What do the presidents below have in common?
They all won the Electoral College vote while losing the popular vote.
Imagine how different the history of the United States might be if these four people had not served, and if their opponents had served instead.
How does this kind of odd election happen? Since a strong majority of presidential elections have produced the same candidate winning both the Electoral College and the popular vote, that's an especially good question. In class, the suggestion was that extremely close elections are more likely to produce that outcome.
In particular, what happened in 2016? In brief, this: Clinton won a number of large states such as California and New York by huge margins. She almost won a number of other large states such as Florida and Pennsylvania. At the same time, Trump won a number of small states (which held fewer voters) by big margins, and just barely won several sizable states (not just Pennsylvania and Florida--also Michigan and Wisconsin). Clinton wracked up huge vote totals in big states; Trump won several key states by just a few thousand votes. Total: nationally about 3 million more people voted for Clinton, but the Electoral College is winner-take-all almost everywhere so Trump's big wins in many small states and narrow wins in several key states was enough to give him the EC edge.
The winner/losers are (clockwise from top left) Rutherford Hayes in 1876, Benjamin Harrison in 1888, Donald Trump in 2016, and George W. Bush in 2000.

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