The Language Log post also emphasizes that mass media reports of such surveys sometimes quote numbers completely different from the actual survey results, presumably to increase the value of the news story. So:
In the passage quoted above, Robin Young states the survey result incorrectly — actually, 73% of respondents, not 28%, were able to name one of the constitutional freedoms – and she spins it in a doubtful direction to boot, because only 65% were able to name one of the Simpsons characters.
In the cited New York Times article, Diane Ravitch is referring to the 2010 NAEP 12th grade U.S. History test, in which 82%, not 2%, of 12th graders correctly identified Brown v. Board of Education.
In addition to discounting "public ignorance" surveys, we should discount surveys and other factual information reported through such media.
This Language Log post gives a much better idea of what's going on. 28% was the number for "more than one" of the constitutional freedoms, which was later commonly misquoted as "one or more". And, of course, there's the matter of picking out a point of the distribution which is the most striking.
In other words: nobody is actually lying about the survey results. Instead, the falsehood is distributed along the chain: the press release states the results in a deliberately misleading way, and subsequent reports on it simply aren't careful to avoid being misled.
How many times have you heard a claim from a somewhat reputable source like "only 28 percent of Americans are able to name one of the constitutional freedoms, yet 52 percent are able to name at least two Simpsons family members"?
Mark Liberman over at Language Log wrote up a post showing how even when such claims are based on actual studies, the methodology is biased to exaggerate ignorance:
If, every time you heard a claim of the form "Only X% of Americans know Y" you thought "there's something strange about that", then you get 1 rationality point. If you thought "I don't believe that", then you get 2 rationality points.