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Simpson’s Paradox is a statistical phenomenon where an association between two variables in a population emerges, disappears or reverses when the population is divided into subpopulations. For instance, two variables may be positively associated in a population, but be independent or even negatively associated in all subpopulations. Cases exhibiting the paradox are unproblematic from the perspective of mathematics and probability theory, but nevertheless strike many people as surprising. Additionally, the paradox has implications for a range of areas that rely on probabilities, including decision theory, causal inference, and evolutionary biology. Finally, there are many instances of the paradox, including in epidemiology and in studies of discrimination, where understanding the paradox is essential for drawing the correct conclusions from the data.

The following article provides a mathematical analysis of the paradox, explains its role in causal reasoning and inference, compares theories of what makes the paradox seem paradoxical, and surveys its applications in different domains.

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