Department of Awful Statistics: Small Schools Edition

It seems even Bill Gates can be taken in by poor understanding of statistics.  Specifically, this one:

The chart at left, for example, shows by size the percentage of schools in North Carolina which were ever ranked in the top 25 of schools for performance.  Notice that nearly 30% of the smallest decile (10%) of schools were in the top 25 at some point during 1997-2000 but only 1.2% of the schools in the largest decile ever made the top 25.

Seeing this data many people concluded that small schools were better and so they began to push to build smaller schools and break up larger schools. Can you see the problem?

The problem is that because small school don't have a lot of students, scores are much more variable.  If for random reasons a few geniuses happen to enroll one year in a small school scores jump up and if a few extra dullards enroll the next year scores fall.   

Thus, for purely random reasons we would expect small schools to be among the best performing schools in any givenyear.  Of course we would also expect small schools to be among the worst performing schools in any given year!  And in fact, once we look at all the data this is exactly what we see.  The figure below shows changes in fourth grade math scores against school size.  Note that small schools have more variable scores but there is no evidence at all that scores on average increase with school size. 

The Gates foundation, which pushed for smaller schools on the basis of this finding, has apparently now acknowledged the mistake.  Victory: math!