probability of exactly . A study by Kock suggests that the probability that Simpson's paradox would occur at random in path models (i.e., models generated by path analysis) with two predictors and one criterion variable is approximately 12.8 percent; slightly higher than 1 occurrence per 8 path models. Simpson's second paradox A second, less well-known paradox was also discussed in Simpson's 1951 paper. It can occur when the "sensible interpretation" is not necessarily found in the separated data, like in the Kidney Stone example, but can instead reside in the combined data. Whether the partitioned or combined form of the data should be used hinges on the process giving rise to the data, meaning the correct interpretation of the data cannot always be determined by simply observing the tables. Judea Pearl has shown that, in order for the partitioned data to represent the correct causal relationships between any two variables, and , the partitioning variables must satisfy a graphical condition called "back-door criterion": They must block all spurious paths between and No variable can be affected by This criterion provides an algorithmic solution to Simpson's second paradox, and explains why the correct interpretation cannot be determined by data alone; two different graphs, both compatible with the data, may dictate two different back-door criteria. When the back-door criterion is satisfied by a set Z of covariates, the adjustment formula (see Confounding) gives the correct causal effect of X on Y. If no such set exists, Pearl's do-calculus can be invoked to discover other ways of estimating the causal effect. The completeness of do-calculus can be viewed as offering a complete resolution of the Simpson's paradox. See also Spurious correlation References Bibliography Leila Schneps and Coralie Colmez, Math on trial. How numbers get used and abused in the courtroom, Basic Books, 2013. . (Sixth chapter: "Math error number 6: Simpson's paradox. The Berkeley sex bias case: discrimination detection"). External links Simpson's Paradox at the Stanford Encyclopedia of Philosophy, by Jan Sprenger and Naftali Weinberger. How statistics can be misleading – Mark Liddell – TED-Ed video and lesson. Pearl, Judea, "Understanding Simpson’s Paradox" (PDF) Simpson's Paradox, a short article by Alexander Bogomolny on the vector interpretation of Simpson's paradox The Wall Street Journal column "The Numbers Guy" for December 2, 2009 dealt with recent instances of Simpson's paradox in the news. Notably a Simpson's paradox in the comparison of unemployment rates of the