Simpson's paradox

An intriguing paradox illustrating how one may be misled when a relevant variable is overlooked. The paradox is illustrated in the following example, which shows a cross-classification of three dichotomous variables, A, B, and C (where, for example, A₁ and A₂ are the two categories of A):

	C1				C2
	B1	B2	Total			B1	B2	Total			B1	B2	Total
A1	95	800	895	+	A1	400	5	405	=	A1	495	805	1 300
A2	5	100	105		A2	400	195	595		A2	405	295	700
Total	100	900	1 000		Total	800	200	1 000		Total	900	1 100	2 000

In the subpopulation corresponding to C₁, there is a strong positive association between A and B. The same is true for the subpopulation corresponding to C₂. However, when the information on these two very dissimilar subpopulations is pooled, the association for the entire population is strongly negative. See ecological fallacy for a diagram that shows how this type of result can occur with continuous data.

---

---

A Dictionary of Statistics. Second edition revised. Copyright © Oxford University Press, 2008. All rights reserved.

---