Myths about menstrual cycles are a great example of why carefully collected data is important. Without that, it's extremely easy to see a pattern where none exists. For instance, there's no good evidence that menstrual cycles have anything to do with cycles of the Moon.
The evidence is also against there being such a thing as ladies synchronizing their menstrual cycles. I know. I know. It totally happened to you in college. I thought it happened to me, too. And there are few scientists who think the phenomenon is real. But the preponderance of evidence seems to be against them. Again (and this cannot be said enough) humans are really good at spotting patterns—even when patterns don't exist.
Kate Clancy, an anthropologist who studies the evolutionary medicine of women’s reproductive physiology, has a post on Scientific American blogs looking at the research that's poked holes in the synchronized periods hypothesis. A couple of the studies she talks about came out this year.
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Maybe we should look to our primate relatives for evidence, then: in fact two papers have come out this year testing this hypothesis in primates! Setchell et al (2011) observed semi free-living mandrills, which is a kind of Old World monkey, a group to which the Great Apes belong. Out of ten observation-years of data, they found a single year that had significant synchrony… only to have that one year fail to be significant once they corrected for multiple testing. Multiple testing corrections are important because of the chance that if you test a hypothesis enough times you will get a spurious significant result (and for a brilliant take on this, see this xkcd comic).