2. Stock 4.18 2.96
17.0,
p < .001
4.12 3.08 p < .05 4.24 2.85 p < .001
3. Wildfire season 3.84 2.77 12.0,
p < .001
3.85 2.69 p < .05 3.83 2.85 p < .05
4. Drought in Spain 4.58 4.38 ns 4.77 4.15 ns 4.41 4.59 ns
5. Delta SS 4.47 4.09 ns 4.38 4.08 ns 4.55 4.11 ns
Average preference across the
5 scenarios
(repeated-measures results of
Scenarios 1–5)
4.31 3.51
18.9,
p < .001
4.35 3.40 p < .001 4.26 3.61 p < .05
Average preference across the
first 3 scenarios that showed
significant framing effects
(repeated-measures results of
Scenarios 1-3)
4.20 3.00
30.5,
p < .001
4.21 2.92 p < .001 4.11 3.11 p < .01
Note: Higher numbers represent greater preference for the risky option.
N = 108; n = 52 and 56 for less and higher numerate, respectively.
low and high numerate groups. Previous studies, how-
ever, have shown that more and less numerate decision
makers appear to use different sources of information in
decisions, setting the stage for our hypotheses concern-
ing different information-processing mechanisms under-
lying risky-choice framing effects for those low and high
in numeracy. Thus, we turn to an analysis of framing in
evaluations of the separate options next.
A repeated-measures ANOVA was conducted of the at-
tractiveness ratings of the remaining three Sure-Thing op-
tions with frame, numeracy (mean-deviated and contin-
uous) and their interaction as predictors. Previous re-
search has demonstrated that individuals lower in numer-
acy show stronger attribute-framing effects than those
higher in numeracy. As stated in Hypothesis 3, we ex-
pected that ratings of the Sure-Thing option would be
similar to an attribute frame. The overall effect of frame
was significant with Sure-Thing options in the positive
frame rated as more attractive than those in the negative
frame (attractiveness means = .53 and –.28, respectively,
F(1, 104) = 10.8, p < .01). As hypothesized, less numer-
ate individuals showed stronger framing effects than did
the highly numerate, interaction F(1, 104) = 3.9, p = .05.
Examination of the means by frame separately within
low and high numerate groups (based on a median split,
the highly numerate scored 10 or 11 correct out of 11 pos-
sible, whereas the low-numerate group scored between
5 and 9 correct3) revealed a significant framing effect
among the less numerate (attractiveness means in the pos-
itive and negative frame were .73 and -.54, respectively,
p < .001) and a non-significant effect for the highly nu-
merate (means = .35 and –.05, respectively, ns). In no
case was the framing effect for a scenario greater for the
highly numerate than for the less numerate. See Table 2
for attractiveness means by frame and numeracy.
3
Individuals in this study were fairly numerate overall, with only
13% of them scoring between 5 and 7 correct, 12% scoring 8 correct,
23% with 9 correct, and 26% each scoring 10 and 11 correct.

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 440
Table 2: Attractiveness means of the Sure-Thing options by frame and numeracy.
Overall Less numerate (5–9) Higher numerate (10–11)
Sure-thing component Neg Pos
Main
effects of
frame in
ANOVA
F (1,106)
Neg Pos
Main
effects of
frame in
ANOVA
Neg Pos
Main
effects of
frame in
ANOVA
1. Sweden - disease – 0.27 0.32 3.8,
p = .06
– 0.65 0.38 p = .02 0.07 0.26 ns
2. Stock – 0.44 0.40
8.2,
p < .01
– 0.69 0.69 p < .01 – 0.21 0.11 ns
3. Wildfire season – 0.13 0.89 12.5,
p < .001
– 0.27 1.12 p < .001 0.00 0.67 ns
Average rating across three
scenarios (repeated-measures
results)
– 0.28 0.53
10.8,
p < .001
– 0.54 0.73 p < .001 – 0.05 0.35 ns
Scenarios not included:
4. Drought in Spain – 0.13 – 0.09 ns – 0.31 – 0.08 ns 0.03 – 0.11 ns
5. Delta SS – 0.62 – 0.40 ns – 0.77 – 0.15 ns – 0.48 – 0.63 ns
Average rating across five
scenarios (repeated-measures
results)
– 0.32 0.22
5.2,
p < .05
– 0.54 0.39 p < .01 – 0.12 0.06 ns
We conducted a similar repeated-measures ANOVA
with attractiveness ratings of the Risky option. In this
case, we were not sure what to expect because two di-
ametrically opposite effects are theoretically possible.
First, negative frames could lead to the gamble being
perceived as more attractive, consistent with Prospect
Theory’s psychological shift towards risk seeking in this
frame compared to the positive frame. Second, negative
frames could lead to poorer evaluations of the gamble
compared to positive frames, consistent with an attribute-
framing effect. Results of the analysis demonstrated no
overall effects of frame or its interaction with numer-
acy. However, an analysis by scenario shown in Table
3 indicated a nonsignificant tendency for the highly nu-
merate to rate the attractiveness of the Risky component
as higher in the negative frame than the positive frame
condition in every scenario, whereas the less numerate
tended to show the opposite pattern. This pattern was
significant for the highly numerate when all five scenar-
ios were considered.
3.3 Full-Scenario risky-choice framing ef-
fects as a function of evaluations of the
separate options
Thus far, we have found different effects of frame for the
more and less numerate in evaluations of the separate op-
tions. This result may explain the overall lack of effect of
numeracy on risky-choice framing. Specifically, the less
numerate showed stronger framing effects in their evalua-
tions of the Sure-Thing option; the highly numerate were
not influenced by frame in their evaluations of the Sure-
Thing option but appeared to be somewhat influenced by
a framing effect consistent with Prospect Theory in their
evaluations of the Risky option. Thus, we turn to an ex-
amination of the extent to which the frame, the evalua-
tions of both options, and their interactions with numer-
acy influenced the full risky-choice framing effect. Hy-
pothesis 4 stated that the frame would have a direct in-
fluence on choice preferences of the less numerate with
little influence from their evaluations of the separate op-

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 441
Table 3: Attractiveness means of the Risky options by frame and numeracy.
Overall Less numerate (5–9) Higher numerate (10–11)
Risky component Neg Pos
Main
effects of
frame in
ANOVA
F (1,106)
Neg Pos
Main
effects of
frame in
ANOVA
Neg Pos
Main
effects of
frame in
ANOVA
1. Sweden - disease 0.00 – 0.36 ns – 0.31 – 0.38 ns 0.28 – 0.33 p = .10
2. Stock – 0.04 – 0.13 ns – 0.19 0.31 ns 0.10 – 0.56 ns
3. Wildfire season 0.02 0.04 ns – 0.27 0.04 ns 0.28 0.04 ns
Average rating across three
scenarios (repeated-measures
results)
– 0.01 – 0.15 ns – 0.26 – 0.01 ns 0.22 – 0.28 ns
Scenarios not included:
4. Drought in Spain 0.13 – 0.15 ns 0.12 0.19 ns 0.14 – 0.48 p = .10
5. Delta SS – 0.04 – 0.51 2.9,
p = .09
– 0.15 – 0.15 ns 0.07 – 0.85 p = .02
Average rating across five
scenarios (repeated-measures
results)
0.01 – 0.22 ns – 0.16 0.00 ns 0.17 – 0.44 p = .05
tions, suggesting a more superficial reaction to the verbal
cues identifying the positive and negative frames. For the
highly numerate, however, we expected that the separate
evaluations of the Sure-Thing and Risky options would
guide their choice preferences, indicating their more ana-
lytical, componential approach to decisions involving nu-
meric information.
To simplify this analysis, we constructed average rat-
ings across the three problems for evaluations of the Sure-
Thing option and the Risky option, as well as average
choice preferences for the Full-Scenario risky choices. A
regression analysis was conducted of the average choice
preference in the Full-Scenario risky choices; indepen-
dent variables were the direct effects of frame, the aver-
age mean-deviated Sure-Thing attractiveness rating, the
average mean-deviated Risky attractiveness rating, nu-
meracy (mean-deviated and continuous), and the three
two-way interactions with numeracy (model F(7, 100)
= 8.3, p < .001, R-squared = .37). Although the frame
was associated with the component ratings, correlations
among the predictors were small enough that multi-
collinearity did not appear to be a problem (the tolerances
were acceptable; range = .72-.96).
The results indicated a main effect of frame such that
individuals in the positive-frame condition tended to pre-
fer the Sure-Thing option more than those in the negative-
frame condition (b = –.51, t(100) = –4.9, p < .001). In
addition, individuals who rated the Sure-Thing option as
more attractive were more likely to prefer it (b = –.22,
t(100) = –2.6, p = .01) and those who rated the Risky
option as more attractive were more likely to prefer the
Risky option (b = .23, t(100) = 2.7, p < .01). Numer-
acy did not have a direct influence on risky-choice prefer-
ences. However, the main effects were qualified by three
interactions. Individuals lower in numeracy were influ-
enced more than those higher in numeracy by the frame
in the risky-choice scenarios (interaction b = .19, t(100)
= 2.6, p = .01). Choice preferences of individuals high
in numeracy were influenced more than those low in nu-
meracy by the attractiveness of the Sure-Thing option as
well as the Risky option (interaction b = –.17, t(100) =
–3.1, p < .01 and b = .13, t(100) = 2.1, p < .05, respec-
tively). A similar pattern of results was shown in each of
the three scenarios with strongest effects in the Swedish
disease problem (see Table 4).
In order to examine these numeracy interactions in
more detail, a median split was performed on numer-
acy. Regression analyses of the average choice prefer-
ence were conducted with frame, the average Sure-Thing
attractiveness rating, and the average Risky attractiveness

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 442
Table 4: Regression results (b-values) predicting the Full-Scenario risky-choice preferences by scenario, on average
across the three scenarios, and on average across the five scenarios.
Overall
model,
F (7, 100)
Frame
Sure-
Thing
option
rating
Risky
option
rating
Numeracy Frame *
numeracy
Sure-
Thing
option
rating *
numeracy
Risky
option
rating *
numeracy
1. Sweden - disease 6.8*** –.55*** –.15 .26** .02 .26** –.24*** .18*
2. Stock 4.1*** –.61*** –.15 .17 –.01 .19 –.17* .02
3. Wildfire season 2.8** –.41** –.25* .08 –.06 .10 –.03 .10
Average of 3 scenarios 8.3*** –.51*** –.22** .23** .00 .19** –.17** .13*
4. Drought in Spain 1.1 –.09 –.17 .19 –.02 .13 –.14 .06
5. Delta SS 2.4* –.10 –.17 .37** .12 –.07 –.09 .01
Average of 5 scenarios 7.1*** –.34*** –.24** .30*** .02 .12
(p=.07)
–.15** .07
Note: * p < .05; ** p < .01; *** p < .001.
rating as predictor variables, separately within each me-
dian split of numeracy (see Table 5 for detailed results).
For individuals high in numeracy, the model strongly pre-
dicted their preferences (model F(3, 52) = 11.1, p < .001,
R-squared = .39). Frame was the smallest influence on
their preferences compared to the ratings of the separate
options; the average attractiveness ratings of the Sure-
Thing and Risky options were both highly predictive.
Among the less numerate, the model was less predic-
tive of their preferences but still highly significant (model
F(3, 48) = 6.8, p < .001, R-squared = .30). Frame was the
strongest predictor by far of their preferences; average at-
tractiveness ratings of the Sure-Thing and Risky options
did not predict their choice preferences.
We partialled out the effects of ACT scores and Need
For Cognition (NFC) from numeracy scores using regres-
sion and then used the resulting numeracy residuals to
conduct again the analyses of the Full-Scenario risky-
choice problems above. The results did not change in any
substantial way. A second analysis, after partialling out
the effects of GPA from numeracy instead, also showed
no substantial changes from the original analysis. Nu-
meracy appears to influence framing effects over and
above these proxy measures of intelligence and prefer-
ence for thinking harder (see Appendix C for more de-
tailed results). A similar set of analyses was conducted
using NFC scores in place of numeracy (with and with-
out partialling out the effects of numeracy, ACT scores,
and GFA). Neither NFC nor its interactions with frame or
ratings of the component options significantly predicted
choices in any analysis.
Another way to examine the extent to which partici-
pants were consistent in their evaluations is to compare
the inferred choice preference calculated from the attrac-
tiveness ratings of the separate options and the actual
choice preference made when the two options were pre-
sented jointly. To do this, for each pair of ratings, we
subtracted the rating for the Sure-Thing option from the
rating for the Risky option. Thus, higher values meant
greater inferred preference for the Risky option over the
Sure-Thing option, just as it does for the actual choice
preference. Correlations by scenario were then calculated
between the choices inferred from these differences and
the actual choice responses, separately within the low-
and high-numerate groups. Finally, the average correla-
tion (computed across the five scenarios) was calculated
within each group.
The results demonstrated greater consistency of in-
ferred and actual preferences among the high numerate
than among the less numerate. Specifically, the aver-
age correlation between inferred and actual choice pref-
erences across the five scenarios was .45 and .09 for
the high and low numerate, respectively (the respective
ranges of correlations were .36 to .63 and .01 to .15).4
4An analysis with participants and (all five) items as crossed random
effects (Baayen, Davidson, & Bates, in press) supported these results.
Specifically, the dependent variable was risky choice, and the fixed-
effect predictors were the Risky-Sure attractiveness difference, frame
(positive/negative), numeracy, and the interactions of numeracy with the
attractiveness difference and with frame. Of primary interest, both in-
teractions with numeracy were significant (as assessed by MCMC sam-
pling, p = .0012 and .0228, respectively). That is, high numeracy
was associated with a stronger relationship between the Full-scenario

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 443
Table 5: Regression results (b-values) predicting the risky-choice preferences in low and high numeracy groups.
Low numerate (n = 52)
(scores = 5–9 correct out of 11)
High numerate (n = 56)
(scores = 10–11 correct out of 11)
Overall
model
F (3, 48)
Frame
Sure-
Thing
option
rating
Risky
option
rating
Overall
model
F (3, 52)
Frame
Sure-
Thing
option
rating
Risky
option
rating
1. Sweden - disease 6.2** –.92*** .18 .01 11.7*** –.10 –.47*** .58***
2. Stock 2.1 –.61* .10 .08 7.7*** –.54** –.43** .27
3. Wildfire season 2.5 –.46 –.16 –.06 3.9** –.35 –.34* .21
Average of 3 scenarios 6.8*** –.66*** .02 .04 11.1*** –.31* –.44*** .43***
4. Drought in Spain 0.7 –.32 .06 .09 2.9* .16 –.41** .34*
5. Delta SS 1.4 –.17 .06 .29 4.5** –.04 –.33* .45**
Average of 5 scenarios 5.7** –.50*** .01 .18 11.3*** –.16 –.44*** .43***
Note: * p < .05; ** p < .01; *** p < .001.
4 Discussion
These results help us understand both the risky-choice
framing phenomenon and the role of numeracy in deci-
sion making. Risky-choice framing effects appear robust
(with the highly numerate demonstrating a tendency to-
wards a smaller effect of frame compared to the less nu-
merate). These effects, however, do not appear to be a
singular effect as previously thought, but rather have dif-
ferent underlying mechanisms for different people.
In particular, in analyses of the less numerate, they
demonstrated a direct effect of frame on their choices
and no significant effects of how attractive they found
the separate options. This lack of influence may be due
to the less numerate attending more superficially to the
individual options and attempting less to evaluate them;
they may focus, for example, on the sure gain in the
Sure-Thing option when choosing in the positive frame
and the possibility of no loss provided by the Risky op-
tion in the negative frame. It may also be, however, that
the less numerate are less able to translate information
about a given option into an attractiveness rating. If this
is the case, then the ratings of the less numerate should
be less reliable (more variable) than those of the highly
numerate. There was some evidence consistent with this
hypothesis, with the standard deviation of responses to
the five Sure-Thing components, five Risky components,
and five Full scenarios being higher for the less numer-
ate compared to the highly numerate in 11 of the 15 re-
sponses (p = .059, based on a sign test using the bino-
risky-choice preference and the attractiveness difference, and with rela-
tively more risky choice in the positive frame (and relatively less in the
negative frame).
mial distribution), but the size of the difference was small
with standard deviations of responses ranging from 1.38
to 1.75 for the less numerate and 1.32 to 1.68 for the
highly numerate. Finally, the less numerate may attend
to the individual-option information in the risky choice
and may translate it appropriately, but fail to integrate the
information in their choices, suggesting that choices are
not always based on integrating judgments of provided
options. This suggestion is supported by the lower ob-
served consistency among the less numerate between in-
ferred preferences based on component ratings and actual
preferences.
The less numerate also demonstrated a larger fram-
ing effect on their evaluations of the Sure-Thing options.
These results are consistent with previous numeracy re-
sults with attribute framing (Peters et al., 2006). In the
analysis of the Full-Scenario framing effect, these Sure-
Thing evaluations influenced their choice preferences less
relative to the preferences of those higher in numer-
acy. At face value, this result was somewhat inconsis-
tent with the robust numeracy effect in attribute-framing
studies; it was, however, consistent overall with Peters
et al.’s (2006) interpretation of their attribute-framing re-
sults. Specifically, they argued that the less numerate re-
spond more to the given frame of information whereas the
highly numerate demonstrated more complex processing
of the same information.
Consistent with this argument, the model was more
predictive of preferences of the highly numerate than the
less numerate. The highly numerate showed only a small
direct effect of frame over and above their evaluations of
the individual options. Evaluations of both options in-
fluenced their choice preferences as if the highly numer-

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 444
ate were able to keep on-line both pieces of information
during the choice process whereas the less numerate may
not be able to keep as much numeric information on-line;
pilot data suggest that the highly numerate have some-
what greater working-memory capacity compared to the
less numerate (Peters, 2006) which could exacerbate this
effect. Overall, the highly numerate nonetheless appear
to consider both the possibility of ensuring some saving
of lives (their evaluations of sure-thing options) and the
possibility of avoiding loss of life (evaluations of risky
options). One caveat on this argument is the correlational
nature of the design combined with all participants re-
sponding to the Full Scenarios first and the component
options second. Given this, it is possible that participants
simply attempted to be consistent from the first choice
response to the later attractiveness ratings, and the highly
numerate were more successful at this consistency. Fur-
ther studies should explore both these working-memory
and order issues.
These results provide additional evidence that the
highly numerate not only understand numbers better, as
shown in past studies, but they use them more frequently
as well. This is important for two reasons. First, it
again highlights the distinction between comprehension
and use of information. Decision makers all may know
that 200 is one-third of 600, but bringing this information
(e.g., evaluations of the Sure-Thing option) to bear on de-
cisions is another matter. Compared to the less numerate,
the highly numerate appeared to use more complex pro-
cessing of given numeric information in the construction
of their preferences. Second, the results are a reminder
that information providers can, and perhaps should, pro-
vide additional assistance to decision makers by paying
careful attention to how numeric information is provided
in order to help the less numerate make better use of nu-
merical information (e.g., Fagerlin et al., 2007). Testing
formats may be critical, as the intuitions of information
providers concerning best ways to present information
may not be adequate. As Fischhoff (in press) argued,
“One should no more release untested communications
than untested pharmaceuticals.”
These results demonstrate that the well-known risky-
choice framing effect is more than one effect. Debiasing
these effects, therefore, may require different methods for
different people. For the less numerate, providing nu-
meric information in formats that are easier to evaluate
(and allow them access into the meaning of the informa-
tion that can then be used in decisions; Peters et al., 2007)
may be best. The highly numerate may already access
the meaning of the information, and debiasing may in-
volve encouragement to think harder about the individual
options.
At a broader level, we feel that the paradigm intro-
duced here can be applied to other phenomena in human
judgment and decision making. In a number of choice
situations with options varying on dimensions such as
complexity, there are opposing models for the decision-
making process that pit an algebraic comparison of the
integrated values of the choice options against a heuris-
tic solution based on which option “feels” better. For
example, performance on ambiguity-probability tradeoff
tasks originating with Ellsberg’s (1961) two-color prob-
lem has been explained as either a process where the un-
known probability for an ambiguous option is estimated
and then compared to the known probability of the unam-
biguous option or as a process where the option with the
most information is preferred (see Lauriola et al., 2007).
By having each respondent rate both ambiguous and un-
ambiguous options separately as well as making a forced
choice, and by including individual-difference measures
such as those used by Lauriola et al., comparisons can be
made between those individuals who are more apt to be
influenced by component evaluations and those who are
more apt to use simplifying heuristics in forced-choice
situations. As another example, the well-known “decoy”
or “asymmetric dominance effect” in consumer behav-
ior involves a change in preference between two options
when a third option is introduced which is inferior to one
option on all attributes but is superior to the other option
on one attribute. The result is an increase in preference
for the dominating option (Huber et al., 1982). Using the
present approach to unravel the processes used by differ-
ent consumers, respondents would rate each of the sepa-
rate options as well as make both the two-option choice
and the three-option choice. In other words, the approach
described in the current paper can be used to catego-
rize individuals in terms of whether choices are based
on the separate evaluations of choice options or whether
new, usually simplifying, factors emerge in head-to-head
forced choices.
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Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 446
Appendix A. Risky-choice scenarios
Positive frames
Options have equal expected value:
1. Sweden is preparing for the outbreak of an unusual
disease, which is expected to kill 600 people. The fol-
lowing alternative programs have been proposed to limit
the spreading of the disease: If Program A is adopted,
200 people will be saved. If Program B is adopted, there
is a 1/3 probability that 600 people will be saved, and a
2/3 probability that no people will be saved.
2. Suppose that you have invested $60,000 in the stock
of a company that just filed for bankruptcy. They have
proposed two alternatives in order to save some of the
invested money: If Program A is adopted, $20,000 will be
saved. If Program B is adopted, there is a 1/3 probability
that $60,000 will be saved, and a 2/3 probability that no
money will be saved.
Sure-thing option A has a higher expected value than
risky option B:
3. The wildfire season is about to start and an old growth
forest in the Northwest of the US will be affected. This
forest is home to 3,600 animals that are endangered by
the fire. Two programs have been proposed to protect the
animals. If Program A is adopted, 1,560 animals will be
saved. If Program B is adopted, there is a 1/3 probability
that 3,600 animals will be saved, and a 2/3 probability
that none will be saved.
Risky option B has a higher expected value than sure-
thing option A:
4. A severe drought is foreseen to hit the South of Spain
this summer. The drought will cause the destruction of
24,000 acres of crops. Two programs of water supply
have been proposed: If Program A is adopted, 5,600 acres
of crops will be saved. If Program B is adopted, there is
a 1/3 probability that 24,000 acres of crops will be saved,
and a 2/3 probability that none will be saved.
5. The country of Delta’s Social Security System is
undergoing a crisis. Economists believe that 12 million
seniors will lose their medical benefits next year. Two
alternative programs have been proposed to mitigate this
problem. If Program A is adopted, 2.8 million senior cit-
izens will keep their medical benefits. If Program B is
adopted, there is a 1/3 probability that 12 million seniors
will keep their medical benefits, and a 2/3 probability that
no senior citizens will keep their medical benefits.
Negative frames
Options have equal expected value:
1. Sweden is preparing for the outbreak of an unusual
disease, which is expected to kill 600 people. The fol-
lowing alternative programs have been proposed to limit
the spreading of the disease: If Program A is adopted,
400 people will die. If Program B is adopted, there is a
1/3 probability that nobody will die, and a 2/3 probability
that 600 people will die.
2. Suppose that you have invested $60,000 in the stock
of a company that just filed for bankruptcy. They have
proposed two alternatives in order to save some of the
invested money: If Program A is adopted, $40,000 will
be lost. If Program B is adopted, there is a 1/3 probabil-
ity that no money will be lost, and a 2/3 probability that
$60,000 will be lost.
Sure-thing option A has a higher expected value than
risky option B:
3. The wildfire season is about to start and an old growth
forest in the Northwest of the US will be affected. This
forest is home to 3,600 animals that are endangered by
the fire. Two programs have been proposed to protect
the animals. If Program A is adopted, 2,040 animals will
perish. If Program B is adopted, there is a 1/3 probabil-
ity that no animals will perish, and a 2/3 probability that
3,600 animals will perish.
Risky option B has a higher expected value than sure-
thing option A:
4. A severe drought is foreseen to hit the South of Spain
this summer. The drought will cause the destruction of
24,000 acres of crops. Two programs of water supply
have been proposed: If Program A is adopted, 18,400
acres of crops will be lost. If Program B is adopted, there
is a 1/3 probability that no crops will be lost, and a 2/3
probability that 24,000 acres of crops will be lost.
5. The country of Delta’s Social Security System is
undergoing a crisis. Economists believe that 12 million
seniors will lose their medical benefits next year. Two
alternative programs have been proposed to mitigate this
problem. If Program A is adopted, 9.2 million senior cit-
izens will lose their medical benefits. If Program B is
adopted, there is a 1/3 probability that zero senior citizens
will lose their medical benefits, and a 2/3 probability that
12 million senior citizens will lose their medical benefits.

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 447
Appendix B. Numeracy items (Lipkus, Samsa, & Rimer, 2001) with percent
correct per item in this sample.
Items ordered hardest to easiest % correct
responses
The chance of getting a viral infection is .0005. Out of 10,000 people, about how many of them are
expected to get infected?
56.5
If Person A’s chance of getting a disease is 1 in 100 in ten years, and person B’s risk is double that
of A, what is B’s risk?
62.0
In the ACME PUBLISHING SWEEPSTAKES, the chance of winning a car is 1 in 1,000. What
percent of tickets of ACME PUBLISHING SWEEPSTAKES win a car?
70.4
If Person A’s risk of getting a disease is 1% in ten years, and Person B’s risk is double that of A’s,
what is B’s risk?
83.3
In the BIG BUCKS LOTTERY, the chances of winning a $10.00 prize are 1%. What is your best
guess about how many people would win a $10.00 prize if 1,000 people each buy a single ticket
from BIG BUCKS?
88.9
If the chance of getting a disease is 10%, how many people would be expected to get the disease: B:
Out of 1000? Answer: ____ people
90.7
Imagine that we roll a fair, six-sided die 1,000 times. Out of 1,000 rolls, how many times do you
think the die would come up as an even number?
91.7
If the chance of getting a disease is 10%, how many people would be expected to get the disease: A:
Out of 100? Answer: ____ people
96.3
If the chance of getting a disease is 20 out of 100, this would be the same as having a ______%
chance of getting the disease
97.2
Which of the following numbers represents the biggest risk of getting a disease? 1 in 100, 1 in 1000,
1 in 10
97.2
Which of the following numbers represents the biggest risk of getting a disease? 1%, 10%, 5% 99.1

Judgment and Decision Making, Vol. 3, No. 6, August 2008 Numeracy and risky-choice framing 448
Appendix C. Regression results (b-values) predicting the average risky-choice
preference across the three scenarios using numeracy and numeracy residu-
als.
Overall
model Frame
Sure-
Thing
option
rating
Risky
option
rating
Numeracy
or residual
Frame*
Numeracy
or residual
Sure-
Thing
option
rating*
Numeracy
or residual
Risky
option
rating*
Numeracy
or residual
Using
numeracy,
F(7,100)
8.3*** –.51*** –.22** .23** .00 .19** –.17** .13*
Using numeracy residuals after partialling out:
Need For
Cognition
(NFC),
F(7,100)
8.2*** –.51*** –.22** .23** .01 .20** –.17** .13*
ACT scores,
F(7,90) 8.0*** –.54*** –.20* .24** –.03 .20* –.14* .14*
NFC and
ACT scores
F(7,90)
8.0*** –.54*** –.20* .24** –.03 .21* –.14* .14*
GPA,
F(7,100) 8.2*** –.51*** –.22** .23** .01 .19* –.16** .13*