Friday, November 15, 2013

The statistics of small numbers: variation of the mean

To illustrate what I wrote in Sowing Confusion and Insecurity, I coded up a random sampling program in Python.  I looked at the California STAR test results for English-learners in 2013. They give a mean, but no standard deviation.  However, I guesstimated one using the distribution of performance levels and the tables of scale score ranges for each performance level and grade.

Not surprisingly, the mean English score for English learners in 8th grade is ~300 while the average for all students is ~360.  I took a WAG (wild-assed guess) at the standard deviation for English learners of 125.  Then I sampled 18 students (blue) and 180 students (green) and computed the sample mean 100 times.  Notice the much larger spread in the smaller sample size mean scores?

This means that the average scores of small groups of kids will jump around more year-to-year than for larger groups of kids.  No amount of wishing or screaming is going to change that.

If we hold schools to the ridiculous standards of NCLB, then schools will either "fail" or learn how to cheat.  The easiest way to game the system is to treat students in the at-risk groups tracked by NCLB like hot potatoes.  If you can push  enough of them out of your school (below the magic threshold of 11 students), they become someone else's problem.  This gives school districts incentive to ghettoize at-risk kids in certain sacrificial lambs schools.

Is a school that pushes out kids who might score low or who belong to small groups tracked by NCLB "growth targets" truly a public school?

Leave a comment if you want a copy of my Python code for your own computer experiments.


  1. Virginia schools have "solved" this problem by having certain groups of kids take the tests. My son claims to have taken more tests than any living human. But they wanted HIM to take the tests - he got a perfect score on his PSAT, SAT and is a very good student.
    So as it stands,Mount Vernon High school gets the boundaries that give it the less accomplished students, and West Potomac gets the more affluent ones. They have ghettoized.

    1. I don't understand how VA can do that.

      In CA, ALL students have to take the state STAR tests unless the parents ask for a test waiver. The school cannot suggest the waiver to parents (as some schools used to do). The # of tested students and total enrollment is listed publicly.

      The students who take the national NAEP test are supposed to be random. Are you saying that VA attains high NAEP scores relative to their peers by cherry-picking test-takers?