The ubiquity of the Simpson’s Paradox
Open accessAbstract
The Simpson’s Paradox is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, it has a deep statistical significance. In this paper, we discuss basic examples in arithmetic, geometry, linear algebra, statistics, game theory, gender bias in university admission and election polls, where we describe the appearance or absence of the Simpson’s Paradox . In the final part, we present our results concerning the occurrence of the Simpson’s Paradox in Quantum Mechanics with focus on the Quantum Harmonic Oscillator and the Nonlinear Schrödinger Equation. We discuss how likely it is to incur in the Simpson’s Paradox and give some concrete numerical examples. We conclude with some final comments and possible future directions.
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The discrepancy between the general trend and colour morph-specific results illustrates Simpson's paradox, where aggregated data masks underlying subgroup patterns.
“The discrepancy between the general trend and the colour morph-specific results illustrates Simpson's paradox, where aggregated data masks underlying subgroup patterns [64].”
From Tiger colour morphs in zoos raise conservation and welfare concerns (2026) · cites this paper as [64]
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The interpretation of interaction in contingency tables
Simpson, E. H. (1951) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1951.tb00088.x · Cited by 2,301 · cited 2× · see citing works
“In his paper (Simpson 1951), Simpson considered a 2 × 2 × 2 contingency table with attributes A, B, and C and illustrated the paradox using a heuristic example of clinic patients, divided into a Treatment Group and in a No-Treatment group.”
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Research works printed but not linked: 15
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- Norris, F: Can Every Group Be Worse Than Average? Yes (2013). https://economix.blogs.nytimes.com/2013/05/01/can- every-group-be-worse-than-average-yes/. Accessed 1 May 2013 Paris, MGA: Two quantum Simpson's Paradoxes. J. Phys. A. 45, 132001 (2012)
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- Selvitella, A: The Simpson's Paradox in quantum mechanics. J. Math. Phys. 58(3), 37 (2017). 032101
- Shi, Y: Quantum Simpson's Paradox and High Order Bell-Tsileron Inequalities (2012). preprint available at arxiv.org/pdf/1203.2675 · arXiv
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Cited by
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Tiger colour morphs in zoos raise conservation and welfare concerns
Erzinçlioğlu, T. S., Piaz, M. T., Chinpilas, T., Hunnestad, A. V., Godfrey, L., & Fourage, A. (2026) · Discover Animals · doi:10.1007/s44338-026-00163-5
- Authors
- Selvitella, A.
- Affiliations
- McMaster University (CA)
- Corresponding author
- Selvitella, Alessandro
- Venue
- Journal of Statistical Distributions and Applications
- Published
- 2017-03-28
- Volume
- 4
- Issue
- 1
- ISSN
- 2195-5832 (ISSN-L)
- OpenAlex
- W2604002910
- Cited by
- 16
- References
- 1
- Citations by year
- 2026: 3 · 2025: 4 · 2024: 2 · 2023: 1 · 2022: 3 · 2021: 2 · 2020: 1
- FWCI
- 0.19
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Locations
- Journal of Statistical Distributions and Applications (publishedVersion, cc-by)
- RePEc: Research Papers in Economics (submittedVersion)
- Funders
- Canadian Statistical Science Institute (CANSSI)
- Central Michigan University
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Statistics Education and Methodologies
- Keywords
- Simpson's paradox, election polls, quantum harmonic oscillator, nonlinear Schrödinger equation
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