[noam87]

For me it's Simpson's paradox: it throws everyone off -- it's caused (and will continue to cause) real-world damage, it's everywhere once you see it -- it's in how newspapers report science, it's in our social policy and how we talk about social issues, it's in court cases --, and finally, it's really hard to explain to a non-math person; so even when it's happening, you sound like the irrational one for pointing it out.

... and don't get cocky once you know about it, because it's so pernicious it'll get you too if you're not careful!

[yiyus]

I do not think this is a particularly difficult concept to explain to anyone. A typical example (not difficult to find) and a simple graph are usually good enough for most people.