On Friday, we began to see reasonably clear signs that President Obama would receive some kind of bounce in the polls from the Democratic convention.

Mr. Obama had another strong day in the polls on Saturday, making further gains in each of four national tracking polls. The question now is not whether Mr. Obama will get a bounce in the polls, but how substantial it will be.

Some of the data, in fact, suggests that the conventions may have changed the composition of the race, making Mr. Obama a reasonably clear favorite as we enter the stretch run of the campaign.

On Saturday, Mr. Obama extended his advantage to three points from two points in the Gallup national tracking poll, and to four points from two in an online survey conducted by Ipsos. He pulled ahead of Mitt Romney by two points in the Rasmussen Reports tracking poll, reversing a one-point deficit in the edition of the poll published on Friday.

A fourth tracking poll, conducted online by the RAND Corporation’s American Life Panel, had Mr. Obama three percentage points ahead of Mr. Romney in the survey it published early Saturday morning; the candidates had been virtually tied in the poll on Friday. (The RAND survey has an interesting methodology — we’ll explore it more in a separate post.)

The gains that Mr. Obama has made in these tracking polls over the past 48 hours already appear to match or exceed the ones that Mr. Romney made after his convention. The odds, however, are that Mr. Obama has some further room to grow.

The reason is that the tracking polls are not turned around instantaneously. The Gallup poll, for instance, now consists of interviews conducted between Saturday, Sept. 1, and Friday, Sept. 7. That means that many of the interviews in the poll still predate the effective start of the Democratic convention on Tuesday night.

That Mr. Obama has made these gains in polls that only partially reflect the Democratic convention suggests that his bounce could be more substantial once they fully do so. Mathematically, Mr. Obama has to have been running well ahead of Mr. Romney in the most recent interviews in these surveys to have made up for middling data earlier in the week.

In fact, it is possible to reverse-engineer an estimate of what Mr. Obama’s numbers look like in the postconvention part of the tracking surveys. Specifically, I will be looking to infer Mr. Obama’s numbers from interviews conducted after Bill Clinton’s speech on Wednesday night, which in my view was the pivotal moment of the convention.

Let’s use the Gallup tracking poll as an example. Mr. Obama now leads in that survey by four percentage points. Conversely, he led by one point in the version of the poll published on Wednesday afternoon, ahead of Mr. Clinton’s speech. What must Mr. Obama’s numbers have looked like in the interviews since the Clinton speech in order for him to make those gains?

This can be determined with a little algebra if we know what percentage of the interviews in the Gallup survey reflect post-Clinton data. Fortunately, this calculation is fairly straightforward.

Gallup’s tracking poll is reported over a seven-day window, and roughly the same number of people are polled each day. The interviews Gallup conducted on Saturday, Sept. 1, and then on Sunday, Monday, and Tuesday, predated the Clinton speech. The interviews it conducted on Thursday and Friday post-dated it. The only day about which there is any ambiguity is Wednesday itself. But, since Mr. Clinton’s speech was made late Wednesday night, only a small fraction of the respondents in the poll would have had the chance to watch it by the time that Gallup called them — probably just the last round or so of interviews that Gallup conducted on the West Coast. (We’ll assume that 20 percent of the Wednesday interviews did reflect Mr. Clinton’s speech, although the fraction was probably a little lower than that in practice.)

Over all, that means that only about 30 percent of the data from the Gallup poll post-dated Mr. Clinton’s remarks.

If you do the math, it implies that Mr. Obama must have been leading Mr. Romney by 10 or 11 points in the minority of the poll conducted since Mr. Clinton’s speech for him to have gained three points in the survey over all.

In the table below, I’ve run through the same calculation for the other tracking polls. The results imply that Mr. Obama has run about nine points ahead of Mr. Romney in the portion of the Ipsos poll conducted since Mr. Clinton’s speech, about eight points ahead in the RAND poll, and about four points ahead in the Rasmussen poll.

On average between the four polls, it appears that Mr. Obama must have held about an eight-point lead since Mr. Clinton’s speech in order to have gained so much ground so quickly.

This method is not perfect — the only way we would know exactly how well Mr. Obama had been doing is if the polling firms published day-by-day results, which none of them do.

But on Friday, I wrote that Mr. Obama might eventually hold about a five-point lead over Mr. Romney once the tracking polls fully rolled over to post-convention data. Now it looks like his advantage could potentially be a bit larger than that, depending on how long the bounce holds. Despite a mediocre jobs report on Friday, there were no signs in the polls that Mr. Obama’s bounce had immediately receded, as he gained further ground in the surveys that were released on Saturday.

Earlier in the week of the convention, before there was any data on the magnitude of Mr. Obama’s bounce, I used a series of golf metaphors to serve as a guide to interpreting the postconvention numbers. By that nomenclature, it now appears that Mr. Obama is on track for a “birdie” convention, meaning that he would exit the conventions in a somewhat stronger position than where he entered them.

The equivalent of a par score remains a possibility if Mr. Obama’s numbers cool off a bit, which they very well may, although that would be better than Mr. Romney’s bogey.

But there is also the possibility of an eagle, with Mr. Obama holding as much as an eight- to nine-point lead over Mr. Romney in the polls once they fully reflect post-convention data. His polls seem to have been about that strong since Mr. Clinton’s speech, at least.

Again, this is just the upside case for Mr. Obama — not the reality yet. But the fact that it seems plausible is a bit surprising to me. Very little has moved the polls much all this year — including Mr. Romney’s convention and his choice of Paul D. Ryan as his running mate, events that typically produce bounces. But Mr. Obama has already made clear gains in the polls in surveys that only partially reflect his convention.

As surprising as it might be, however, I do not see how you can interpret it as anything other than a good sign for Mr. Obama. All elections have turning points. Perhaps Mr. Obama simply has the more persuasive pitch to voters, and the conventions were the first time when this became readily apparent.

Polls conducted after the incumbent party’s convention typically inflate the standing of the incumbent by a couple of points, but not usually by more than that. Otherwise, they have predicted the eventual election outcome reasonably well.

Since 1968, the largest post-convention polling deficit that a challenger overcame to win the race was in 2000, when George W. Bush trailed Al Gore by about four points after the Democratic convention but won the Electoral College — although Mr. Bush lost the popular vote.

And unlike Mr. Bush, who at least led in the polls after his own convention that year, Mr. Romney did not, essentially only bringing the race to a tie in polls conducted early in the week of his convention.

In fact, Mr. Romney has never held a lead over Mr. Obama by any substantive margin in the polls. The Real Clear Politics average of polls put Mr. Romney ahead by a fraction of a percentage point at one point in October 2011, and he pulled into an exact tie at one point late in the week of his convention, after it was over, but he has never done better than that.

That makes this an extremely odd election. You would figure that at some point over the past year, Mr. Romney would have pulled into the lead in the polls, given how close it has usually been. John McCain held occasional leads in 2008; John Kerry led for much of the summer in 2004; and Michael Dukakis had moments where he was well ahead of George H.W. Bush in the spring and summer of 1988. But Mr. Romney, if there have been moments when his polls were ever-so-slightly stronger or weaker, has never really had his moment in the sun.

Instead, the cases where one candidate led essentially from wire to wire have been associated with landslides: Bill Clinton in 1996, Ronald Reagan in 1984, Richard Nixon in 1972 and Dwight D. Eisenhower in 1952 and 1956.

There is almost no chance that Mr. Obama will win by those sort of margins. But this nevertheless seems like an inauspicious sign for Mr. Romney. If even at his high-water mark, he can only pull the race into a rough tie, what pitch can he come up with in October or November to suddenly put him over the top?

Conservatives sometimes cite Ronald Reagan’s win in 1980 as a favorable precedent for Mr. Romney, because the polls showed him in a tight race with Jimmy Carter in October and early November, 1980. Nevertheless, Mr. Reagan had shown much clearer signs of upside potential earlier in the race — most conspicuously, in leading Mr. Carter by nearly 30 points after the Republican convention in Detroit.

Because of demographic changes, the Republican base is probably just a bit too narrow to win the election for Mr. Romney on its own, even with a strong Republican turnout.

Certainly, Mr. Romney will win his fair share of independent voters because of the economy — and if there are substantive signs of economic decline in October and November, probably enough to win him the election.

But unless there is some change of course, it looks increasingly as though he lacks the appeal to the voting blocks that might allow him to win 51 percent of the vote rather than 49 percent.

Mr. Romney’s tenure at Bain Capital and his opposition to the auto bailouts may be a negative with the Reagan Democrats of the Midwest. His conservative stances on immigration will likely prevent him from having George W. Bush’s comparative appeal among Hispanic voters. He is mostly playing defense among the so-called security moms with whom Mr. Bush performed fairly well in 2004, having largely sidestepped discussions of national security.

Highly educated voters who are moderate on social policy but conservative on fiscal policy might be a natural constituency for Mr. Romney, and they were critical to his election as governor in Massachusetts in 2002. But Mr. Romney has rarely broken from the Republican orthodoxy on social issues. By contrast, it was Democrats who were much more forthright about touting their support for abortion rights, gay rights, and less rigid immigration policy at their convention in Charlotte, N.C.

I will acknowledge that there is the risk of jumping the gun with this analysis. Our forecast model began to see Mr. Romney’s subpar convention bounce as a bearish indicator for him early during his convention week. Now that Mr. Obama appears to be making gains when Mr. Romney did not, it has become more entrenched in seeing Mr. Obama as the favorite — enough so that it now gives him almost a 4-in-5 chance of victory. Taking the temperature of voters around the party conventions is tricky: it is a period when a lot of undecided voters start to tune in for the first time, but it is also associated with volatile polling. Every election is different, and no statistical method to analyze them is beyond reproach.

But in the immediate term, it seems like the upside case for Mr. Romney is that Mr. Obama’s polls cool off quickly — and soon revert to where they were before the conventions, with Mr. Obama about two points ahead in the polling average. That’s certainly a very winnable election for Mr. Romney, but nevertheless one where he is the modest underdog.

And Mr. Romney’s downside case is that Mr. Obama’s polling bounce will be a little stickier, and that Mr. Obama will already be fairly close to having achieved 50 percent of the vote with precious few undecided voters left in the race. That would make Mr. Romney a clear underdog — perhaps even one who needs some foreign policy or economic crisis to intervene to give him much of a chance at winning.