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Inferring an unobservable population size from observable samples


Success in the physical and social worlds often requires knowledge of population size. However, many populations cannot be observed in their entirety, making direct assessment of their size difficult, if not impossible. Nevertheless, an unobservable population size can be inferred from observable samples. We measured people’s ability to make such inferences and their confidence in these inferences. Contrary to past work suggesting insensitivity to sample size and failures in statistical reasoning, inferences of populations size were accurate—but only when observable samples indicated a large underlying population. When observable samples indicated a small underlying population, inferences were systematically biased. This error, which cannot be attributed to a heuristics account, was compounded by a metacognitive failure: Confidence was highest when accuracy was at its worst. This dissociation between accuracy and confidence was confirmed by a manipulation that shifted the magnitude and variability of people’s inferences without impacting their confidence. Together, these results (a) highlight the mental acuity and limits of a fundamental human judgment and (b) demonstrate an inverse relationship between cognition and metacognition.

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  1. 1.

    Fifteen (s1 + s2o = 10 + 5 − 0 = 15) is the logical minimum number of objects in the population when s1 is 10, s2 is 5, and o is 0/5. Thus, any population size estimates that fall below this value are invalid. In the Supplemental Materials, analyses show that including these participants does not change the findings.

  2. 2.

    Statistical tests are abbreviated and reported in accordance with guidelines of the Publication Manual of the American Psychological Association (6th ed.). For additional effects, please refer to data and code posted on OSF (

  3. 3.

    Experiments 2 and 3 were conducted after summer 2018 when researchers observed a drop in data quality from Amazon Mechanical Turk (Bai, 2018). To guard against this concern, far more participants than necessary were recruited and stringent manipulation checks were included, resulting in the exclusion of many data points from the analysis. Although Experiments 1 and 4 do not contain these manipulation checks, this is not a concern because data quality was assessed in accordance with Bai (2018) and the results are robust and replicate.

  4. 4.

    Some readers may wonder about manipulating the presence versus absence of Nmax in addition to Nmin. This manipulation would not be feasible because computing the theoretical posterior distribution requires a bounded prior over N, meaning Nmax must be specified.


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This work was supported by a National Science Foundation Graduate Research Fellowship (DGE 1144152) to J.C.

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Correspondence to Jack Cao.

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Cao, J., Banaji, M.R. Inferring an unobservable population size from observable samples. Mem Cogn (2019).

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  • Numerical cognition
  • Population estimates
  • Accuracy
  • Confidence
  • Sampling processes