Many Americans have strong opinions about policy issues shaping the presidential campaign, from immigration to Social Security. But their grasp of numbers that underlie those issues can be tenuous.

Americans vastly overestimate the percentage of fellow residents who are foreign-born, by more than a factor of two, and the percentage who are in the country illegally, by a factor of six or seven. They overestimate spending on foreign aid by a factor of 25, according to a 2010 survey. And more than two-thirds of those who responded...