What causes headaches?
—MIKE A., WILMINGTON, DEL.

Dawn A. Marcus, associate professor at the University of Pittsburgh School of Medicine's department of anesthesiology, offers this answer:

Although they may feel as if they emanate from the brain, headaches actually arise as a result of irritation in nearby structures: skin, joints, muscles, nerves or blood vessels. Brain tissue, encased in the protective coating of the skull, has not evolved the ability to respond to pressure sensations.

Clinicians classify all headaches as either secondary or primary. Secondary headaches, which appear as symptoms of an underlying disorder, have no uniform cause. Anything from a pinched nerve to a sinus infection can lead to secondary head pain.

Most headaches, however, are primary, meaning that the headache is not a symptom of another condition but the problem itself. Research suggests that this type—which includes tension headaches and migraines—may derive from a single, identifiable pathway.

The chain begins when pain centers in the brain are activated, at which point they produce neurotransmitters such as serotonin and norepinephrine. These chemicals call for expansion of meningeal blood vessels enveloping the brain, resulting in increased blood flow. As the vessels swell, they stretch the nerves that surround them. These nerves, in turn, convey signals to the trigeminal system, an area of the brain that relays pain messages for the head and face, and we perceive pain.

Why the pathway is initiated at all is still an open question, although some circumstances seem to make headache onset more likely. These triggers may be internal (for example, hormonal changes during menstruation) or environmental (such as stress or sleep deprivation). So far, however, most evidence for what factors are directly responsible is anecdotal, and the mechanism by which the triggers are converted to chemical signals is little understood.

How can a poll of only 1,004 Americans represent 260 million people?
—C. BURES, WELLESLEY, MASS.

Andrew Gelman, professor in the departments of statistics and political science at Columbia University, explains:

You can learn a lot about a large population from a smaller cross section—but that does not make the technique flawless.

Mathematically, the margin of error depends inversely on the square root of the number of those sampled; however, the margin of error is an abstraction based on tacit assumptions. In practice, actual errors may be larger than advertised.

One assumption is that the queried group is truly random—that respondents have been chosen one at a time, with everyone in the U.S. equally likely to have been picked. To approximate this ideal, polls use telephone numbers generated randomly by a computer. But if you do not have a phone, you will not be in the survey, and if you have two lines, you have two chances to be included. Another confounding factor is that women, whites, older people and college graduates are more likely to agree to be interviewed. Statistical weighting helps pollsters match the sample to the population, but they can counter only known biases.

Finally, any margin of error is an understatement, because opinions change. For instance, surveying 4,000 people would improve the margin of error to 1.5 percent. Although this sounds appealingly precise, it is generally a waste of time, because public views vary enough day to day that it is meaningless to attempt too exact an estimate. It would be like getting on a scale and measuring your weight as 173.26 pounds; after you drink a glass of water five minutes later, your precise weight would have changed but to an unimportant degree.

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