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Simpson’s Paradox (Stanford Encyclopedia of Philosophy)
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Simpson’s Paradox<br>First published Wed Mar 24, 2021; substantive revision Sat Jun 6, 2026
Simpson’s Paradox is a statistical phenomenon where an<br>association between two variables in a population emerges, disappears<br>or reverses when the population is divided into subpopulations. For<br>instance, two variables may be positively associated in a population,<br>but be independent or even negatively associated in all<br>subpopulations. Cases exhibiting the paradox are unproblematic from<br>the perspective of mathematics and probability theory, but<br>nevertheless strike many people as surprising. Additionally, the<br>paradox has implications for a range of areas that rely on<br>probabilities, including decision theory, causal inference, and<br>evolutionary biology. Finally, there are many instances of the<br>paradox, including in epidemiology and in studies of discrimination,<br>where understanding the paradox is essential for drawing the correct<br>conclusions from the data.
The following article provides a mathematical analysis of the paradox,<br>explains its role in causal reasoning and inference, compares theories<br>of what makes the paradox seem paradoxical, and surveys its<br>applications in different domains.
1. Introduction
2. Definition and Mathematical Characterization
2.1 Varieties of Simpson’s Paradox
2.2 Necessary and Sufficient Conditions
3. Simpson’s Paradox and Causal Inference
3.1 Probabilistic Causality and Simpson’s Paradox
3.2 Specific Debates: Causal Interaction, Average Effects, Mediators
3.3 DAGs and Causal Identifiability
3.4 Confounding and Pearl’s Analysis of the Paradox
3.5 Implications
4. What Makes Simpson’s Paradox Paradoxical?
5. Applications
5.1 Non-Categorical Data and Linear Regression
5.2 Epidemiology and Meta-Analysis
5.3 Decision Theory and the Sure-Thing Principle
5.4 Philosophy of Biology and Natural Selection
5.5 Policy Questions: Interpreting Data on Discrimination
5.6 Using Statistics to Evaluate Task Performance
6. Conclusions
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1. Introduction
We begin with an illustration of the paradox with concrete data. The<br>numbers in<br>Table 1<br>summarize the effect of a medical treatment for the overall<br>population (N = 52), and separately for men and women:
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Full Population, \(\bf N=52\)<br>Men \(\bf(\r{M})\), \(\bf N=20\)<br>Women \(\bf(\neg \r{M})\), \(\bf<br>N=32\)
Success \(\bf(\r{S})\)<br>Failure \(\bf(\neg \r{S})\)<br>Success Rate<br>Success<br>Failure<br>Success Rate<br>Success<br>Failure<br>Success Rate
Treatment (T)<br>20<br>20<br>50%<br>≈ 61%<br>12<br>15<br>≈ 44%
Control
(¬T)<br>50%<br>≈ 57%<br>≈ 40%
Table 1: Simpson’s Paradox: the<br>type of association at the population level (positive, negative,<br>independent) changes at the level of subpopulations. Numbers taken<br>from Simpson’s original example (1951).
For matters of exposition, we assume that these frequencies are<br>unbiased estimates of the underlying probabilities. The treatment<br>looks ineffective at the level of the overall population, but it leads<br>to higher success percentages than the control both for men and for<br>women (61% vs. 57% for men and 44% vs. 40% for women). Writing these<br>proportions as conditional probabilities, with \(\r{T}\)=treatment,<br>\(\r{S}\)=success/recovery, and \(\r{M}\)=male subpopulation, we<br>obtain
\[ p(\r{S}\mid \r{T}) = p(\r{S}\mid \neg \r{T}) \]
but at the same time,
\[\begin{align*}<br>p(\r{S}\mid \r{T}, \r{M}) & \gt p(\r{S}\mid \neg \r{T}, \r{M} ) \\<br>p(\r{S}\mid \r{T}, \neg \r{M}) &\gt p(\r{S}\mid \neg \r{T}, \neg \r{M})<br>\end{align*}\]
Should we use the treatment or not? When we know the gender of the<br>patient, we would presumably administer the treatment, whereas it does<br>not look like the right thing to do when we don’t know the<br>patient’s gender—although we know that the patient is<br>either male or female!
This phenomenon was first pointed out in papers by Karl G. Pearson<br>(1899) and George U. Yule (1903), but it was Simpson’s short<br>paper “The interpretation of interaction in contingency<br>tables” (1951), discussing the interpretation of such<br>association reversals, that led to the phenomenon being labeled as<br>“Simpson’s Paradox”. The phenomenon is, however,<br>broader than independence in the overall population and positive<br>association in the subpopulations; for example, the associations may<br>also be reversed. Nagel and Cohen (1934: ch. 16) provide an...