Simpson's paradox fools many. Percentages can favor women over men across each of several subgroups but then reverse, favoring men over women when the subgroups are combined into one. At one level this seems illogical. We seem to expect that patterns observed consistently for portions of a whole should also apply when the portions are aggregated together into one. This simple view misses lurking variables. In a famous example, graduate admission to Berkeley seemed biased against women when considered overall, but when the admissions were considered by individual departments there was no bias or bias in favor of women. The lurking variable is that "not all departments are equally easy to enter." and "the proportion of women applicants tends to be high in departments that are hard to get into and low in those departments that are easy to get into".
Lewis Lehe and Victor Powell at UC Berkeley have produced interactive applets to illustrate Simpson's paradox. As Flowing Data mentions "Sometimes when you zoom in, you see a completely opposite trend of what you saw overall".
We've considered Simpson's paradox before where even microbes can be used to illustrate it.