Insensitivity to Sample Size¶
Elicited judgments differentiate sample sizes too little relative to how a stated sampling model changes a sample statistic's probability or spread.
Core Idea¶
Insensitivity to sample size is a judgment error: estimates of a sample result's probability or spread change too little across sample-size conditions compared with what a stated sampling model requires. In comparable, sufficiently independent samples, a proportion or mean ordinarily varies less when the sample is larger. The concept concerns a person's judgment about that relation, not the variability of small samples itself. A study may compare one person's answers across sizes or compare answers from separately assigned groups; those designs support different claims about individuals.[ref-982cbcd68618][ref-9ad9423bd48d]
Scope of Application¶
The error can be tested when a reference population, a statistic, different sample sizes, a valid model comparison, and elicited human judgments are all specified. Kahneman and Tversky used a hospital birth-proportion question and a task asking separate groups for daily sample-mean height distributions at \(N=10,100,1000\). The later replication found little separation among elicited distributions but a more mixed pattern of equal-likelihood answers in extreme-outcome comparisons.[ref-982cbcd68618][ref-9ad9423bd48d]
A small county's extreme rate or a small school's extreme score shows why denominators matter, but neither plot by itself measures someone committing this cognitive error.[^ref-7ce80db46723] Changing populations, dependent observations, or different selection methods also require a new normative comparison before the bias label can be applied.
Clarity¶
Distinguish the statistical fact from the psychological finding. A model predicts how the chosen statistic's distribution changes with sample size. Researchers then compare that prediction with what people report. A weak or absent size adjustment in the reported judgments is the error. The original height experiment elicited probabilities across seven height bins, not a single observed answer about a mean above six feet.[^ref-982cbcd68618]
Manages Complexity¶
A proposed case becomes manageable through five questions: What population generated the samples? What were the sample sizes? Which statistic and event were judged? What does the appropriate model predict? What did a person or separately assigned groups actually answer? This structure works for both the hospital's binary birth proportion and the height task's continuous sample mean without treating their response formats as identical.[^ref-982cbcd68618]
Abstract Reasoning¶
For a boy-birth rate near one half, the binomial model gives the smaller hospital with about 15 daily births more days above 60% boys than the larger hospital with about 45. In the original more-than-60% question, “about the same” was the modal participant answer. That answer underweights the size difference for this task. It does not prove that the same respondent would always ignore sample size elsewhere.[^ref-982cbcd68618]
Knowledge Transfer¶
The diagnostic comparison transfers within judgment research from binary proportions to sample means: hold the generating process comparable, establish the modeled size effect, and inspect the elicited responses. The mathematics also helps interpret small-area rates and school-score extremes, but a psychological claim in those settings needs evidence about the observers' judgments.[ref-982cbcd68618][ref-7ce80db46723]
Example¶
Hospital births. The small hospital has about 15 births each day and the large one about 45; participants decide which records more days with over 60% boys. The reference population is about half boys, the two daily birth counts are the sample sizes, the proportion-above-threshold is the event, and the participant's “about the same” answer diverges from the binomial comparison.[^ref-982cbcd68618]
Mean height. Separate participant groups estimated the distribution of the daily mean height of 10, 100, or 1,000 men across seven bins. The population mean is the reference, the group assignment supplies the different \(N\) conditions, and the elicited distributions differentiated them too little. This is a between-group pattern, not proof that one participant compared all three sizes.[ref-982cbcd68618][ref-9ad9423bd48d]
Neighborhood in Abstraction Space¶
Insensitivity to Sample Size sits in a sparse region of the domain-specific corpus (97th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Descriptive Statistics & Correlation Measures (40 abstractions)
Nearest neighbors
- Genetic Load — 0.77
- Null distribution — 0.76
- Sampling error — 0.76
- Standardized Rate — 0.76
- Student's t-Test — 0.76
Computed from structural-signature embeddings · 2026-10-17
Not to Be Confused With¶
- Representativeness Heuristic is a broader proposed typicality-based explanation; the observed size error does not prove that mental cause in every person.[^ref-982cbcd68618]
- Sampling (Representativeness) concerns selection and population coverage, not the judgment about size-dependent variability.
- Bias in the live Prime requires a persistent process-level directional offset. A wrong answer in one sample-size task need not establish that broader structure.
- Small-sample variation is a property of a sampling process, not an observed human error.[^ref-7ce80db46723]
References¶
[^ref-982cbcd68618]: Daniel Kahneman and Amos Tversky (1972), “Subjective probability, A judgment of representativeness,” Cognitive Psychology 3: 430–454, doi:10.1016/0010-0285(72)90016-3. Complete original article consulted, especially printed pp. 437–442, Figure 1c, and the hospital problem on p. 443. The scan is hosted by Radboud University; the publisher identifies the work. The printed title uses a colon after “probability”; the comma transcription preserves the complete title for reference binding.
[^ref-9ad9423bd48d]: Lewend Mayiwar, Kai Hin Wan, Erik Løhre, and Gilad Feldman (2025), “Revisiting representativeness heuristic classic paradigms, Replication and extensions of nine experiments in Kahneman and Tversky (1972),” Quarterly Journal of Experimental Psychology 78: 707–730, doi:10.1177/17470218241255916. Full author manuscript consulted; PDF viewer pp. 43–47 give the Problem 6 distributions and Problems 7–9 comparisons, and pp. 57–58 discuss their mixed implications. The printed title uses a colon after “paradigms”; the comma transcription preserves the complete title for reference binding.
[^ref-7ce80db46723]: Howard Wainer and Harris L. Zwerling, “Evidence That Smaller Schools Do Not Improve Student Achievement,” Phi Delta Kappan 88(4) (2006): 300–303, doi:10.1177/003172170608800411. Complete publisher-archive article consulted; pp. 300–302 show small-denominator county and school extremes. County maps reprint Gelman and Nolan's material; this article does not measure a person's probability judgment, and its school findings differ by grade.