The purpose isn’t to unnecessarily shift anything or normalize anything; in fact, it has broad applications because you can model random (e.g., non-random individual effects,) and fixed effects (e.g., treatment effects).

For our case, we are discussing schools but we may have individual student data. Therefore, we know that a score will be dependent on differences within-schools (i.e., student level differences) and between-schools (the purpose of this discussion). We would “nest” the students within their schools and identify the variability that occurs within- and between-schools. Individual student scores can then vary within their schools because they are dependent on school-specific variables (dependence can quantified by the intra-class correlation coefficient).

By then partialing out the within-school error, we can more accurately identify the quality of schools (at least based on our outcome variability). Valued-added modeling, which is used to measure school and teacher performance, is essentially a special case of this model.

In the case of your discussion on normality, it is true you can find examples (especially with an N of 4) of non-normal data. We shouldn’t just blindly assume a sample is normal and check this assumption. That being said, not only have the variables used to measure school quality are not only theoretically normal, the extensive studies of these variables across fields (economics, psychology, education, etc.) have verified the distribution. Therefore, we are using the theory that as a sample size increases it will converge on its probability distribution of that variable which in this case is the normal distribution.

Overall, I think it is important to know that there will be exceptions to any rule. But your insistance on arguing the exception as if it is the norm does not take away from the legitimacy of the general statement that half the schools will be below average.

This post was edited on 10/2/14 at 9:23 am