Simpson's paradox is the data science equivalent of looking through one eye and seeing a very clear trend, and then looking through the other eye and seeing the very clear opposite trend.  In one case, you see a trend one way in a group, but then breaking the group into subgroups gives the exact opposite trend.  Confused?  Scratching your head?  Welcome to the tricky world of ecological inference.

Relevant links:
https://gking.harvard.edu/files/gking/files/part1.pdf
http://blog.revolutionanalytics.com/2013/07/a-great-example-of-simpsons-paradox.html

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