Contradictory statistics in news source? Here are some interesting statistics at the bottom of a news article from today:

"Up to 6% of children have food allergies, ..."
  "Nearly nine in 10 schools nationally had one or more students with food allergies."

I readily found a 15-year-old source that showed that, for public schools at least, the average population of schools was generally high enough that most schools have a very high likelihood of food allergies, being on the order of as few as 160 and probability of allergy around 0.00005. Of course that table doesn't say anything about the distribution other than the mean or anything about non-public schools, which are likely to be smaller.
I find it very unlikely that there could be that many schools without allergies if the incidence rate is that high and a typical school has a few hundred students. Is there any relatively simple ways to (in)validate the plausibility of those two figures?
 A: I had the same initial reaction.  However, I think that a few very small schools can account for this.
Suppose that school size were distributed uniformly in log-space.  I don't know school size distribution statistics, but it wouldn't surprise me if this were the case.
I did a simple mathematical experiment, letting school sizes range from 5 to 5000 using 30 samples.  Then I took the mean probability over the 30 of having no children with allergies.  I got about 14%, which wasn't too far off the 1 in 10.


schls=5*10.^[0:0.1:3];
    mean(.94.^schls)


ans =
0.1357

A: These results are highly consistent for moderately sized schools.
prob of school with at least 1 allergic child = 0.9
prob of school having no allergic child = 0.1

prob of allergic child = 0.06
prob of nonallergic child = 0.94
prob of 10 non allergic children = 0.94^10 
prob of 100 non allergic children = 0.94^100 = 0.002

Verifying other assumptions about school size, etc. is beyond the scope of the report. You wouldn't be able to falsify any findings without obtaining those data.
