Simpson Paradox Question I am trying to understand if the following statement is an example for Simpson paradox:
"In the US elections a certain candidate got more votes than the other, but the other one was elected".
I think it's not an instance of Simpson paradox, but I am having trouble formalizing it. Am I right? if so, how do I explain it formally?
 A: I guess it will depend on how you define Simpson's paradox.
I suppose the essence of the paradox is that it occurs when the result of looking at a data set broken into groups gives the opposite impression of looking at the data set as a whole.
The electoral college case given in the question is somewhat similar to the batting average example on Wikipedia, but I don't think your example follows the paradox as I roughly sketched it out above.
You could probably come up with a scenario that is more Simpson's-like.
Perhaps if it were the case that one candidate loses the popular vote in each state, but still wins the popular vote in aggregate (similar to batting average example).  Or one in which one candidate loses the electoral college in a majority of states, but still wins the overall college (which probably happens in reality on occasion †).

† I thought this might be a common occurrence in U.S. presidential elections, with perhaps the Democrat winning fewer, but more populous, states. Working backward, the first occurrence I found was 1976, with Carter winning only 23 states + DC, (out of 50 +DC), but winning 297 electoral votes (out of 537) [usually 538].
A: Simpson's Paradox is considered to happen when estimating for some statistical effect between and within groups. In this case there is no factor effect to be estimated, instead you measure a proportion, so I woudn't say that Simpson's Paradox is involved here, but you may consider this as an example of a wider generalized paradox about grouped data. 
To show how there is a relation, let me make a simpler example:
Suppose you have a nation with various electoral district, where which of them elects one candidate (majority system), but some districts are more populous than others. If there is a general tendency within each district population to vote candidate A, but also there is a trend among districts where more populous ones tend to express more votes for candidate B, you could have candidate B take more votes while the victory goes to candidate A.
In Simpson's Paradox usual cases you are interested in within group effects, which are not reflected by analysis that don't take account of grouping, in this case you could say you are interested in elections outcome, which is not reflected by analysis that don't take into account the electoral districts, as if the overall majority decided, which is not.
USA electoral system makes it quite more complicate, but I think I have expressed the gist already. Also, if you think the system should be changed, for what is worth I actually agree, but that's a different story, Simpson's paradox is about data analysis, not about changing the grouping of the data, or changing any electoral system.
A: To simplify, suppose that each state has one electoral vote, which goes to the candidate with the most votes in that state. If A wins more than half the states by a slim majority and B wins fewer states with overwhelming majorities, then the overall 'popular' vote for B could be greater. 
The actual situation in the US is that states have widely varying numbers of electoral votes (only very roughly proportionate to population), but the same idea that some states are won by slim margins and others by greater margins still applies.   
A: This is not an example of Simpson's Paradox.
It is directly related to the weighting of the Electoral College and its winner-take-all nature.
Simpson's Paradox generally requires an inbalance in the participation in two categories. To generate that, you would need something where candidates can participate unequally across states. For example, we could get a Simpson's Paradox by looking at rally attendance if the candidates travel to different types of locations/venues.
For example:
========  =============================  ===========================  ================
Candiate  Avg. in Small States (number)  Avg. in Big States (number)  Total Attendance
========  =============================  ===========================  ================
A         10,000 (5)                     40,000 (10)                  **450,000**     
B         **15,000 (10)**                **50,000 (5)**               400,000         
========  =============================  ===========================  ================

Candidate B has larger average attendance in both types of states, but visits small states more often than big states. If you look at total rally attendace (or average, since they have the same number of rallies), Candidate A appears to draw larger crowds.
Because the candidates participate in the same states for the election (assuming you don't have a case where candidates skip states altogether), you can't get this kind of unbalanced design. 
