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Statistical Thinking: No One Left Behind

  • Björn MederEmail author
  • Gerd Gigerenzer
Part of the Advances in Mathematics Education book series (AME)

Abstract

Is the mind an “intuitive statistician”? Or are humans biased and error-prone when it comes to probabilistic thinking? While researchers in the 1950s and 1960s suggested that people reason approximately in accordance with the laws of probability theory, research conducted in the heuristics-and-biases program during the 1970s and 1980s concluded the opposite. To overcome this striking contradiction, psychologists more recently began to identify and characterize the circumstances under which people—both children and adults—are capable of sound probabilistic thinking. One important insight from this line of research is the power of representation formats. For instance, information presented by means of natural frequencies, numerical or pictorial, fosters the understanding of statistical information and improves probabilistic reasoning, whereas conditional probabilities tend to impede understanding. We review this research and show how its findings have been used to design effective tools and teaching methods for helping people—be it children or adults, laypeople or experts—to reason better with statistical information. For example, using natural frequencies to convey statistical information helps people to perform better in Bayesian reasoning tasks, such as understanding the implications of diagnostic test results, or assessing the potential benefits and harms of medical treatments. Teaching statistical thinking should be an integral part of comprehensive education, to provide children and adults with the risk literacy needed to make better decisions in a changing and uncertain world.

Keywords

Bayesian reasoning Statistical thinking Probabilistic reasoning Bayes rule Probability formats Natural frequencies Mammography problem Heuristics Risk communication 

Notes

Acknowledgements

BM was supported by Grant ME 3717/2 from the Deutsche Forschungsgemeinschaft (DFG) as part of the priority program “New Frameworks of Rationality” (SPP 1516). We thank Christel Fraser and Rona Unrau for editing the manuscript.

References

  1. Akl, E. A., Oxman, A. D., Herrin, J., Vist, G. E., Terrenato, I., Sperati, F., Costiniuk, C., Blank, D., & Schünemann, H. (2011). Using alternative statistical formats for presenting risks and risk reductions. Cochrane Database of Systematic Reviews, 3. Google Scholar
  2. Ancker, J. S., Senathirajah, Y., Kikafka, R., & Starren, J. B. (2006). Design features of graphs in health risk communication: a systematic review. Journal of the American Medical Informatics Association, 13, 608–618. CrossRefGoogle Scholar
  3. Arkes, H. R., & Gaissmaier, W. (2012). Psychological research and the prostate-cancer screening controversy. Psychological Science, 23, 547–553. CrossRefGoogle Scholar
  4. Bar-Hillel, M. (1980). The base-rate fallacy in probability judgments. Acta Psychologica, 44, 211–233. CrossRefGoogle Scholar
  5. Bastian, L. A., Nanda, K., Hasselblad, V., & Simel, D. L. (1998). Diagnostic efficiency of home pregnancy test kits: a meta-analysis. Archives of Family Medicine, 7, 465–469. CrossRefGoogle Scholar
  6. Bodemer, N., Meder, B., & Gigerenzer, G. (2012). Communicating relative risk changes with baseline risk: the effect of presentation format and numeracy. Manuscript under revision. Google Scholar
  7. Bramwell, R., West, H., & Salmon, P. (2006). Health professionals’ and service users’ interpretation of screening test results: experimental study. British Medical Journal, 333, 284–289. CrossRefGoogle Scholar
  8. Brase, G. L. (2009). Pictorial representations and numerical representations in Bayesian reasoning. Applied Cognitive Psychology, 23, 369–381. CrossRefGoogle Scholar
  9. Brainerd, C. J. (1981). Working memory and the developmental analysis of probability judgment. Psychological Review, 88, 463–502. CrossRefGoogle Scholar
  10. Casscells, W., Schoenberger, A., & Graboys, T. B. (1978). Interpretation by physicians of clinical laboratory results. The New England Journal of Medicine, 299, 999–1001. CrossRefGoogle Scholar
  11. Christensen-Szalanski, J. J. J., & Beach, L. R. (1982). Experience and the base-rate fallacy. Organizational Behavior and Human Performance, 29, 270–278. CrossRefGoogle Scholar
  12. Christensen-Szalanski, J. J. J., & Bushyhead, J. B. (1981). Physicians’ use of probabilistic information in a real clinical setting. Journal of Experimental Psychology. Human Perception and Performance, 7, 928–935. CrossRefGoogle Scholar
  13. Cole, W. G., & Davidson, J. E. (1989). Graphic representation can lead to fast and accurate Bayesian reasoning. Proceedings of the Annual Symposium on Computer Application in Medical Care, 8, 227–231. Google Scholar
  14. 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. CrossRefGoogle Scholar
  15. Covey, J. (2007). A meta-analysis of the effects of presenting treatment benefits in different formats. Medical Decision Making, 27, 638–654. CrossRefGoogle Scholar
  16. Daston, L. J. (1988). Classical probability in the enlightenment. Princeton: Princeton University Press. Google Scholar
  17. Djulbegovic, M., Beyth, R. J., Neuberger, M. M., Stoffs, T. L., Vieweg, J., Djulbegovic, B., & Dahm, P. (2010). Screening for prostate cancer: systematic review and meta-analysis of randomized controlled trials. British Medical Journal, 341, c4543. CrossRefGoogle Scholar
  18. Eddy, D. M. (1982). Probabilistic reasoning in clinical medicine: problems and opportunities. In D. Kahneman, P. Slovic, & A. Tversky (Eds.), Judgment under uncertainty: heuristics and biases (pp. 249–267). Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  19. Edwards, W. (1968). Conservatism in human information processing. In B. Kleinmuntz (Ed.), Formal representation of human judgment (pp. 17–52). New York: Wiley. Google Scholar
  20. Edwards, W., & von Winterfeldt, D. (1986). On cognitive illusions and their implications. In H. R. Arkes & K. R. Hammond (Eds.), Judgment and decision making. Cambridge: Cambridge University Press. Google Scholar
  21. Edwards, A., Elwyn, G., Covey, J., Matthews, E., & Pill, R. (2001). Presenting risk information—a review of the effects of “framing” and other manipulations on patient outcomes. Journal of Health Communication, 6, 61–82. CrossRefGoogle Scholar
  22. Edwards, A., Elwyn, G., & Mulley, A. (2002). Explaining risks: turning numerical data into meaningful pictures. British Medical Journal, 324, 827–830. CrossRefGoogle Scholar
  23. Evans, J., Handley, S. J., Perham, N., Over, D. E., & Thompson, V. A. (2000). Frequency versus probability formats in statistical word problems. Cognition, 77, 197–213. CrossRefGoogle Scholar
  24. Fagerlin, A., Wang, C., & Ubel, P. A. (2005). Reducing the influence of anecdotal reasoning on people’s health care decisions: is a picture worth a thousand statistics? Medical Decision Making, 25, 398–405. CrossRefGoogle Scholar
  25. Fischbein, E., Pampu, I., & Minzat, I. (1970). Comparison of ratios and the chance concept in children. Child Development, 41, 377–389. CrossRefGoogle Scholar
  26. Galesic, M., García-Retamero, R., & Gigerenzer, G. (2009). Using icon arrays to communicate medical risks: overcoming low numeracy. Health Psychology, 28, 210–216. CrossRefGoogle Scholar
  27. Gigerenzer, G. (1996). On narrow norms and vague heuristics: a reply to Kahneman and Tversky. Psychological Review, 103, 592–596. CrossRefGoogle Scholar
  28. Gigerenzer, G. (2002). Calculated risks: how to know when numbers deceive you. New York: Simon & Schuster. Google Scholar
  29. Gigerenzer, G. (2007). Gut feelings: the intelligence of the unconscious. New York: Viking Press. Google Scholar
  30. Gigerenzer, G., & Gaissmaier, W. (2011). Heuristic decision making. Annual Review of Psychology, 62, 451–482. CrossRefGoogle Scholar
  31. Gigerenzer, G. & Gray, J. A. M. (Eds.) (2011). Better doctors, better patients, better decisions: envisioning health care 2020. Cambridge: MIT Press. Google Scholar
  32. Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: frequency formats. Psychological Review, 102, 684–704. CrossRefGoogle Scholar
  33. Gigerenzer, G., & Hoffrage, U. (1999). Overcoming difficulties in Bayesian reasoning: a reply to Lewis and Keren (1999) and Meilers and McGraw (1999). Psychological Review, 106, 425–430. CrossRefGoogle Scholar
  34. Gigerenzer, G., Hertwig, R., & Pachur, T. (Eds.) (2011). Heuristics: the foundations of adaptive behavior. New York: Oxford University Press. Google Scholar
  35. Gigerenzer, G., Mata, J., & Frank, R. (2009). Public knowledge of benefits of breast and prostate cancer screening in Europe. Journal of the National Cancer Institute, 101, 1216–1220. CrossRefGoogle Scholar
  36. Gigerenzer, G., Todd, P. M., & the ABC Research Group (Eds.) (1999). Simple heuristics that make us smart. New York: Oxford University Press. Google Scholar
  37. Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2007). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8, 53–96. Google Scholar
  38. Girotto, V., & Gonzalez, M. (2001). Solving probabilistic and statistical problems: a matter of information structure and question form. Cognition, 78, 247–276. CrossRefGoogle Scholar
  39. Girotto, V., & Gonzalez, M. (2008). Children’s understanding of posterior probability. Cognition, 106, 325–344. CrossRefGoogle Scholar
  40. Hammerton, M. (1973). A case of radical probability estimation. Journal of Experimental Psychology, 101, 252–254. CrossRefGoogle Scholar
  41. Hertwig, R., Hoffrage, U., & the ABC Research Group (2012). Simple heuristics in a social world. New York: Oxford University Press. CrossRefGoogle Scholar
  42. Hoffrage, U., & Gigerenzer, G. (1998). Using natural frequencies to improve diagnostic inferences. Academic Medicine, 73, 538–540. CrossRefGoogle Scholar
  43. Hoffrage, U., Gigerenzer, G., Krauss, S., & Martignon, L. (2002). Representation facilitates reasoning: what natural frequencies are and what they are not. Cognition, 84, 343–352. CrossRefGoogle Scholar
  44. Hoffrage, U., Lindsey, S., Hertwig, R., & Gigerenzer, G. (2000). Communicating statistical information. Science, 290, 2261–2262. CrossRefGoogle Scholar
  45. 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. CrossRefGoogle Scholar
  46. Kahneman, D., & Tversky, A. (1972). Subjective probability: a judgment of representativeness. Cognitive Psychology, 3, 430–454. CrossRefGoogle Scholar
  47. Kahneman, D., & Tversky, A. (1973). On the psychology of prediction. Psychological Review, 80, 237–251. CrossRefGoogle Scholar
  48. Kahneman, D., & Tversky, A. (1996). On the reality of cognitive illusions. Psychological Review, 103, 582–591. CrossRefGoogle Scholar
  49. Kahneman, D., Slovic, P., & Tversky, A. (Eds.) (1982). Judgment under uncertainty: heuristics and biases. Cambridge: Cambridge University Press. Google Scholar
  50. Kaye, D. H., & Koehler, J. J. (1991). Can jurors understand probabilistic evidence? Journal of the Royal Statistical Society. Series A, 154, 75–81. CrossRefGoogle Scholar
  51. Kleiter, G. (1994). Natural sampling: rationality without base rates. In G. H. Fischer & D. Lang (Eds.), Contributions to mathematical psychology, psychometrics, and methodology (pp. 375–388). New York: Springer. CrossRefGoogle Scholar
  52. Knight, F. H. (1921/2006). Risk, uncertainty and profit. Boston: Houghton Mifflin. Google Scholar
  53. Koehler, J. J. (1996a). The base rate fallacy reconsidered: descriptive, normative and methodological challenges. Behavioral and Brain Sciences, 19, 1–54. CrossRefGoogle Scholar
  54. Koehler, J. J. (1996b). On conveying the probative value of DNA evidence: frequencies, likelihood ratios and error rates. University of Colorado Law Review, 67, 859–886. Google Scholar
  55. Koehler, J. J. (2006). Train our jurors. In G. Gigerenzer & C. Engel (Eds.), Heuristics and the law (pp. 303–326). Cambridge: MIT Press. Google Scholar
  56. Kurz-Milcke, E., Gigerenzer, G., & Martignon, L. (2008). Transparency in risk communication: graphical and analog tools. Annals of the New York Academy of Sciences, 1128, 18–28. CrossRefGoogle Scholar
  57. Kurz-Milcke, E., & Martignon, L. (2006). Lebendige Urnen und ereignisreiche Bäume: Überlegungen und Versuche zu einer Didaktik der Stochastik in der Grundschule [Lively urns and eventful trees: developing a didactics of stochastics in elementary school]. Anregungen zum Stochastikunterricht, 3, 182–203. Google Scholar
  58. Kurzenhäuser, S., & Hoffrage, U. (2002). Teaching Bayesian reasoning: an evaluation of a classroom tutorial for medical students. Medical Teacher, 24, 516–521. CrossRefGoogle Scholar
  59. Labarge, A. S., McCaffrey, R. J., & Brown, T. A. (2003). Neuropsychologists’ abilities to determine the predictive value of diagnostic tests. Archives of Clinical Neuropsychology, 18, 165–175. Google Scholar
  60. Lewis, C., & Keren, G. (1999). On the difficulties underlying Bayesian reasoning: comment on Gigerenzer and Hoffrage. Psychological Review, 106, 411–416. CrossRefGoogle Scholar
  61. Lindsey, S., Hertwig, R., & Gigerenzer, G. (2003). Communicating statistical DNA evidence. Jurimetrics, 43, 147–163. Google Scholar
  62. Lipkus, I. M., & Hollands, J. G. (1999). The visual communication of risk. Journal of the National Cancer Institute Monographs, 25, 149–163. CrossRefGoogle Scholar
  63. Lipkus, I. M., Samsa, G., & Rimer, B. K. (2001). General performance on a numeracy scale among highly educated samples. Medical Decision Making, 21, 37–44. CrossRefGoogle Scholar
  64. Martigon, L. (2013). Fostering children’s probabilistic reasoning and their risk evaluation. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: presenting plural perspectives. Dordrecht: Springer. This volume. Google Scholar
  65. Martignon, L., & Krauss, S. (2009). Hands on activities with fourth-graders: a tool box of heuristics for decision making and reckoning with risk. International Electronic Journal for Mathematics Education, 4, 117–148. Google Scholar
  66. Martignon, L., & Krauss, S. (2007). Gezinkte Würfel, Magnetplättchen und Tinker Cubes für eine Stochastik zum Anfassen in der Grundschule. Stochastik in der Schule, 27, 21–30. Google Scholar
  67. Meder, B., & Nelson, J. D. (2012). Information search with situation-specific reward functions. Judgment and Decision Making, 7, 119–148. Google Scholar
  68. Meder, B., Mayrhofer, R., & Waldmann, M. R. (2009). A rational model of elemental diagnostic inference. In N. A. Taatgen & H. van Rijn (Eds.), Proceedings of the 31st annual conference of the Cognitive Science Society (pp. 2176–2181). Austin: Cognitive Science Society. Google Scholar
  69. Moyer, V. A., on behalf of the U.S. Preventive Services Task Force (2012). Screening for prostate cancer: U.S. Preventive Services Task Force recommendation statement. Annals of Internal Medicine, 157, 120–134. CrossRefGoogle Scholar
  70. Multmeier, J. (2012). Representations facilitate Bayesian reasoning: computational facilitation and ecological design revisited. Unpublished doctoral dissertation, Free University, Berlin. Google Scholar
  71. Nelson, J. D., McKenzie, C. R. M., Cottrell, G. W., & Sejnowski, T. J. (2010). Experience matters: information acquisition optimizes probability gain. Psychological Science, 21, 960–969. CrossRefGoogle Scholar
  72. Peters, E. (2008). Numeracy and the perception and communication of risk. Annals of the New York Academy of Sciences, 1128, 234–267. CrossRefGoogle Scholar
  73. Peterson, C. R., & Beach, L. R. (1967). Man as intuitive statistician. Psychological Bulletin, 68, 29–46. CrossRefGoogle Scholar
  74. Peterson, C. R., & Miller, A. J. (1965). Sensitivity of subjective probability revision. Journal of Experimental Psychology, 70, 117–121. CrossRefGoogle Scholar
  75. Peterson, C. R., Schneider, R. J., & Miller, A. J. (1965). Sample size and the revision of subjective probabilities. Journal of Experimental Psychology, 69, 522–527. CrossRefGoogle Scholar
  76. Phillips, L. D., & Edwards, W. (1966). Conservatism in a simple probability inference task. Journal of Experimental Psychology, 72, 346–354. CrossRefGoogle Scholar
  77. Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children. New York: Norton. (Original work published 1951). Google Scholar
  78. Rouanet, H. (1961). Études de décisions expérimentales et calcul de probabilités [Studies of experimental decision making and the probability calculus]. In Colloques Internationaux du Centre National de la Recherche Scientifique (pp. 33–43). Paris: Éditions du Centre National de la Recherche Scientifique. Google Scholar
  79. Schwartz, L., Woloshin, S., Black, W. C., & Welch, H. G. (1997). The role of numeracy in understanding the benefit of screening mammography. Annals of Internal Medicine, 127, 966–972. CrossRefGoogle Scholar
  80. Sedlmeier, P. (1999). Improving statistical reasoning: theoretical models and practical implications. Mahwah: Erlbaum. Google Scholar
  81. Sedlmeier, P., & Gigerenzer, G. (2001). Teaching Bayesian reasoning in less than two hours. Journal of Experimental Psychology. General, 130, 380–400. CrossRefGoogle Scholar
  82. Simon, H. (1969). The science of the artificial. Cambridge: MIT Press. Google Scholar
  83. Steurer, J., Fischer, J. E., Bachmann, L. M., Koller, M., & ter Riet, G. (2002). Communicating accuracy of tests to general practitioners: a controlled study. British Medical Journal, 324, 824–826. CrossRefGoogle Scholar
  84. Thompson, W. C., & Schumann, E. L. (1987). Interpretation of statistical evidence in criminal trials. Law and Human Behavior, 11, 167–187. CrossRefGoogle Scholar
  85. Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: heuristics and biases. Science, 185, 1124–1131. CrossRefGoogle Scholar
  86. Wegwarth, O., Schwartz, L. M., Woloshin, S., Gaissmaier, W., & Gigerenzer, G. (2012). Do physicians understand cancer screening statistics? A national survey of primary care physicians. Annals of Internal Medicine, 156, 340–349. CrossRefGoogle Scholar
  87. Wegwarth, O., Gaissmaier, W., & Gigerenzer, G. (2011). Deceiving numbers: survival rates and their impact on doctors’ risk communication. Medical Decision Making, 31, 386–394. CrossRefGoogle Scholar
  88. Weir, B. S. (2007). The rarity of DNA profiles. Annals of Applied Statistics, 1, 358–370. CrossRefGoogle Scholar
  89. Yost, P. A., Siegel, A. E., & Andrews, J. M. (1962). Non-verbal probability judgments by young children. Child Development, 33, 769–780. Google Scholar
  90. Zhu, L., & Gigerenzer, G. (2006). Children can solve Bayesian problems: the role of representation in mental computation. Cognition, 98, 287–308. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Center for Adaptive Behavior and Cognition (ABC)Max Planck Institute for Human DevelopmentBerlinGermany
  2. 2.Harding Center for Risk LiteracyMax Planck Institute for Human DevelopmentBerlinGermany

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