OHIO — Both presidential campaigns this week made one thing clear—they’re intent on wooing this state’s women voters.

First Lady Michelle Obama made two appearances on Tuesday, first in the Columbus suburb of Westerville and later in Dayton, and more women than men filled those audience seats.

On the very same day—and probably not by coincidence—Mitt Romney launched a statewide campaign entitled Ohio’s Women for Mitt. That effort will be co-chaired by the state’s first lady Karen Kasich and Jane Portman, wife of US Sen. Rob Portman, who is oft-mentioned as a potential Romney running mate.

The state’s media dutifully covered both events, but the Dayton Daily News, in a piece that drew on contributions from several staff writers, gets a nod over [the others for focusing on the ways the rival campaigns are reaching out to women here: Democrats with a greater focus on health and social issues, Republicans by repeating their general emphasis on the economy.

Jon Ward, writing Tuesday night at The Huffington Post, captured the same distinction:

The contrast between the messaging at the Obama and Romney campaigns could not have been starker. The warm-up speakers at the Obama event emphasized a host of issues around health care and education. The president’s health care overhaul got big cheers. So did his moves to help college students pay for school, and to make it easier for women to sue employers for workplace gender-based discrimination.

At the Romney event, only a few comments by [former House Speaker Jo Ann] Davidson, and a sign on the lectern reading “Women for Mitt” distinguished the event as being in any way gender-specific. The other speakers besides Davidson, including [Lt. Gov. Mary] Taylor, focused their remarks almost exclusively on the economy and jobs.

This is interesting stuff. And it’s also interesting to know, as the Dayton paper reports, that women voters rank the economy as by far the most important issue in this campaign.

That said, both local and national media outlets should tread cautiously when framing this subject with poll numbers, and these stories showed why. The Dayton Daily News, explaining why the campaigns are fighting over women here, noted that Obama fared much better among women in 2008 than John Kerry had in 2004.

But Obama did better than Kerry among men, too—by the same amount. In fact, as Brendan Nyhan has argued at CJR, “presidential election outcomes can almost never be attributed to a shift in a single demographic group,” so journalists covering the horse race “should keep their eye on the big picture.” That’s because most of the time, different groups—and different states—will move in unison.

While that’s the rule, it does make sense for journalists to keep their eyes open for exceptions. And the Daily News story does note that “the gap [among women voters] has gotten bigger over the past four years, according to most polls.” The article cites a June Quinnipiac poll that found Obama leading 50-35 among Ohio women, and a recent Purple Strategies poll that gives Obama an 11-point advantage with women in 12 swing states, including Ohio. In HuffPo, Ward also cited the Purple Strategies poll, including the fact that 56 percent of Ohio women said they viewed Romney unfavorably, with only 32 percent liking him.

But though these polls probably are good news for Obama, they offer limited information about the gender gap. In both 2004 and 2008, the Democratic presidential nominee did seven points better among women than men nationally. In the June Quinnipiac poll, Obama did only about five points better among women than men in Ohio—but that amounted to a blowout, because he did well across the board.

The Quinnipiac poll at least had a sample of more than 1,200 Ohioans. The Purple Strategies poll surveyed about 2,400 “likely voters,” but only 600 of them were from Ohio; assuming the survey was evenly divided between the genders, about 300 Ohio women participated. That’s a big enough sample to have some value, but, as this chart shows, small enough that it creates a pretty large margin of error—which makes the poll a bit of a shaky foundation for arguments about a growing gender gap. (CJR’s Brian Crowley has previously written about the problems with reporting on polling subgroups.)

This doesn’t mean that reporters shouldn’t pay attention to the specific ways in which campaigns court women voters. (As Ward wrote in his lede, “The war over women is on.”) But we shouldn’t get carried away by flashy poll numbers. And as for the idea that Ohio women will be a key voter demographic in this election, or that the gender gap is growing, only time—and some more data—will provide proof.

T.C. Brown covered government and politics in the Ohio Statehouse Bureau for The Plain Dealer of Cleveland for more than 17 years, and he has also written for other local, state and national publications. Brown is a founding partner in Webface, a social media communication company.