Thursday, February 28, 2008

Average Class Size - Are We Looking at the Wrong Measure?


The DOE’s recently updated average class size reports undoubtedly have their share of errors and flawed categorizations at the individual school level. However, these much-awaited reports also offer an occasion to ask a more fundamental question: Is “average class size” a meaningful measure of students’ learning environment, especially in large high schools? Based on class size data from my son’s high school, the answer is a resounding “NO!”

The reasons arise largely from the non-discriminating nature of simple averages and their unfortunate tendency to disguise information when the data is not concentrated closely together. Consider the situation where ten guys who each make $40,000 per year are sitting in a bar when Alex Rodriguez walks in (with his $25 - $40 million or more annual income). The average income of the customers in the bar is now somewhere around $3 million. This example shows how averages are easily pulled toward extreme values (mathematicians call them “outliers”) while other measures like the median (the value of the middle data point when they are all placed in ranked order) stays the same (in this case, $40,000).

A single (and by no means the worst) example from my son’s school’s data will help illustrate the problem and demonstrate why the DOE’s report is so misleading in their favor. Based on the school’s class master list for English 9, the DOE would have reported an average class size of 29.4, with 14 classes serving a register of 411 students. However, our actual class sizes were (in descending order): 34, 34, 33, 33, 33, 33, 33, 31, 31, 28, 26, 24, 23, and 15. What can we say about this 9th grade English learning environment?

A. The DOE’s reported class size average would have been 29.4 (due to timing differences, they reported it as 28.5 as of 1/23/08 - my data is from late February, after the Spring Semester began).
B. The median (middle) class size was 32 (the average of the seventh and eighth classes in the list).
C. The median (middle) student in this group (the 206th student in descending order) was in a class of 33.
D. 90% of these students were in classes whose average size was 31.1.
E. 80% of these students were in classes whose average size was 32.3.
F. 71.8% of these students were in classes of 31 or more.

All of these very reasonable mathematical ways of looking at the data suggest that our school’s 14 English 9 classrooms are substantially more crowded than implied by the DOE’s simple arithmetic average. From a parent’s viewpoint, the picture is much more disturbing than being told the average class size is 29.4. However, we would never know this because the DOE does not release the underlying data from which the averages are derived.

This pattern holds for every core academic course in our school, as shown below (the column labels A – F correspond to the measures in the example above).

Course --------- A ------- B ------- C ------- D --------- E --------- F

English 9 ----- 29.4 ---- 32 ------ 33 ------ 31.1 ----- 32.3 ----- 71.8%
English 10 --- 28.9 ----- 34 ------ 34 ----- 33.8 ----- 33.9 ----- 91.9%
English 11 ---- 30.0 ---- 33.5 ---- 34 ----- 32.8 ----- 33.5 ----- 87.1%
English 12 ---- 28.7 ---- 30 ------ 30 ----- 29.7 ----- 31.4 ----- 46.3%
Global 9 ------ 28.6 ---- 32 ------ 32 ----- 31.7 ------ 31.9 ----- 67.8%
Global 10 ----- 27.0 ---- 30 ------ 30 ----- 30.0 ----- 30.1 ----- 24.6%
US History --- 28.1 ---- 30 ------ 30 ----- 30.3 ----- 30.7 ----- 30.5%
Biology ------- 29.5 ----- 31 ------ 32 ----- 31.3 ----- 32.0 ----- 56.2%
Earth Sci. ----- 26.6 ---- 29.5 ---- 30 ----- 29.7 ----- 30.4 ----- 36.5%
Chemistry ---- 32.7 ---- 33 ------- 33 ----- 33.1 ----- 33.3 ----- 91.9%
Physics ------- 33.4 ----- 34 ------ 34 ----- 33.6 ----- 33.8 ----- 100%
Math B (1) --- 26.5 ------ 31 ------ 33 ----- 29.8 ----- 32.3 ----- 66.1%
Math B (3) --- 30.7 ----- 32 ------ 32 ----- 31.6 ----- 32.0 ----- 58.1%
Algebra 1 ---- 23.9 ----- 28 ------- 30 ----- 26.1 ----- 29.1 ----- 47.3%

These numbers tell me, as a parent leader, that 80% of the students in my son’s school are in classes whose average size exceeds 30 (often substantially exceeding, as shown in Column E) in every single core course except ninth grade Algebra 1. THAT is what the typical day looks like for the vast majority of these young people, and THAT is what the DOE does not want us to know. The DOE’s happy-go-lucky numbers, typically in the high 20’s, only occur because a few pull-out classes of 15 or 18 or 20 have dragged the average downward. The effect in smaller schools is even greater -- the DOE will report a school with (say) 4th grade classes of 30, 30, 30, and 18 as having an average class size of 27, a figure no reasonable person would accept as reflective of that school's classroom reality.

Of course, we want every measure of class size to be reduced, but as it now stands, the measure we are most often given by the DOE stands as the least representative and most misleading through understatement of all the possible class size measures. It is a classic example of a pickpocket's misdirection: "Keep looking here while the real action (emptying your pockets) is somewhere else." Shouldn’t we be demanding a truer picture of the classroom environment in our schools for the “average student?”

1 comment:

jd2718 said...

Add a class in each subject with no students, and you would see a real fall in average class size.

More seriously, multiply each kid by the size of his class, divide the sum by the number of kids, and you'll get the average size of a class from the student's point of view (short version, square each class size, add'em up, divide by kids). I get 30.3 for your example.

Jonathan