Devlin's Angle

July/August 2005

When numbers matter

"Power cables linked to cancer" was the attention grabbing headline in Britain's Guardian newspaper on June 3. "Children living near high-voltage power lines are substantially more likely to develop leukemia, researchers from Oxford University and the national grid report today in the British Medical Journal," the article began. "Those living within 200 metres of the overhead cables were 70% more likely to develop the disease than similar children living more than 600 metres away."

Now, Britain's Guardian newspaper is hardly a supermarket tabloid that deals in sensational stories of dubious authenticity, but a highly respectable national daily newspaper. And the claims it reported had been accepted for publication in the British Medical Journal. So when I read the article, I had little doubt that the facts reported were true, at least in a literal sense. But that 70% increased risk figure seemed way, way too high to be true, in the sense that I, and presumably most other readers, initially understood it.

To get a better appreciation of that dramatic - and scary - 70% figure, let's consider a hypothetical, numerically simplified scenario. Suppose we have a population of 100,000 children, half living near a power line, half not. Suppose that 10 of the children who do not live near a power line develop leukemia over a given period while 17 who do live near a power line also develop leukemia over the same period. Thus, there are 70% more leukemia cases among the children living near power lines than among those who do not live near power lines. (I took the figures I did precisely to get me that 70% increase; 17 is 70% greater than 10. In a moment I'll check to see how realistic my estimates are.)

Now look not at the percentages in my example but the actual numbers of leukemia cases involved. The 27 cases in total amount to a mere 0.027% of the population. Breaking this figure down into the two groups, the risk factor for children not living near power lines is 0.010%, the risk for children who do live near power lines is 0.017%. True enough, the latter figure is 70% greater than the first, but these are tiny risks. How realistic are my figures? A Google search reveals that a typical incidence figure for leukemia is 4 to 5 new cases a year per 100,000 people. So if we set the time period in my example at five years, my estimates - guesses would be more accurate - are pretty realistic. In short, the incidence of leukemia is truly tiny, regardless of where you live in relation to power lines.

This is not to say that the difference is not statistically significant. (And it is certainly not to diminish the anguish of the small number who do fall victim to the disease.) Even if we assume that there is a significant correlation between living near power lines and the incidence of leukemia, however, what conclusions can be drawn from the figures? The Guardian article was careful not to conclude that exposure to high voltage power lines can cause leukemia, but many readers will surely make that assumption. Yet as anyone who has taken a college statistics course will know, correlation is not causation. While there may remain the possibility that long-term exposure to the electromagnetic fields surrounding high voltage power lines can lead to cancer, there is another possibility that is, to my mind, far more likely.

What kind of people live close to power lines? By and large, poor people. The pylons are ugly and the lines emit a constant hum, particularly in wet weather. Property values are dramatically lower near power lines. Consequently, people who can afford to live elsewhere, generally choose to do so. Personally, I would not live near a power line (and I don't) if I had a choice (which I do). Yes, the mere possibility that power line exposure might be detrimental to my health is one of the factors that I take into consideration. But that is part of a whole range of lifestyle choices aimed at living a long, physically healthy life, surrounded by psychologically invigorating scenic beauty (free of nearby power lines), eating good food, and exercising regularly. If I had to put my money on it - and unlike many people who live near power lines I have money to spare that I could waste on such a wager - I would say that it is likely to be one or more factors associated with poverty that give rise to an increased incidence of leukemia, not the effects of an electromagnetic field. To provide even more perspective, let me add that, given the rare overall incidence of leukemia anyway, from a numerical point of view the disease is likely to be one of the least significant effects of poverty on growing children, and if we do want to take action, it should be directed against poverty, not power distribution companies.

There may indeed be a significant message regarding child health in the Guardian article. But I think it's about poverty, not power lines.

In fact, the Guardian article carries another message, the one that this column is directed toward. Namely, that numbers, being powerful, are dangerous unless handled properly. What looks on the face of it like an alarming statistic that demands immediate and drastic action turns out after a few moments reflection to be nothing of the kind. Now, when I made my initial estimate of the risks, I had no idea what the real leukemia incidence figures are for a randomly chosen population of 100,000 people. I simply picked numbers that I guessed were roughly of the right order, and which got me quickly to that reported 70% increase figure. I could have looked up the real data behind the Guardian story. But that's not the point - which is why I didn't. The real issue as I see it is that it generally doesn't require more than a few minutes with some sensibly chosen hypothetical figures in order to get a general sense of what a particular statistic might really mean. Do we teach students - tomorrow citizens and politicians - to be able to approach in this way the statistics they read? I hope we do. But I worry that we don't.

Let me leave you with another statistical problem to ponder over the summer months, particularly if your vacation plans call for air travel, when you can pass your time reflecting on the matter while you stand in the security line at the airport. Have the enormous expenditure and inconvenience of the heightened security measures we have put into place led to a net saving of American lives?

You'll have to do some guesswork here to come up with numbers to play with, just as I did above. You'll need to come up with a plausible figure for the number of lives that would have been lost through airborne terrorist incidents if we just had the pre- 9/11 security measures, and then you'll need to produce an estimate for the increased number of road accident deaths caused by the greater numbers of people who, faced with the increased hassle of flying in today's climate, now choose to drive to their destination rather than fly, whenever they have a choice.

A variant of the same problem: would more lives be saved if the money we now spend on airport security were channeled instead into fitting every new automobile with a device that immobilizes it if the person sitting behind the wheel has more than a certain level of alcohol in his or her breath?

Or suppose - echoes of the post World War II Marshall Plan - we divert funds from increased security and fighting overseas wars to making the lives of people in the known terrorist breeding grounds sufficiently rewarding and full of hope that they have reasons not to be suicide terrorists.

I don't know the answer to these questions. I'm not sure there are correct answers. There may be too many unknowns. And I'm certainly not advocating any particular actions. (My suspicion is that locking all cockpit doors eliminated most of the risk of further aircraft hijackings, and all that highly visible stuff at the airport is more for show than anything else. Still, I typically fly around 100,000 miles a year, so I personally have a vested interest in keeping the skies free of terrorists. As for Iraq, I can imagine at least one possibly valid reason for invading Iraq, though it's not the bogus one the nation was given at the time, any more than the dropping of the two atomic bombs in 1945 was really about ending the war in Japan, a case that seems to me to have several similarities to the invasion of Iraq). But that's not the point. The point - at least, my point - is that I'm not sure anyone has really thought about the issues from a simple numerical angle. Not even those who make the decisions on our behalf. (If they have, then why haven't they told us about their considerations?) If no one has made those simple, yet highly informative calculations, that's a great pity. Humankind has had numbers for ten thousand years now. Using numbers, we can make better decisions. Sometimes, that requires the use of sophisticated mathematics. But quite often, a few quick calculations on the back of an envelope are enough. Not necessarily to get "the right answer." Rather, just to get a better grasp of the issues. Maybe there's a shortage of envelopes?

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's newest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs) was published recently by Thunder's Mouth Press.