Paul Krugman is a talented economist, but a boring columnist, so I do not read him very often. But I get annoyed when he says things that an economist ought to know better than to say. Today, he comments on Bush's fluctuating polls.

But anyone who takes the trouble to look at the numbers knows that the thrill is gone. Mr. Bush's approval ratings have plunged over the last two months. A year ago he was, indeed, immensely popular; right now he's not significantly more popular than he was before Sept. 11.

Let me explain with an example. Using Excel, I produced an imaginary set of people, all of whom put, on average, a value of 0.5 (on a scale of 0 to 1) of their approval of George Bush. Their feelings fluctuate randomly and evenly, however, between 0.4 and 0.6. When the pollster comes around, however, they are asked only the yes-no question of whether they approve of Bush. If at the time they are asked, their answer is greater than 0.5, they are recorded as 1; if their answer is less than 0.5, they are recorded as 0.

I then used Excel to simulate running this poll twenty times. Using the yes-no answers, the twenty polls generate approval ratings from 44.2% up to 53.1%. If instead, I asked people to give their actual feeling, on a 0-1 scale, the poll results range from 49.5% to 50.3%.

Always be leery of someone who says, "Look at the facts". He is likely leading you to "Look at my interpretation of the facts."

I should mention that this idea comes from Charles Manski at Northwestern University, in a paper titled "Probabilistic Polling", which is not online but was published in Navigating Public Opinion: Polls, Policy, and the Future of American Democracy.

That's interesting. Thanks for sharing, as they say.