Skip to main content
SpringerLink
Log in

From reading numbers to seeing ratios: a benefit of icons for risk comprehension

  • Original Article
  • Published:
Psychological Research Aims and scope Submit manuscript

Abstract

Promoting a better understanding of statistical data is becoming increasingly important for improving risk comprehension and decision-making. In this regard, previous studies on Bayesian problem solving have shown that iconic representations help infer frequencies in sets and subsets. Nevertheless, the mechanisms by which icons enhance performance remain unclear. Here, we tested the hypothesis that the benefit offered by icon arrays lies in a better alignment between presented and requested relationships, which should facilitate the comprehension of the requested ratio beyond the represented quantities. To this end, we analyzed individual risk estimates based on data presented either in standard verbal presentations (percentages and natural frequency formats) or as icon arrays. Compared to the other formats, icons led to estimates that were more accurate, and importantly, promoted the use of equivalent expressions for the requested probability. Furthermore, whereas the accuracy of the estimates based on verbal formats depended on their alignment with the text, all the estimates based on icons were equally accurate. Therefore, these results support the proposal that icons enhance the comprehension of the ratio and its mapping onto the requested probability and point to relational misalignment as potential interference for text-based Bayesian reasoning. The present findings also argue against an intrinsic difficulty with understanding single-event probabilities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. We are aware that the response “30%” also coincides with the literal representation of the false alarm rate for the PE group. Nevertheless, given the extremely low percent accuracy for this group (0% according to the strict criterion; 5% considering the rounded responses), the reported effects are independent of the interpretation of this ambiguity.

  2. We considered the use of 100 or 10 (only observed in IA responses) in the denominator as an attempt to normalize the response. Nevertheless, as discussed below, most of the NF responses using the 100 might not be “true” normalization attempts, but rather a consequence of misleading associations.

  3. In Experiment 2, most of the Bayesian (misaligned) ratios used 10 or 100 as denominator (66 and 80% for IA and NF formats, respectively). Therefore, although simplifications were fewer than in Experiment 1, the overall percentage of correct responses to IA problems expressed as equivalent ratios was indeed higher (49 vs 67% for Experiments 1 vs 2).

  4. In the pilot experiment, different groups received the same IA and NF problems of Experiment 1 (see “Appendix”), but were asked for frequencies (e.g., “of the women who test positive, how many have breast cancer?”), in the IA (N = 20) and NF (N = 22) formats. For the IA group, the mean number of correct responses was similar as in the present experiments (1.42). For the NF group, it was higher than in present experiments (0.82), but still lower than for the IA group (p = .02).

  5. Although a default frequency-based representation is defended by these authors, it is also claimed that a frequentist mechanism might produce subjective confidence for single event probabilities: “even though it might initially output a frequency, and perhaps even store the information as such, other mechanisms may make that frequentist output consciously accessible in the form of a subjective degree of confidence” (Cosmides & Tooby, 1996, p. 66, note 19).

  6. For NF problems, the mean number of correct posterior probability estimates, among participants who correctly estimated both probabilities of the datum, was 0.5. For IA problems, this mean was 1.6.

References

  • Barbey, A. K., & Sloman, S. A. (2007). Base-rate respect: From ecological rationality to dual processes. Behavioral and Brain Sciences, 30, 241–297.

    Article  PubMed  Google Scholar 

  • Brase, G. L. (2009). Pictorial representations and numerical representations in Bayesian reasoning. Applied Cognitive Psychology, 23(3), 369–381.

    Article  Google Scholar 

  • Brase, G. L. (2014). The power of representation and interpretation: Doubling statistical reasoning performance with icons and frequentist interpretations of ambiguous numbers. Journal of Cognitive Psychology, 26, 81–97.

    Article  Google Scholar 

  • Brase, G. L., & Hill, W. T. (2017). Adding up to good Bayesian reasoning: Problem format manipulations and individual skill differences. Journal of Experimental Psychology: General, 146(4), 577.

    Article  Google Scholar 

  • Chapman, G. B., & Liu, J. (2009). Numeracy, frequency, and Bayesian reasoning. Judgment and Decision Making, 4(1), 34.

    Google Scholar 

  • Cosmides, L., & Tooby, J. (1996). Are humans good intuitive statisticians after all? Rethinking some conclusions of the literature on judgment under uncertainty. Cognition, 58, 1–73.

    Article  Google Scholar 

  • Evans, J. S. B. T., Handley, S. J., Perham, N., Over, D. E., & Thompson, V. A. (2000). Frequency versus probability formats in statistical word problems. Cognition, 77(3), 197–213.

    Article  PubMed  Google Scholar 

  • Galesic, M., Garcia-Retamero, R., & Gigerenzer, G. (2009). Using icon arrays to communicate medical risks: Overcoming low numeracy. Health Psychology, 28(2), 210.

    Article  PubMed  Google Scholar 

  • Garcia-Retamero, R., & Hoffrage, U. (2013). Visual representation of statistical information improves diagnostic inferences in doctors and their patients. Social Science and Medicine, 83, 27–33.

    Article  PubMed  Google Scholar 

  • Garcia-Retamero, R., Cokely, E. T., & Hoffrage, U. (2015). Visual aids improve diagnostic inferences and metacognitive judgment calibration. Frontiers in Psychology, 6, 932.

    Article  PubMed  PubMed Central  Google Scholar 

  • Gigerenzer, G. (1991). How to make cognitive illusions disappear: Beyond “heuristics and biases”. European Review of Social Psychology, 2(1), 83–115.

    Article  Google Scholar 

  • Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102, 684–704.

    Article  Google Scholar 

  • Gillies, D. (2000). Varieties of propensity. The British Journal for the Philosophy of Science, 51(4), 807–835.

    Article  Google Scholar 

  • Girotto, V., & Gonzalez, M. (2001). Solving probabilistic and statistical problems: A matter of information structure and question form. Cognition, 78, 247–276.

    Article  PubMed  Google Scholar 

  • Hafenbrädl, S., & Hoffrage, U. (2015). Toward an ecological analysis of Bayesian inferences: How task characteristics influence responses. Frontiers in Psychology, 6, 939.

    Article  PubMed  PubMed Central  Google Scholar 

  • Hoffrage, U., Krauss, S., Martignon, L., & Gigerenzer, G. (2015). Natural frequencies improve Bayesian reasoning in simple and complex inference tasks. Frontiers in Psychology, 6, 1473.

    PubMed  PubMed Central  Google Scholar 

  • Holyoak, K. J., & Koh, K. (1987). Surface and structural similarity in analogical transfer. Memory and Cognition, 15(4), 332–340.

    Article  PubMed  Google Scholar 

  • Johnson, E. D., & Tubau, E. (2015). Comprehension and computation in Bayesian problem solving. Frontiers in Psychology, 6, 938.

    PubMed  PubMed Central  Google Scholar 

  • Johnson, E. D., & Tubau, E. (2017). Structural mapping in statistical word problems: A relational reasoning approach to Bayesian inference. Psychonomic Bulletin and Review, 24(3), 964–971.

    Article  PubMed  Google Scholar 

  • Johnson-Laird, P. N., Legrenzi, P., Girotto, V., Legrenzi, M. S., & Caverni, J. P. (1999). Naive probability: A mental model theory of extensional reasoning. Psychological Review, 106, 62–88.

    Article  PubMed  Google Scholar 

  • Khan, A., Breslav, S., Glueck, M., & Hornbæk, K. (2015). Benefits of visualization in the mammography problem. International Journal of Human-Computer Studies, 83, 94–113.

    Article  Google Scholar 

  • Lesage, E., Navarrete, G., & De Neys, W. (2013). Evolutionary modules and Bayesian facilitation: The role of general cognitive resources. Thinking and Reasoning, 19(1), 27–53.

    Article  Google Scholar 

  • Mandel, D. R., & Navarrete, G. (2015). Editorial: Improving Bayesian reasoning: What works and why? Frontiers in Psychology, 6, 1872.

    PubMed  PubMed Central  Google Scholar 

  • Markman, A. B., & Gentner, D. (1993). Structural alignment during similarity comparisons. Cognitive Psychology, 25(4), 431–467.

    Article  Google Scholar 

  • McDowell, M., & Jacobs, P. (2017). Meta-analysis of the effect of natural frequencies on Bayesian reasoning. Psychological Bulletin, 143(12), 1273.

    Article  PubMed  Google Scholar 

  • Navarrete, G., Correia, R., Sirota, M., Juanchich, M., & Huepe, D. (2015). Doctor, what does my positive test mean? From Bayesian textbook tasks to personalized risk communication. Frontiers in Psychology, 6, 1–6.

    Google Scholar 

  • Ottley, A., Peck, E. M., Harrison, L. T., Afergan, D., Ziemkiewicz, C., Taylor, H. A., et al. (2016). Improving Bayesian reasoning: The effects of phrasing, visualization, and spatial ability. IEEE Transactions on Visualization and Computer Graphics, 22(1), 529–538.

    Article  PubMed  Google Scholar 

  • Pennycook, G., & Thompson, V. A. (2012). Reasoning with base rates is routine, relatively effortless, and context dependent. Psychonomic Bulletin and Review, 19(3), 528–534.

    Article  PubMed  Google Scholar 

  • Pighin, S., Gonzalez, M., Savadori, L., & Girotto, V. (2016). Natural frequencies do not foster public understanding of medical test results. Medical Decision Making, 36(6), 686–691.

    Article  PubMed  Google Scholar 

  • Pighin, S., Tentori, K., & Girotto, V. (2017). Another chance for good reasoning. Psychonomic Bulletin & Review, 24(6), 1995–2002.

    Article  Google Scholar 

  • Reyna, V. F. (2004). How people make decisions that involve risk: A dual-processes approach. Current Directions in Psychological Science, 13(2), 60–66.

    Article  Google Scholar 

  • Sedlmeier, P., & Gigerenzer, G. (2001). Teaching Bayesian reasoning in less than two hours. Journal of Experimental Psychology: General, 130(3), 380.

    Article  Google Scholar 

  • Sirota, M., Juanchich, M., & Hagmayer, Y. (2014a). Ecological rationality or nested sets? Individual differences in cognitive processing predict Bayesian reasoning. Psychonomic Bulletin and Review, 21(1), 198–204.

    Article  PubMed  Google Scholar 

  • Sirota, M., Kostovičová, L., & Juanchich, M. (2014b). The effect of iconicity of visual displays on statistical reasoning: Evidence in favor of the null hypothesis. Psychonomic Bulletin and Review, 21(4), 961–968.

    Article  PubMed  Google Scholar 

  • Sloman, S. A., Over, D., Slovak, L., & Stibel, J. M. (2003). Frequency illusions and other fallacies. Organizational Behavior and Human Decision Processes, 91(2), 296–309.

    Article  Google Scholar 

Download references

Funding

ET was funded by Secretaría de Estado de Investigación, Desarrollo e Innovación of Spanish Ministerio de Economía y Competividad (PSI2013-41568-P). AC and ET were also funded by the Catalan Government (2014-SGR-79).

Author information

Authors and Affiliations

  1. Departament de Cognició, Desenvolupament i Psicologia de l’Educació, Facultat de Psicologia, Universitat de Barcelona, Passeig de la Vall d’Hebron, 171, 08035, Barcelona, Catalonia, Spain

    Elisabet Tubau, Javier Rodríguez-Ferreiro, Itxaso Barberia & Àngels Colomé

  2. Institute of Neurosciences (UBNeuroscience), Universitat de Barcelona, Passeig de la Vall d’Hebron, 171, 08035, Barcelona, Catalonia, Spain

    Elisabet Tubau, Javier Rodríguez-Ferreiro & Àngels Colomé

Authors
  1. Elisabet Tubau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Javier Rodríguez-Ferreiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Itxaso Barberia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Àngels Colomé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elisabet Tubau.

Ethics declarations

Conflict of interest

All authors declare that they have no conflict of interest.

Ethical approval

Experimental procedure was in accordance with the ethical standards of the University of Barcelona’s Bioethics Commission, and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Appendix

Appendix

Problems presented in each format in each experiment (original in Spanish).

Mammogram problem (Experiment 1)

Icon array (IA) The following Figure1 shows the prevalence of breast cancer among women over 50 who participate in routine screening, as well as the results of the mammogram for women who have and do not have breast cancer

Natural frequencies (NF) Among 100 women over 50 who participate in routine screening, four have breast cancer. Three of the four women with breast cancer and 12 of the 96 women without breast cancer receive a positive mammogram

Percentages (PE) Among the women over 50 who participate in routine screening, 4% have breast cancer. 75% of the women with breast cancer and 12% of the women without breast cancer receive a positive mammogram

Posterior probability question Imagine a friend at that age receives a positive mammogram. Based on the above information, what is the probability of her having breast cancer? (NF and IA versions prompted a “X of Y” response; PE version prompted a “%” response)

Hypertension problem (Experiment 1)

IA The following Figure1 shows the prevalence of hypertension among women over 40 who participate in routine screening, as well as the type of diet followed by women who have and do not have hypertension

NF Among 100 women over 40 who participate in routine screening, 20 have hypertension. 12 of the 20 women with hypertension and 24 of the 80 women without hypertension follow a sodium-rich diet

PE Among the women over 40 who participate in routine screening, 20% have hypertension. 60% of the women with hypertension and 30% of the women without hypertension follow a sodium-rich diet

Posterior probability question Imagine a friend at that age follows a sodium-rich diet. Based on the above information, what is the probability of her having hypertension? (NF and IA versions prompted a “X of Y” response; PE version prompted a “%” response)

Mammogram problem (Experiment 2)

IA The following Figure1 shows the prevalence of breast cancer among women over 50 who participate in routine screening, as well as the results of the mammogram for women who have and do not have breast cancer

NF Among 100 women over 50 who participate in routine screening, four have breast cancer and 96 have not breast cancer. Three of the women with breast cancer and 12 of the women without breast cancer receive a positive mammogram

Aligned question2(probability of the datum) Based on the above data, what is the probability of a woman at that age receiving a positive mammogram? (X of Y)

Misaligned question (posterior probability) (the same as in Experiment 1)

Hypertension problem (Experiment 2)

IA The following Figure1 shows the prevalence of hypertension among women over 40 who participate in routine screening, as well as the type of diet followed by women who have and do not have hypertension

NF Among 100 women over 40 who participate in routine screening, tem have hypertension and 90 have not hypertension. Eight of the women with hypertension and 16 of the women without hypertension follow a sodium-rich diet

Aligned question2(probability of the datum) Based on the above data, what is the probability of a woman at that age following a rich-sodium diet? (X of Y)

Misaligned question (posterior probability) (the same as in Experiment 1)

  1. 1Corresponding icon array was presented (see an example in Fig. 2)
  2. 2Alignment was manipulated regarding the relational match between presented and requested relations in verbal formats (see further details in “Experiment 2”)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tubau, E., Rodríguez-Ferreiro, J., Barberia, I. et al. From reading numbers to seeing ratios: a benefit of icons for risk comprehension. Psychological Research 83, 1808–1816 (2019). https://doi.org/10.1007/s00426-018-1041-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00426-018-1041-4

Search

Navigation