The Book of Why: The New Science of Cause and Effect (Penguin Science)

by Judea Pearl

Pearson discovered possibly the most interesting kind of “spurious correlation” as early as 1899. It arises when two heterogeneous populations are aggregated into one. Pearson, who, like Galton, was a fanatical collector of data on the human body, had obtained measurements of 806 male skulls and 340 female skulls from the Paris Catacombs (Figure 2.5). He computed the correlation between skull length and skull breadth. When the computation was done only for males or only for females, the correlations were negligible—there was no significant association between skull length and breadth. But when the two groups were combined, the correlation was 0.197, which would ordinarily be considered significant. This makes sense, because a small skull length is now an indicator that the skull likely belonged to a female and therefore that the breadth will also be small. However, Pearson considered it a statistical artifact. The fact that the correlation was positive had no biological or “organic” meaning; it was just a result of combining two distinct populations inappropriately.

This is an interesting thing to know, and somethng i didn't realise... So if there is population heterogeneity, this can result in a correlation betwen two variables due to their common cause (one of the populations). But when correlated within the group, they are no correlated, because there is no variation in the common cause.