YouGov's final survey in America shows Barack Obama ahead in enough states to secure a second term as President, but by a narrower margin than in his first victory four years ago.

In one of the most extensive polls ever conducted, YouGov questioned more than 36,000 Americans in 27 states, between last Wednesday and Saturday. Our survey included all the battleground states, and also the largest states such as New York, California and Texas. Together, they put Obama just 2% ahead in the popular vote, but with enough support in the key states to make him the clear favourite to win the 270 Electoral College votes he needs.

There has been little doubt for some time that Obama will win 18 states comfortably. (This counts Washington DC as a status quo; as it contributes three Electoral College votes, even though it does not belong to any of the 50 true states in the US.) Together, these give Obama an absolute minimum of 237 Electoral College votes; 33 short of his target.

Mitt Romney holds clear leads in 24 seats.They give him 191 Electoral College votes, 79 short of victory.

That leaves nine states up for grabs. They provide 110 electoral votes. Obama needs to win just under a third of them; Romney needs almost three-quarters of them.

These are YouGov's final results for these nine states.

If all our figures are exactly right, then Obama will win 66 Electoral College votes from these battleground states and win overall by 303-235.

However, all polling is prone to two types of error; random and systematic. Let us deal with these in turn.

Suppose every poll is conducted perfectly, in that it contacts a perfectly random sample of voters. Because it is a sample, there is a margin of error. It's rather like tossing a good coin a large number of times. It should come up heads around the same number of times as it comes up tails. But statistical theory tells us we can't be sure the EXACT number of heads and tails. All we can do is predict the likely range. A calculation for margin of error allows us to determine that range.

If we assume that YouGov's samples in each state are subject only to random error, then we can be reasonably confident that Obama will win Nevada, New Hampshire and Wisconsin. Their 20 Electoral College votes bring Obama's total to 257. He needs another 13 for victory.

This means he will be re-elected president if he wins any one of four closely contest states; Ohio or Virginia (where he is narrowly ahead), or Florida or North Carolina (where he is narrowly behind). Indeed, he could lose all four of these states and still reach his 270 vote target if he wins both Iowa and Colorado.

In contrast, Romney must win all four of the bigger states AND either Iowa or Colorado.

The upshot? Although probability theory tells us that, individually, each of these six states could go either way, it is very unlikely that Romney will win ALL the states he needs to win the Electoral College. If YouGov's figures are subject to only random error, the chances of Obama remaining President are well over 90%. *

Romney's real chances of winning this week rest on our figures (and those of other pollsters, who paint a broadly similar picture of the state of the race) being systematically adrift of what voters actually do. There are four ways this could be the case:

1. Late swing. Fieldwork for our survey ended on Saturday. Some people wait until the last minute before deciding how to vote. If these break more for Romney, he will do better than the polls indicate.
2. Differential turnout. Like all polling companies, YouGov counts those people we think will actually vote (or, in the case of tens of millions of Americans tell us they have already voted). If, on the day, Romney's supporters are more enthusiastic, or his workers in the key states do better at getting their vote out, he could, again, outperform his poll ratings.
3. Poor methodology. All polls seek to match their samples to the characteristics of the US as a whole. But what if our understanding of those characteristics is slightly wrong; say, because our demographic data is slightly out-of-date? Our samples may contain a slight bias towards the kind of people who support one candidate or the other.
4. Response rates. Suppose Obama's supporters are fractionally keener to answer pollsters' questions than Romney's supporters. Again, our data will overstate Obama's support and, hence, his chances of winning.

None of us can be really certain whether any, or all, of these factors are present. In 2008 the pollsters as a whole had a good election. YouGov got the share of the national vote right to within one percentage point, and called the right winning in all bar one very closely-fought state. However, as British pollsters know only too well when they recalling the 1992 general election (when the polls pointed to a Conservative defeat, yet John major led them to victory), a good past record provides no immunity against new challenges. And, of course, these systematic factors, if they exist, might work in the opposite direction, and propel Obama to a more emphatic victory than now seems likely.

What we can measure is broadly the extent to which Romney needs to beat our figures in order to defeat Obama. Our central prediction is that Obama will win the national popular vote by 2% and the Electoral College by 303-235. This is how our prediction would move if we move the figures in each of our states in Romney's direction.

If we move the Obama-Romney balance ONE POINT in Romney's direction in every state, then Obama's lead would be down to 1%. He would still win 288 Electoral College votes and so win the election. Our figures would make the two candidates level in Iowa and Colorado, so we couldn't say who would win them.

If we move the balance TWO POINTS, the two candidates would be level in the national popular vote. On our central prediction, he would win Iowa and Colorado, and draw level in Virginia. But Obama would still have 275 Electoral College votes and remain president.

If we move the balance THREE POINTS, Romney would move into a one point lead. On our central projection, he would win Virginia. This would take his Electoral College tally to 263, while Obama would be down to 257. Ohio would be impossible to call. Whoever won the buckeye state would win the presidency.

In other words, for Romney to win, not only must our surveys (and polls by almost every other company) be systematically exaggerating Obama's support, they must exaggerate his state-by-state lead by around three points.

To put it another way, Romney could win one million more votes than Obama across America, and still not be sure of victory. There have been elections when the winner of the popular vote has lost the Electoral College. The most recent time was Al Gore twelve years ago when he lost to George W Bush. But no sitting president seeking re-election has lost the popular vote and retained the presidency. Given the near-certainty that the Republicans will keep their majority in Congress, Obama might find it even harder to implement his policies than if he outpolls Romney nationally as well as in the key states.

* Statistical footnote. When a coin is tossed a hundred times, there is roughly an evens chance that it will come down heads slightly more often than tails. But if you do the exercise five times, there is only a 3% chance that heads will appear more often than tails on all five occasions. That is why the odds overall are stacked against Romney unless our figures are systematically adrift. Winning all the states he needs is like a coin coming up heads five times in a row.

Find the detailed results here

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