1 Background: Visualizing Uncertainty
In communicating the results of statistical analysis, visualization authors commonly represent uncertainty as a range of possible values as recommended by numerous experts (e.g., [Cumming2014, Manski2018b, taylor_guidelines_1994]). Other conventional uncertainty representations commonly used in statistical analysis include aggregate encodings of distributions such as boxplots [Tukey1977], histograms [Pearson1895], and densities [Spiegelhalter1999, Barrowman2003]. Frequencybased uncertainty visualizations build on a large body of work suggesting that framing probabilities as frequencies of events improves statistical reasoning [Chance2000, Galesic2009, Gigerenzer1995, Hoffrage1998, Hogarth2011, Hullman2015, Kay2016, Kale2019hops, Kim2019, Micallef2012, Fernandes2018]. These include hypothetical outcome plots (HOPs) [Hullman2015], which encode possible outcomes as frames in an animation, and quantile dotplots [Kay2016], which quantize a distribution of possible outcomes and represent each quantile as a discrete dot. A growing body of work suggests that lay and expert audiences commonly misinterpret interval representations of uncertainty [Belia2005, Soyer2012, Hoekstra2014] and that other uncertainty visualization formats such as gradient plots [Correll2014], violin plots [Correll2014, Hullman2015], HOPs [Hullman2015, Kale2019hops], and quantile dotplots [Kay2016, Fernandes2018] lead to more accurate interpretation and performance on various tasks.
In our study, we compare two frequencybased visualizations, quantile dotplots and HOPs, with two more conventional uncertainty representations, intervals and densities. By testing each with and without added means, we investigate the extent to which users of these uncertainty visualizations differ in their tendency to ignore uncertainty.
When chart users don’t know how to interpret uncertainty, prior work [Hullman2015] suggests that they may substitute a judgment of the mean difference between distributions for more complicated judgments about the reliability of effects. This visual distance heuristic motivates design principles, for example, that the quantitative axis on a bar chart should always start at zero [Brinton1939, Huff1993], or that axis scales should align visual distance with effect size [Witt2019]. Axis scale impacts the perceived importance of effect size regardless of chart type (e.g., lines versus bars) and despite attempts to signal that an axis does not start at zero (e.g., breaking the axis) [Correll2020]
. Rescaling the axis on a chart that displays inferential uncertainty (e.g., 95% confidence intervals) to the scale implied by descriptive uncertainty (e.g., 95% predictive intervals) can reduce bias in impressions of effect size
[Hofman2020]. In our study, we investigate the visual distance heuristic by asking users to compare distributions with different levels of variance on a common scale (Fig. 1).2 Method
We tested how adding means to different uncertainty visualizations impacts users’ estimates and incentivized decisions from effect size.
2.1 Tasks & Procedure
Our task was like a fantasy sports game. We showed participants charts comparing the predicted number of points scored by their team with and without adding a new player (e.g., Fig. 1). Participants estimated the effect size of adding the new player and decided whether or not to pay to add the new player to their team.
Effect Size Estimation: We asked participants to estimate a measure of effect size called probability of superiority or common language effect size [McGraw1992]: “How many times out of 100 do you estimate that your team would score more points with the new player than without the new player?” We elicited probabilities as “times out of 100” based on literature in statistical reasoning (e.g., [Gigerenzer1995, Hoffrage1998]) suggesting that people reason more accurately with probabilities when they are framed as frequencies. Probability of superiority, the percent of the time that outcomes for one group exceed outcomes for another group , is a proxy for standardized mean difference [Coe2002, Cummings2011], the difference between two group means relative to uncertainty in the estimates. Using synthetic data (see Section 3.5), we evaluated bias in effect size estimates compared to a known ground truth.
Intervention Decisions: We also asked participants to make binary decisions indicating whether they would “Pay for the new player,” or “Keep [their] team without the new player.” On each trial, the participant’s goal was to win an award worth $3.17M, and they could pay $1M to add a player to their team if they thought the new player improved their chances of winning enough to be worth the cost. There were four possible payouts in each trial:

[noitemsep,nolistsep]

The participant won without paying for a new player (+$3.17M).

The participant paid for a new player and won (+$2.17M).

They failed to win without paying for a new player ($0).

The participant paid for a new player and failed to win ($1M).
The user could only lose money if they paid for the new player.^{1}^{1}1In pilot studies, we tested how framing outcomes as winning versus losing awards impacted user behavior and found that participants had greater preference for intervention when it was described as increasing the certainty of gains, consistent with prior work by Tversky and Kahneman [Kahneman1979, Tversky1981]. We set up the incentives for our task so that a riskneutral chart user should pay for a new player only when effect size was larger than 74% probability of superiority or Cohen’s d of 0.9, the average effect size in a recent survey of studies in experimental psychology [Schafer2019]. This enabled us to evaluate intervention decisions compared to a utilityoptimal standard.
Feedback: At the end of each trial we told users whether or not their team scored enough points to win an award, using a Monte Carlo simulation to generate a win and loss based on the participant’s decision. We split feedback into two tables. One showed the change in account value for the current trial. The other showed cumulative account value and how this translated into a bonus in real money. By showing probabilistic outcomes, instead of the expected value of decisions, feedback gave participants a noisy signal of how well they were doing, mirroring realworld learning conditions for decisions under uncertainty.
Payment: Participants received a guaranteed reward of $1 plus a bonus of , where $0.08 per $1M was the exchange rate from account value to real dollars, was the value of their fantasy sports account at the end of the experiment, and $150M was a cutoff account value below which they receive no bonus. These values were carefully chosen to result in bonuses between $0 and $3, such that participants who guessed randomly and experienced unlucky probabilistic outcomes would receive no bonus, and participants who responded optimally would be guaranteed a bonus.
User Strategies: To supplement our quantitative measures with qualitative descriptions of users’ visual reasoning, at the end of each of the two block of trials, we asked users, “How did you use the charts to complete the task? Please do your best to describe what sorts of visual properties you looked for and how you used them.”
2.2 Formalizing a Class of Decision Problems
Our decision task represents a class of decision problems where one makes a binary decision about whether or not to invest in an intervention that changes the probability of an allornothing outcome. For example, this class of problems includes medical decisions about treatments that may save someone’s life or cure them of a disease, organizational decisions about hiring personnel to reach a contract deadline, and personal decisions such as paying for education to seek a promotion. Previous decisionmaking literature examines similar problems in the context of salting the road in freezing weather [Joslyn2012, Joslyn2013], voting in presidential elections [Westwood2019], and willingness to pay for interventions in a fictional scenario [Hofman2020]. The key similarity between these decision problems is that their incentive structures imply a common utility function.
A utility function defines optimal (i.e., utility maximizing [vonNeumann1944]) decisions for a riskneutral observer, providing a normative benchmark used to measure bias in decisionmaking. Comparing behavior to a riskneutral benchmark is a common practice in judgment and decisionmaking studies [Baron2008], often used to measure risk preferences [Weber1994] or attitudes that make a person more or less inclined to take action than they should be based on a costbenefit analysis. In the class of decision problems we investigate, the implied utility function depends on both the amount of money one stands to win or lose (e.g., the value of an award and the cost of a new player) and the effect size (e.g., the difference in team performance with versus without a new player).
Let be the value of an award. Let be the cost of adding a new player to the team. The utilityoptimal decision rule is to intervene if
where is the probability of winning an award without a new player, and is the probability of winning an award with a new player. Assuming a constant ratio between the value of the award and the cost of intervention , we express the decision rule in terms of the difference between the probabilities of winning an award with versus without a new player:
The threshold level of effect size above which one should intervene depends on the incentive ratio and the probability of a payout without intervention . In our study, we fixed the incentives and the probability of winning an award without a new player so that users would not have to keep track of changing incentives, and effect size alone was the signal that users should base decisions on.^{2}^{2}2In pilot studies, we tried manipulating and and found that these changes had little impact on the effectiveness of different uncertainty visualizations for supporting utilityoptimal decisionmaking. In light of prior work showing that Mechanical Turk workers do not respond to changes of incentives [Stoycheff2016], we suspect that these manipulations might have an impact in realworld settings which is difficult to measure on crowdsourcing platforms. This enabled a controlled evaluation of how users translate visualized effect size into a sense of utility. By modeling a functional relationship between effect size and utility, we go beyond prior work which either does not vary the effectiveness of interventions (e.g., [Joslyn2012, Joslyn2013, Westwood2019]) or examines only two levels of effect size as a robustness check for statistical tests (e.g., [Hofman2020]).
2.3 Experimental Design
We assigned each user to one of four uncertainty visualization conditions at random, making comparisons of uncertainty visualizations betweensubjects. On each trial, users made a probability of superiority estimate and an intervention decision. We asked users to make repeated judgments for two blocks of 16 trials each. In one block, we showed the users visualizations with means added, and in the other block there were no means. We counterbalanced the order of these blocks across participants. Each of the 16 trials in a block showed a unique combination of ground truth effect size (8 levels) and variance of distributions (2 levels), making our manipulations of ground truth, variance, and adding means all withinsubjects. The order of trials in each block was randomized. In the middle of each block, we inserted an attention check trial, later used to filter participants who did not attend to the task. Users always saw an attention check at 50% probability of superiority with means and at 99.9% without means. Hence, each participant completed 17 trials per block and 34 trials total.
2.4 Uncertainty Visualization Conditions
We evaluated visualizations intended to span a design space characterized by the visual salience of the mean, expressiveness of uncertainty representation, and discrete versus continuous encodings of probability. As described above, we showed four uncertainty visualization formats—intervals, hypothetical outcome plots (HOPs), density plots, and quantile dotplots—with and without separate (i.e., extrinsic) vertical lines encoding the mean of each distribution. We expected that adding means would bias effect size estimates toward discounting uncertainty and that this effect would be most pronounced for uncertainty visualizations in which the mean is not intrinsically salient.
Intervals: We showed users intervals representing a range containing 95% of the possible outcomes (Fig. Visual Reasoning Strategies and Satisficing: How Uncertainty Visualization Design Impacts Effect Size Judgments and Decisions, left column). In the absence of a separate mark for the mean, the mean was not intrinsically encoded, and the user could only find the mean by estimating the midpoint of the interval. Intervals were not very expressive of probability density since they only encoded lower and upper bounds on a distribution.
Hypothetical Outcome Plots (HOPs): We showed users animated sequences of strips representing 20 quantiles sampled from a distribution of possible outcomes (Fig. Visual Reasoning Strategies and Satisficing: How Uncertainty Visualization Design Impacts Effect Size Judgments and Decisions, left center column), matching the data shown in quantile dotplots. Animations were rendered at 2.5 frames per second with no animated transitions (i.e., tweening or fading) between frames, looping every 8 seconds. We shuffled the two distributions of 20 quantiles using a 2dimensional quasirandom Sobol sequence [Sobol1976] to minimize the apparent correlation between distributions. Like intervals, HOPs did not make the mean intrinsically salient, as means were implicitly encoded as the average position of an ensemble of strips shown over time. However, HOPs were more expressive of the underlying distribution than intervals and expressed uncertainty as frequencies of events, so they conveyed an experiencebased sense of probability.
Densities: We showed users continuous probability densities where the height of the area marking encoded the probabilities of corresponding possible outcomes on the xaxis (Fig. Visual Reasoning Strategies and Satisficing: How Uncertainty Visualization Design Impacts Effect Size Judgments and Decisions
, right center column). Unlike intervals and HOPs, the mean was explicitly represented as the point of maximum mark height because distributions were symmetrical, so means were intrinsically salient. Densities were also the most expressive of the underlying probability density function among the uncertainty visualizations we tested.
Quantile Dotplots: We showed users dotplots where each of 20 dots represented a 5% chance of a corresponding possible outcome on the xaxis (Fig. Visual Reasoning Strategies and Satisficing: How Uncertainty Visualization Design Impacts Effect Size Judgments and Decisions, right column). Like densities, because distributions were symmetrical and dots were stacked in bins to express this symmetry, the mean was explicitly represented as the point of maximum height and was thus intrinsically salient.
2.5 Generating Stimuli
We generated synthetic data covering a range of effect size such there were an equal number of trials where users should and should not intervene. Recall that 50% corresponded to a new player who did not improve the team’s performance at all, 100% corresponded to a definite improvement in performance, and 74% was the utilityoptimal decision threshold. We sampled eight distinct levels of ground truth probability of superiority, four values between 55% and 74% and four values between 74% and 95%, such that there are an equal number of trials above and below the utilityoptimal decision threshold. Prior work in perceptual psychology [Gonzalez1999, Zhang2012]
suggests that the brain represents probability on a log odds scale. For this reason, we converted probabilities into log odds units and sampled on this logittransformed scale using linear interpolation between the endpoints of the two ranges described above. We added two attention checks at probabilities of superiority of 50% and 99.9%, where the decision task should have been very easy, to allow for excluding participants who were not paying attention.
To derive the visualized distributions from ground truth effect size, we made a set of assumptions. We assumed equal and independent variances for the distributions with and without a new player such that where
was the variance of the difference between distributions. We tested two levels of variance, setting the standard deviation of the difference between distributions
to a low value of 5 or a high value of 15. These levels produced distributions that looked relatively narrow or wide compared to the width of the chart, making visual distance between distributions an unreliable cue for effect size such that at low variance large effect sizes corresponded to distributions that looked close together.We determined the distance between distributions, or mean difference , using the formula where were ground truth values as standardized mean differences (i.e., Cohen’s d [Coe2002, Cummings2011]). The mean number of points scored without the new player was held constant , which corresponded to a 50% chance of winning the award. We calculated mean for the team with a new player . We rendered our chart stimuli using the parameters , , and to define the two distributions on each chart. Holding the chance of winning without a new player constant at 50% (Fig. 1, blue distributions) is an experimental control that enables us to compare a user’s preference for new players across trials using a coin flip gamble as the alternative choice, which is common in judgment and decisionmaking studies [Baron2008].
2.6 Modeling
We wanted to measure how much users underestimate effect size in their probability of superiority responses, how much they deviate from a utilityoptimal criterion in their decisions, and how sensitive they are to effect size for the purpose of decisionmaking. To measure underestimation bias, we fit a linear in log odds model [Gonzalez1999, Zhang2012] to probability of superiority responses, and we derive slopes describing users’ responses as a function of the ground truth (Fig. 2
). To measure bias and sensitivity to effect size in decisionmaking, we fit a logistic regression to intervention decisions, and we derive
points of subjective equality and justnoticeable differences describing the location and scale of the logistic curve as functions of effect size (Fig. 3).2.6.1 Approach
We used the brms package [Burkner2020]
in R to build Bayesian hierarchical models for each response variable: probability of superiority estimates and decisions of whether or not to intervene. We started with simple models and gradually added predictors, checking the predictions of each model against the empirical distribution of the data. This process of
model expansion [Gabry2019] enabled us to understand the more complex models in terms of how they differ from simpler ones.We started with a minimal model, which had the minimum set of predictors required to answer our research questions, and built toward a maximal model, which included all the variables we manipulated in our experiment. We specified the minimal and maximal models for each response variable in our preregistration.^{3}^{3}3 https://osf.io/9kpmb
Expanding models gradually helped us determine priors oneatatime. Each time we added a new kind of predictor to the model (e.g., a random intercept per participant), we honed in on weakly informative priors using prior predictive checks [Gabry2019]. We centered the prior for each parameter on a value that reflected no bias in responses. We scaled each prior to avoid predicting impossible responses and to impose enough regularization to avoid issues with convergence in model fitting. We documented priors and model expansion in Supplemental Materials.^{4}^{4}4 https://github.com/kalealex/effectsizejdm
2.6.2 Linear in Log Odds Model:
We use the following model (WilkinsonPinheiroBates notation [Wilkinson1973, Burkner2020, Pinheiro2020]) for responses in the probability of superiority estimation task:
Where is the user’s probability of superiority response, is the ground truth probability of superiority, is an index of trial order, is an indicator for whether or not extrinsic means are present, is an indicator for low versus high variance,
is a dummy variable for uncertainty visualization condition,
is an indicator for block order, and is a unique identifier for each participant used to model random effects. Note that there are submodels for the mean and standard deviation of user responses.Motivation: We apply a logittransformation to both and , changing units from probabilities of superiority into log odds, because prior work suggests that the perception of probability should be modeled as linear in log odds (LLO) [Gonzalez1999, Zhang2012]. We model effects on both and because we noticed in pilot studies that the spread of the empirical distribution of responses varies as a function of the ground truth, visualization design, and trial order. However, we are most interested in effects on mean response. The term tells our model that the slope of the LLO model varies as a joint function of whether or not means were added, the level of variance, uncertainty visualization, and block order (i.e., all of these factors interacted with each other). This enables us to answer our core research questions, while controlling for order effects. The term models learning effects, so we isolate the impact of uncertainty visualizations. In both submodels, we added withinsubjects manipulations as random effects predictors as much as possible without compromising model convergence.
2.6.3 Logistic Regression
We use this model to make inferences about intervention decisions:
Where is the user’s choice of whether or not to intervene, is the probability that they intervene, and is a logittransformation of the utilityoptimal decision rule (see Section 3.2):
This gives us a uniformly sampled scale of evidence where zero represents the utilityoptimal decision threshold. All other factors are the same as in the LLO model (see Section 3.6.2).
Motivation: We logittransform our evidence scale because internal representations of probabilities are thought to be on a log odds scale [Gonzalez1999, Zhang2012], such that linear changes in log odds appear similar in magnitude. The term tells our model that the location and scale of the logistic curve vary as a joint function of whether or not means were added, the level of variance, uncertainty visualization, and block order. Mirroring an analogous term in the LLO model, this enables us to answer our core research questions, while controlling for order effects. The term models learning effects. As with the LLO model, we specify random effects per participant through model expansion by trying to incorporate as many withinsubjects manipulations as possible.
2.7 Derived Measures
From our models, we derive estimates for three preregistered metrics that we use to compare visualization designs.
Linear in log odds (LLO) slopes measure the degree of bias in probability of superiority estimation (Fig. 2). A slope of one indicates unbiased performance, and slopes less than one indicate the degree to which users underestimate effect size.^{5}^{5}5LLO slopes less than one represent bias toward the probability at the intercept, , which is close to in our study. We measure LLO slopes because they are very sensitive to the expected pattern of bias in responses, giving us greater statistical power than simpler measures like accuracy. Specifically, LLO slope is the expected increase in a user’s logittransformed probability of superiority estimate, , for one unit of increase in logittransformed ground truth, . Using a linear metric (i.e., slope in logitlogit space) to describe an exponential response function in probability units comes from a theory that the brain represents probabilities on a log odds scale [Gonzalez1999, Zhang2012]. The LLO model [Gonzalez1999, Zhang2012] can be thought of as a generalization of the cyclical power model [Hollands2000] that allows a varying intercept or a modification of Stevens’ power law [Stevens1957] for proportions.
Points of subjective equality (PSEs) measure bias toward or against choosing to intervene in the decision task relative to a utilityoptimal and riskneutral decision rule (see Section 3.2). PSEs describe the level of evidence at which a user is expected to intervene 50% of the time (Fig. 3). A PSE of zero is utilityoptimal, whereas a negative value indicates that a user intervenes when there is not enough evidence, and a positive value indicates that a user doesn’t intervene until there is more than enough evidence. In our model, PSE is where and come from the linear model in logistic regression.
Just noticeabledifferences (JNDs) measure sensitivity to effect size information for the purpose of decisionmaking (Fig. 3). They describe how much additional evidence for the effectiveness of an intervention a user needs to see in order to increase their rate of intervening from 50% to about 75%. A JND in evidence units is a difference in the log probability of winning the award with the new player. We chose this scale for statistical inference because units of log stimulus intensity are thought to be approximately perceptually uniform [Stevens1957, Varshney2013]. In our model, JND is where is the same as for PSE.
2.8 Participants
We recruited users through Amazon Mechanical Turk. Workers were located in the US and had a HIT acceptance rate of 97% or more. Based on the reliability of inferences from pilot data, we aimed to recruit 640 participants, 160 per uncertainty visualization. We calculated this target sample size by assuming that variance in posterior parameter estimates would shrink by a factor of roughly if we collected a larger data set using the same interface. Since we based our target sample size on betweensubjects effects (e.g., uncertainty visualization), our estimates of withinsubjects effects (e.g., adding means) were very precise.
We recruited 879 participants. After our preregistered exclusion criterion that users needed to pass both attention checks, we slightly exceeded our target sample size with 643 total participants. However, we had issues fitting our model for an additional 21 participants, 17 of whom responded with only one or two levels of probability of superiority and 4 of whom had missing data. After these nonpreregistered exclusions, our final sample size was 622 (with block order counterbalanced). All participants were paid regardless of exclusions, on average receiving $2.24 and taking 16 minutes to complete the experiment.
2.9 Qualitative Analysis of Strategies
Using the two strategy responses we elicited from each user, we conducted a qualitative analysis to characterize users’ visual reasoning strategies based on heuristics they used with different visualization designs (with and without means) and whether they switched strategies.
The first author developed a bottomup open coding scheme for how users described their reasoning with the charts. Since some responses were uninformative about what visual properties of the chart a user considered (e.g., “I used the charts to estimate the value added by the new player.”), we omitted participants for whom both responses were uninformative from further analysis. Excluding 180 such participants resulted in a final sample of 442 for our qualitative analysis.
We used our open codes to develop a classification scheme for strategies based on what visual features of charts users mention, whether they switched strategies, and whether they were confused by the chart or task. We coded for the following uses of visual features:

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Relative position of distributions

Means, whether users relied on or ignored them

Spread of distributions, whether users relied on variance, ignored it, or erroneously preferred high or low variance

Reference lines, whether users relied on imagined or real vertical lines (e.g., the annotated decision threshold in Fig. Visual Reasoning Strategies and Satisficing: How Uncertainty Visualization Design Impacts Effect Size Judgments and Decisions & 1)

Area, whether users relied on the spatial extent of geometries

Frequencies, whether users of quantile dotplots or HOPs relied on frequencies of dots or animated draws
Thus, we generated a spreadsheet of quotes, open codes, and categorical distinctions which enabled us to provide aggregate descriptions of patterns and heterogeneity in user strategies.
4.3 Discussion
Among the uncertainty visualizations we tested, quantile dotplots lead to the least biased probability of superiority estimates. This is not surprising given previous work (e.g., [Gigerenzer1995, Hoffrage1998, Hullman2015, Kale2019hops, Kay2016]) showing that frequencybased visualizations are effective at conveying probabilities. However, it is surprising that users do not perform reliably differently with frequencybased HOPs than with intervals or densities. HOPs directly encode probability of superiority by how often the draws from the two distributions change order, whereas in all other conditions users would need to calculate effect size analytically from visualized means and variances to arrive at the “correct” inference, although we doubt that users engage in such explicit mathematical reasoning. In Section 5, we present descriptive evidence of heuristics that users employ with different visualization designs, which helps to explain these results.
In most cases, the small effects on LLO slopes when adding means to uncertainty visualizations are probably negligible. However, they are consistent with the pattern of behavior we expect if users rely on visual distance between distributions as a proxy for effect size. When variance is lower relative the axis scale, distances between distributions look small even for large effects (Fig. 1, top), and users tend to underestimate effect size more when means are added. When variance is higher relative the axis scale, distances between distributions roughly correspond to effect size (Fig. 1, bottom), and users tend to underestimate effect size less when means are added, at least for densities and intervals.
Our results suggest that the best visualization design for utilityoptimal decisionmaking probably depends on the level of variance relative to the axis scale. At lower variance, when multiple levels of variance are shown on a common scale, densities without means or quantile dotplots with means lead to the least bias in decisions. At higher variance, users are biased toward intervening in all conditions, and both densities without means and intervals without means lead to the least bias. The impact of means also depends on variance and axis scaling, such that when we average across uncertainty visualizations, adding means exacerbates biases that exist when means are absent. The effect of variance on PSEs (see Supplemental Materials) is large, such that users intervene more often at higher variance than at lower variance. One possible explanation for this is that users rely on distance between distributions as a proxy for effect size and make decisions as if effects are larger when distributions are further apart (Fig. 1).
Reported effects of visualization design on JNDs may not be practically important. All differences in JNDs between visualization designs are smaller than the difference between high versus low variance (see Supplemental Material). Smaller JNDs at high variance may reflect the fact that our high variance charts use white space more efficiently.
4.4 Comparing Magnitude Estimation & DecisionMaking
Different visualization designs lead to the best performance on our magnitude estimation and decisionmaking tasks. To explore this decoupling of performance across tasks, we calculate average posterior estimates of our derived measures—LLO slope, PSE, and JND—for each individual user and compare them. Figure 5 shows that many individuals who are poor at magnitude estimation (i.e., LLO slopes below one) do well on the decision task (i.e., PSEs and JNDs near zero).
One possible explanation for this decoupling of performance on our two tasks is that users may rely on different heuristics to judge the same data for different purposes. This is consistent with Kahneman and Tversky’s [Kahneman1979] distinction between perceiving the probability of an event to be and weighting the probability of an event in decisionmaking as , which suggests that decision weights reflect preferences based on probabilities and risk attitudes [Weber1994]. Recent work in behavioral economics [Khaw2017] suggests that biases in decisionmaking are partially attributable to imprecision in an individual’s subjective perception of numbers (i.e., “number sense”). Since JNDs reflect the precision of perceived effect size implied by one’s decisions and PSEs represent bias in decisionmaking, we can investigate this relationship within individual users in our study (Fig 5). In agreement with prior work, we see that greater sensitivity to effect size for decisionmaking (i.e., JNDs close to zero) predicts more utilityoptimal decisions (i.e., PSEs close to zero). Although, based on the decoupling of LLO slopes and JNDs, it also seems clear that a user’s internal sense of effect size is not necessarily identical when they use the same information for different tasks. We should be mindful that perceptual accuracy may not feed forward directly into decisionmaking.
5 Visual Reasoning Strategies
We use qualitative analysis of reported strategies to identify ways that users judge effect size by comparing distributions, giving us a vocabulary for how visualization design choices impact their interpretations.
5.1 Prevalent Strategies
The strategies we identify are not mutually exclusive. We count a user as employing a strategy if they mention it in either of their responses.
Only Distance: About 62% of users (275 of 442) rely on “how far to the right” the red distribution is compared to the blue one without mentioning that they incorporate the variance of distributions into their judgments (Fig. 1). Roughly 69% of these users (190 of 275) describe making a gist estimate of distance between distributions, with 46% (126 of 275) saying they rely on the mean difference specifically, and 13% (36 of 275) saying they rely on both gist distance and mean difference. Strategies which involve only the distance between distributions should result in a large bias toward underestimating effect size, which is what we see in our aggregated quantitative results.
Distance Relative to Variance: Only about 8% of users (35 of 442) mention that their interpretations of distance depend on the spread of distributions, suggesting that perhaps very few untrained users are sensitive to the impact of variance on effect size. If users estimate standard deviation and mean difference between distributions, they could use this information to calculate effect size analytically. However, we think it is far more likely that these users judge the distance between distributions relative to the spatial extent of uncertainty visualizations, which should result in underestimation bias which is similar to but less pronounced than with judgments of only distance.
Cumulative Probability: A substantial 36% of users (160 of 442) estimate the cumulative probability of winning the award with and/or without the new player. This strategy involves judging the distance, proportion of area, or frequency of markings across the threshold number of points to win (e.g., Fig. 7). These users may be confusing cumulative probability of winning the award, which is the best cue in the decision task, with probability of superiority (i.e., probability that team does better with the new player than without), which is what we ask for in the estimation task. However, since the probability of winning increases monotonically with probability of superiority, this strategy should theoretically result in milder underestimation bias than distancebased strategies.
Distribution Overlap: About 7% of users (31 of 442) describe judging the overlap between distributions. While similar to distancebased strategies, users conceptualize this strategy in terms of area rather than the gap between distributions (Fig. 7). For example, one user said they use HOPs “only to see how much of an overlap [there is] between the two areas,” suggesting that they imagine contours of distributions over the sets of animated draws. This strategy probably results in underestimation bias similar to judging distance relative to variance.
Frequency of Draws Changing Order: This strategy is only relevant to the HOPs condition, where only about 16% of users (19 of 121) employed it. It involves judging the number of animated frames in which that the draws from the two distributions switch order (Fig. 8). This is the best way to estimate probability of superiority from HOPs [Hullman2015]. If we think of the user as accumulating information across frames, the precision of their inference is mostly limited by the number of frames they watch. For example, in Figure 8 red scores higher than blue in 6 of the 8 frames, and watching only 8 frames limits the precision of this inference to increments of . The fact that only a handful of HOPs users employ this strategy helps to explain why the performance of HOPs users is worse than expected.
Switching Strategies: A substantial 29% of users (129 of 442) switch between strategies in the middle of the task. For example, one user of intervals without means described a mix of cumulative probability and distribution overlap strategies: “If the red [distribution] was completely past the dotted line then I would buy the new player no matter what. If there were overlaps with blue I would just risk assess to see if it was worth it to me or not.” While more of a metastrategy, our observation that a significant proportion of users switch is important because it suggests that judgment processes involved in graphical perception may not be consistent within each user.
Strategy  Intervals  HOPs  Densities  Dotplots  Overall 

Distance  73  77  61  64  275 
Rel. to Var.  11  9  10  5  35 
Cumulative  34  50  30  46  160 
Overlap  17  2  9  3  31 
Draw Order  0  19  0  0  19 
Switching  35  48  23  23  129 
Total  112  121  99  110  442 
5.2 Impacts of Visualization Design Choices
Users rely on visual features (Section 3.9) and strategies (Section 5.1) to varying degrees depending on visualization design (Table 1).
Intervals: Roughly 75% of intervals users (85 of 112) rely on relative position as a visual cue for effect size compared to 69% with densities (68 of 99), 61% with HOPs (74 of 121), and 59% with quantile dotplots (65 of 110). Of intervals users who look at relative position, about 87% (74 of 85) employ an only distance strategy, while only about 13% (11 of 85) judge distance relative to variance . In other words, only about 10% of intervals users (11 of 112) incorporate variance into their judgments of distance. About 28% of intervals users (31 of 112) report looking at area, with about 55% of these users (17 of 31) employing a distribution overlap strategy.
HOPs: About 61% of HOPs users (74 of 121) look at relative position to judge effect size. Of HOPs users who rely on relative position, merely 3% (2 of 74) use a distance relative to variance strategy. However, looking at relative position is not mutually exclusive with looking at frequency of draws, which 45% of HOPs users (54 of 121) rely on as a visual feature. Among HOPs users who rely on frequencies, about 69% (37 of 54) employ a cumulative probability strategy, while about 35% (19 of 54) rely on the optimal strategy of counting the frequency of draws changing order. Roughly 40% of HOPs users (48 of 121) mention switching strategies compared to 31% with intervals (35 of 112), 23% with densities (23 of 99), and 21% with quantile dotplots (23 of 110). Among HOPs users who switch strategies, about 81% (39 of 48) rely on the mean as a cue. Strategy switching involves the mean for about 30% of HOPs users who rely on relative position (22 of 74) compared to 43% of HOPs users who rely on frequency (23 of 54). That most HOPs users rely on relative position, and that those who do rely on frequency are more likely to switch to or from relying on the mean, helps to explain poor performance with HOPs.
Densities: About 69% of densities users (68 of 99) rely on relative position as a visual cue. Of densities users who look at relative position, only about 13% (9 of 68) employ a distance relative to variance strategy. As one might expect, a substantial 36% densities users (36 of 99) rely on area as a cue, compared to 10% of quantile dotplots users (11 of 110). Among densities users who rely on area, about 53% (19 of 36) employ a cumulative probability strategy, while about 28% (10 of 36) employ a distribution overlap strategy. Interestingly, about 27% of densities users (27 of 99) mention relying on the spread of distributions as a cue, more than the 21% of users with intervals (24 of 112), 21% with HOPs (25 of 121), and 10% with quantile dotplots (11 of 110) who report relying on the same cue.
Quantile Dotplots: Roughly 59% of quantile dotplots users (65 of 110) describe looking at relative position to judge effect size, similar to 61% of users with HOPs (74 of 121) and less than the 69% of densities users (68 of 99) and 76% of intervals users (85 of 112) who report using the same cue. Merely 6% of quantile dotplots users who rely on relative position (4 of 65) employ a distance relative to variance strategy. 37% of quantile dotplots users (41 of 110) rely on frequency as a visual cue by counting dots. About 81% of quantile dotplots users who rely on frequency (33 of 41) employ a cumulative probability strategy.
Adding Means: A substantial 35% of users (155 of 442) describe relying on the mean as a cue for effect size. If we split users based on whether or not they start the task with means, about 31% of users (67 of 218) switch strategies when means are added to the charts halfway through the task, compared to 10% (23 of 224) who switch strategies when means are removed. This asymmetry in strategy switching suggests that means are “sticky” as a cue: Among the 15% of users (67 of 442) who start with and rely on means, about 66% (44 of 67) attempt to visually estimate means after means are removed from charts, almost twice as many as the 34% (23 of 67) who switch to relying on other cues. However, the impact of adding means on performance depends on what other strategies a user is switching between. Among the 20% of users (90 of 442) who rely on means and switch strategies, about 44% (40 of 90) just incorporate the mean into judgments of relative position without relying on other visual cues. Other groups of users switch between relying on means and less similar visual cues, with 34% (31 of 90) also mentioning frequency and 12% (11 of 90) mentioning area. That many users switch between relying on relative position and means, and that strategies are heterogeneous, helps to explain why the average impact of means on performance is small in our results.
6 General Discussion
Our results suggest that design guidelines for visualizing effect size should depend on the user’s task, the variance of distributions, and design choices about axis scales. To provide concrete design guidelines while acknowledging the inherent complexity of our results, we present highlevel takeaways for designers alongside relevant caveats.
Quantile dotplots support the most perceptually accurate distributional comparisons, at least among the visualization designs we tested. Caveat: Asking users to perform two tasks may have led users to rely on relatively simple strategies like cumulative probability more than strategies which require more mental energy like frequency of draws changing order. Conditions of high cognitive load seem to favor uncertainty visualizations like quantile dotplots over HOPs.
Densities without means seem to support the best decisionmaking across levels of variance. On a fixed axis scale, densities without means and quantile dotplots with means perform best at lower variance, while densities without means and intervals without means perform best at higher variance. No visualization design we tested eliminated bias in decisionmaking at higher variance. Caveats: The visualization design that leads to the least bias in decisionmaking depends on the variance of distributions relative to axis scale. Future work should investigate bias in decisionmaking over a gradient of variances shown on a common scale, including charts with heterogeneous variances, as this would enable more exhaustive design recommendations.
Adding means leads to small biases in magnitude estimation and decisionmaking from distributional comparisons, leading users to underestimate effect size and make less utilityoptimal decisions in most in most cases we tested. Caveats: Although the biasing effects of means are mostly negligible, our estimates of these biases are probably very conservative for two reasons: (1) added means were only highly salient in the HOPs condition; and (2) in the absence of added means, users already tend to rely on relative position, a cue which the mean merely reinforces. The effects of adding means on decision quality reverse at high versus low variance, so these biases may disappear for specific combinations of variance and axis scale.
Users rely on distance between distributions as a proxy for effect size, so designers should note when this will be misleading and encourage more optimal strategies. Our quantitative analysis shows that adding means induces small but reliable biases in magnitude estimation, consistent with distancebased heuristics. Our qualitative analysis of strategies verifies that the majority of users (357 of 442; 80.8%) rely on distance between distributions or mean difference to judge effect size. Caveats: Subtle design choices probably impact the tendency to rely on distance heuristics versus other strategies. For example, including a decision threshold annotation on our charts (Fig. 1) may have encouraged users to judge effect size as cumulative probability, rather than probability of superiority, contributing to underestimation bias.
6.1 Limitations
We only tested symmetrical distributions, and this may limit the generalizability of our inferences. Although we speculate that chart users may rely on central tendency regardless of the family of a distribution, reasoning with multimodal distributions in particular may involve different strategies not accounted for in the present study.
Because we rely on selfreported strategies in our qualitative analysis, our findings only reflect conscious strategies. This leaves out implicit or automatic information processing such as visual adaptation [Kale2019adaptation] and ensemble processing [szafir2016], except in rare cases where users report trying to “roughly average” predictions presented as HOPs.
Our choice to incentivize the decisionmaking task but not magnitude estimation may have contributed to the decoupling of performance on our two tasks. We cannot disentangle this possible explanation from evidence corroborating Kahneman and Tversky’s [Kahneman1979] distinction between perceived probabilities and decision weights (see Section 4.4).
We control the incentives for our decision task rather than manipulating them, in part because it is not feasible to test dramatically different incentives on Mechanical Turk. As such the risk preferences that we measure as PSEs are representative of users optimizing small monetary bonuses, and they may not capture how people respond to visualized data in crisis situations when lives, careers, or millions of dollars are at stake. However, by devising a task that is representative of a broad class of decision problems (see Section 3.2), we make our results as broadly applicable as possible. We speculate that the relative impacts of visualization designs on risk preferences should generalize to decision problems with similar utility functions.
6.2 Satisficing and Heterogeneity
The visual reasoning strategies that chart users rely on when making judgments from uncertainty visualizations may not be what visualization designers expect. We present evidence that, in the absence of training, users satisfice by using suboptimal heuristics to decode the signal from a chart. We also find that not all users rely on the same strategies and that many users switch between strategies. Satisficing and heterogeneity in heuristics make it difficult both to anticipate how people will read charts and to study the impact of design choices. Conventionally, visualization research has characterized visualization effectiveness by ranking visualization designs based on the performance of the average user (e.g., [Cleveland1984]). However, in cases like the present study where users are heterogeneous in their strategies, these averages may not account for the experience of very many users and are probably an oversimplification. Visualization researchers should be mindful of satisficing and heterogeneity in users’ visual reasoning strategies, attempt to model these strategies, and try to design ways of training users to employ more optimal strategies.
6.3 Toward Better Models of Visualization Effectiveness
Because some users seem to adopt suboptimal strategies or switch between strategies when presented with an uncertainty visualization, models of visualization effectiveness which codify design knowledge and drive automated visualization recommendation and authoring systems should represent these strategies. We envision a new class of behavioral models for visualization research which attempt to enumerate possible strategies, such as those we identify in our qualitative analysis, and learn how often users employ them to perform a specific task when presented with a particular visualization design. Previous work [Jardine2020] demonstrates a related approach by calculating expected responses based on a set of alternative perceptual proxies for visual comparison and comparing these expectations to users’ actual responses. Like the present study, this work describes the correspondence between expected patterns and user behavior. Instead, we propose incorporating functions representing predefined strategies into predictive models which estimate the proportion of users employing a given strategy.
In a pilot study, we attempted to build such a model: a Bayesian mixture model of alternative strategy functions. However, because multiple strategies predict similar patterns of responses, we were not able to fit the model due to problems with identifiability. This suggests that the kind of model we propose will only be feasible if we design experiments such that alternative strategies predict sufficiently different patterns of responses. The approach of looking at the agreement between proxies and human behavior [Jardine2020] suffers the same limitation, but there is no analogous mechanism to identifiability in Bayesian models to act as a failsafe against unwarranted inferences. Future work should continue pursuing this kind of strategyaware behavioral modeling.
We want to emphasize that the proposed modeling approach is not strictly quantitative, as the definition of strategy functions requires a descriptive understanding of users’ visual reasoning. As such this approach offers a way to formalize the insights of qualitative analysis and represent the gamut of possible user behaviors inside of visualization recommendation and authoring systems.
7 Conclusion
We contribute findings from a mixed design experiment on Mechanical Turk investigating how visualization design impacts judgments and decisions from effect size. Our results suggest that visualization designs which support the least biased estimation of effect size do not necessarily support the best decisionmaking. We discuss how a user’s sense of the signal in a chart may not necessarily be identical when they use the same information for different tasks. We also find that adding means to uncertainty visualizations induces small but reliable biases consistent with users relying on visual distance between distributions as a proxy for effect size. In a qualitative analysis of users’ visual reasoning strategies, we find that many users switch strategies and do not employ an optimal strategy when one exists. We discuss ways that canonical characterizations of graphical perception in terms of average performance gloss over possible heterogeneity in user behavior, and we propose opportunities to build strategyaware models of visualization effectiveness which could be used to formalize design knowledge in visualization recommendation and authoring systems beyond contextagnostic rankings of chart types.
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