Just a quick followup to yesterday's post about NAEP reading scores. Over at The Quick and the Ed, Chad Aldeman suggests that reading scores have improved more than I suggested — possibly due to a statistical anomaly called Simpson's Paradox1 that can produce different results for an entire group than it does for all its subgroups. But that's not what's going on here. The main issue is that he's looking at different test scores than I am: he uses the Long Term NAEP for 4th graders to look at progress since the mid-70s, while I used the Main NAEP for 8th graders to look at progress since 2002. The former shows significant progress for all ethnic groups, while the latter is virtually flat for every group except Asians.
Click the link for more. I don't have any special axe to grind about the use of either version of NAEP or about which timeframe is appropriate to look at. It all depends on what you're investigating. But it's worth knowing that different data is out there.
1A simple example is below the fold. I either wrote this myself about a decade ago or else copied it from somewhere else. I'm not sure which.
A sex discrimination case in California a while ago has become a sort of classic illustration of Simpson's Paradox. Looking at the proportion of women admitted to the graduate school at the University of California, some women sued the university claiming they were being discriminated against by the graduate school. When administrators looked for which departments were most guilty, however, they were a little astonished to find that there was actually a positive bias for women.
To keep things simple, let's suppose there were only two departments in the graduate school, economics and psychology. Making up numbers, let's further assume that 70 of 100 men (70 percent) who applied to the economics department were admitted and that 15 of 20 women (75 percent) were. Assume also that five out of 20 men (25 percent) who applied to the psychology department were admitted and 35 of 100 women (35 percent) were. Note that in each department a higher percentage of women was admitted.
If we amalgamate the numbers, however, we see what prompted the lawsuit: 75 of the 120 male applicants (62.5 percent) were admitted to the graduate school as a whole whereas only 50 of the 120 female applicants (41.7 percent) were.
Such odd results can surface in a variety of contexts. For example, a certain medication X may have a higher success rate than another medication Y in several different studies and yet medication Y may have a higher overall success rate. Or a baseball player may have a lower batting average than another player against left-handed pitchers and also have a lower batting average than the other player against right-handed pitchers but have a higher overall batting average than the other player.