When comparing groups, typically, the following formula is used:
The difference between the means is divided by the pooled standard deviations.
Often, sample sizes are not presented directly. But usually, they can be determined. For example, the NAEP explorer gives percentages which when multiplied by 100 can be used as N(a) and N(b) in our formula above. For example, below were the average and subtest Math TIMSS 2007 scores for grade 8:
Whites made up 55% of the sample and Blacks made up 12%. The pooled standard deviations then were: SQRT((55*SD^2+12*SD^2)/(55+12). And the standardized differences were (White score – Black score)/pooled SD. These are presented on the right hand side of the figure above.
It will be noticed that the average B/W difference is larger than the average of the subtest differences. This is because the average is a composite score and because composite scores are calculated thusly:
Importantly, the magnitude of the composite score is a function of the correlation between predictors/subtests.
Taking our sample above, there are 7 subtests (Numbers, Algebra …Reasoning). The average magnitude of the B/W difference on these is 0.95. The summed magnitude is 6.66. The average correlation between subtests is about 0.7. So we get:
Composite d = 6.66 / SQRT(7+7(7-1)*0.7)
Which should come out to about to a Cohen’s d of 1.1.