Barack Obama got some good news to start his new year: his Gallup approval rating ticked up to 50 percent yesterday, against 42 percent disapproval. That’s the first time it has been at the 50 percent threshold since early June.

Mr. Obama had some successes to point to in December, like the accomplishments of the lame-duck session of Congress, and some moderately encouraging economic numbers.

Still, it’s unclear whether the movement is real or reflects statistical noise; most polls still have Mr. Obama slightly below 50 percent. And it probably isn’t worth going through very much effort to find out, provided that our focus is on 2012.

Historically, there has been almost no relationship between a President’s approval rating following the midterms and how he eventually fared at the next presidential election. I’ve alluded to this in the past, but the data illustrate the point a little bit more forcefully.

Below are the Gallup approval ratings at a comparable point in time — early January just after the midterms — for the 11 prior incumbent presidents who stood for election (not re-election, technically speaking, in cases like Gerald Ford). The chart also shows what percentage of the vote the president ultimately received, and by what margin he won or lost by, once the election took place.

Franklin Roosevelt in 1943 and Dwight Eisenhower in 1955 had the highest approval ratings at this point in time and indeed won their elections fairly easily. On the other hand, Ronald Reagan was tied with Gerald Ford for having the lowest one — 37 percent — and he won in a blowout. Bill Clinton’s approval rating was 47 percent, and he won. George H.W. Bush’s was 58 percent, but he lost. His son George W. had a 63 percent approval rating and won, but only barely.

As you can see, there’s not much of a relationship there. The correlation between a president’s approval rating in the January after the midterms and the share of the vote he received at the subsequent election (shown in the first of the two graphs below) is .10. The correlation between his approval rating and the margin that he won or lost by (shown in the second chart) is .09.

Neither of these approach statistical significance. If we used this data to predict what a president’s fate would eventually be, our “model” would show that a president with a 35 percent approval rating at this point in time would be expected to win 49.8 percent of the vote, whereas one with a 65 percent approval rating would get 51.7 percent of the vote. In other words, it makes almost no difference, and the margins of error on the forecasts would overwhelm any insight they might provide. It’s just too early.

This is not to say that you should pay no attention at all to presidential approval numbers. It may be that the true relationship between these variables is modestly stronger than this, and that we’ve been a bit unlucky that it hasn’t done a good job of presenting itself. We have only 11 data points to look at, and when you have that few, they aren’t always inclined to be cooperative.

To the extent that changes in Mr. Obama’s approval rating perceive clues about how the public is apportioning credit or blame for the performance of the economy, or how it is refereeing the fights between Mr. Obama and the Republicans in Congress, they might theoretically be interesting — but such relationships are usually difficult to tease out in practice.

Mostly, you’ll just need to be patient. The good news is that the closer we get to the election, the more meaningful the numbers become. By the end of a president’s third year, for instance, the correlation between a president’s approval rating and his performance at the election picks up to around .3 or .4 — still quite crude, but enough to give it some predictive power.