Abstract
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.
Highlights
1 Introduction In 1973, the Associate Dean of the graduate school of the University of California Berkeley worried that the university might be sued for sex bias in the admission process (Bickel et al 1975)
The Chi-square statistics for this test has one degrees of freedom with value χ 2 = 111.25 and corresponding p-value basically = 0, while the Chi-square statistics with Yates continuity correction for this test has a value of χ 2 = 110.849 and corresponding p-value again approximately 0
A naïve conclusion would be that men were much more successful in admissions than women, which would clear be understood as a bad episode of gender bias
Summary
In 1973, the Associate Dean of the graduate school of the University of California Berkeley worried that the university might be sued for sex bias in the admission process (Bickel et al 1975). The authors concluded that “Measuring bias is harder than is usually assumed, and the evidence is sometimes contrary to expectation” (Bickel et al 1975) This episode is one of the most celebrated real examples of what is called Simpson’s Paradox: the trend of aggregated data might be reversed in the pooled data. 2.1 Definitions First, we define the Process of Amalgamation of contingency tables ti, i = 1, . Given the definition of Measure of Amalgamation, we can formally define the Simpson’s Paradox. Definition 3 We say that the Simpson’s Paradox occurs for the Measure of Amalgamation α if max α(ti) < α(T)or min α(ti) > α(T), i i with α defined on the set of contingency tables and real valued, as in Definition 2.
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