Read enough polling commentary, and you’re sure to encounter arguments like this one:

And a staggeringly high number of Democratic incumbents are … sitting below that magic 50 percent number. That number should send shivers down the spines of Democratic strategists. In 2008, when Democrats coasted to victory across the board, 32 House Republican incumbents were under the 50 percent mark in the last poll of the cycle, and 14 of them lost — a 44 percent mortality rate.

That was from The Hotline’s Josh Kraushaar. Here is Real Clear Politics’ Sean Trende making a version of the same claim.

When dealing with incumbents, it is much more important to look at the incumbents’ number than the challengers’ number, since undecideds usually break against the incumbent.

And here is Michael Barone of the Washington Examiner:

Another metric is daunting for Democrats. Polls in House races almost always show incumbents ahead of challengers, because incumbent members of Congress are usually much better known than their opponents. An incumbent running below 50 percent is considered potentially in trouble; an incumbent running behind a challenger is considered in deep doo-doo.

The argument, then, is that the emphasis on the margin in a particular poll (“so-and-so leads by 8 points”) is misplaced. Instead, these analysts say, if there is an incumbent in the race, you should focus (mostly) on his or her number in the polls — and in particular, whether or not it is below 50 percent. If, for instance, the incumbent is at 46 percent in the polls and the opponent is at 34 percent, the incumbent might be more vulnerable than the 12-point margin suggests. Undecided voters could still swing the race toward the incumbent, the argument goes — and indeed the undecided vote usually breaks toward the challenger, it is claimed.

The three analysts I just cited are among the smartest in the business. But their emphasis here is a little off the mark.

I hope we can all agree that, as Mr. Barone puts it, an incumbent who trails in the polls is in “deep doo-doo.” You don’t need advanced analysis to tell you that. And plenty of Democratic incumbents this cycle find themselves in such a position: few will salvage their races.

But what about an incumbent who holds a lead but, nevertheless, stands below the so-called magic number of 50 percent? How often do such candidates lose?

I identified a total of 83 incumbents in our database of House, Senate and gubernatorial polls who — with 30 days to go until the election — led their opponents (by any margin) in an average of nonpartisan polls, but who had under 50 percent of the vote. A bit of housekeeping: our database goes back to 1998 and excludes Zogby Interactive polls; the “average” I’m referring to here is a “lo-fi” version that weights for sample size and recentness of the poll, but not for pollster quality.

Let’s first look at the incumbent candidates for Senate. How many candidates who met this definition — leading in the polls, but with less than 50 percent of the vote — were upended in their re-election bid?

Actually, the percentage is pretty high. Of 25 such candidates, 9 of them lost, or 36 percent:

So far, things look pretty good for the “magic number” theory: more than one in three of these Senate incumbents lost, in spite of holding the polling lead. Since, in recent years, polls have rarely called the winner of the race wrong, this is pretty impressive.

What about gubernatorial incumbents? Did they suffer a similar fate?

No; the theory holds up less well here. Of the 23 gubernatorial incumbents who sat at under 50 percent in the polls — but held a lead over their opponents — 19 held on for the victory. That’s a success rate of 83 percent.

How about House candidates? It stands to reason that, to the extent this effect exists, it would be felt the most among House incumbents, since their opponents — who usually lack the pedigrees that challengers in Senate or governors’ races do — will often have poor name recognition until late in the cycle.

In fact, however, the House incumbents in our sample did well: 31 out of 35 of them won their races, or 89 percent:

If you combine the three types of incumbents — House, Senate, governor — they had a record of 66-17, which equates to a winning percentage of 80 percent. Depending on how you define the term, they may not have been “safe” for re-election — but certainly, most were favorites.

There is also not any particular evidence that, as Mr. Trende suggests, the majority of the undecided vote broke against the incumbents. On average, the incumbent candidates led by 8.1 points in the polls with 30 days to go; they won their elections by an average of 7.2 points. That’s not a huge difference, needless to say.

So, is there anything at all to the theory? A couple of qualifiers are in order:

First, as I stated above, if an incumbent is trailing in the polls, that’s a whole different matter. An incumbent is usually going to lose if he’s at 43 percent in the polls, and his opponent is at 48 percent. His problem, however, is less that he’s polling at 43 percent — and more that his opponent has a better number!

Although incumbents who sit below 50 percent in the polls may or may not be vulnerable, those who are at 50 percent or above will rarely lose — particularly if that 50 percent figure is achieved across several reliable polls. So it is not so much that being below 50 percent necessarily puts a candidate in a “danger zone” but that being above 50 percent puts him in a “safety zone.” Yes, sometimes things can go horribly wrong. But there generally has to be a peculiar dynamic in a race for a candidate to fail to convert in the general election once he’s hit 50 percent in the consensus of polls.

So, that’s sort of the Polling 101 explanation. Focus on the margin between the candidates, just as you might instinctively do.

For our advanced readers, however, I’ll also offer the Polling 201 explanation, which redeems the theory to some extent.

It turns out, we have found, that the number of undecideds in the poll does have some informational value. In particular, it has a fairly clear relationship with the error in the forecast. A candidate who leads 53 percent to 43 percent — meaning, he has a 10-point lead, and just 4 percent of voters are undecided — is usually going to come really close to matching that 10-point margin on Election Day. Maybe he’ll win by 8 points, or 9, or 13 — but it’s going to be somewhere in that vicinity, and it would be a real shock if he lost his race.

On the other hand, things are a little more unpredictable if the candidate’s lead is 43 percent to 33 percent: which is also a 10-point lead, but with many voters undecided. Such a candidate will lose, on occasion. But there are also times that he’ll wind up winning by 20 or 25 points, far outperforming his polling margin on Election Day.

Undecided voters, in other words, are the equivalent of the clock in a football game. If an N.F.L. team holds a 10-point lead at halftime, it is the favorite to go on and win the game. But there is plenty of football left to play, and it will lose some of the time (whereas other times, meanwhile, it will wind up winning by considerably more than 10 points). This is the equivalent of holding a 43-to-33 lead in a political poll, with lots of undecided voters.

On the other hand, an N.F.L. team that holds a 10-point lead with two minutes to play in the fourth quarter will almost never lose. (Nor, for that matter, is it likely to win by much more than 10 points.) This is analogous to having a 53-to-43 lead in the polls: barring the political equivalent of an onside kick and a Hail Mary, such a candidate can start picking out his office furniture.

So, if you’re willing to do so carefully, it is worth looking at the number of undecided voters in a poll. Given a lead of a certain size (say, 10 points), an incumbent (or a nonincumbent, for that matter) is more likely to lose the lead if there are more undecideds rather than fewer. This is not because the undecideds are especially likely to break for the challenger — something which just hasn’t been true to any meaningful extent in recent elections. It’s simply because there are more undecideds, period, and that implies greater volatility and means there is more campaigning left to do.

By the way, the theory espoused by Mr. Kraushaar and others isn’t coming out of nowhere: there is solid evidence that it used to be true, 20 or 25 years ago. Back then, the undecideds in a race usually could be counted upon to break toward the challenger: the name given to this phenomenon was the “incumbent rule.”

But polling has changed since then — as have social norms. On the one hand, pollsters have become more inclined to “push” voters toward an answer — if a voter declines to state a candidate preference initially, the pollster may ask her which candidate she is leaning toward, which may bring implicit preferences to the fore. On the other, voters have perhaps become more willing to advance a candidate preference based on information as thin as party identification. A conservative voter who is unhappy with the Democratic incumbent in their district, for instance, may be willing to note their support for the Republican opponent even if they have never heard of him or her before.

One last Polling 201 clarification: anti-incumbent sentiment may be unusually strong this year — and so perhaps the incumbent rule will make a reappearance (as it arguably did, for instance, in the New Jersey governor’s race in 2009). If I were an incumbent Democrat holding something like a 44-to-39 lead, I certainly wouldn’t feel overly sanguine about my position.

Nevertheless, even in the Democratic wave of 2006, 11 of the 12 Republicans who were under 50 percent in the polling with a month to go, but who did hold a lead, managed to hang on for the victory. So while this is a reasonable hypothesis, it isn’t one that will have much evidentiary support until after the election.