When designing each school sample, there are five objectives that underlie the process of determining the probability of selection for each school and how many students are to be sampled from each selected school containing the respective grade:
The goal in determining the school's measure of size is to optimize across the last four objectives in terms of maintaining the accuracy of estimates and the cost effectiveness of the sample design.
In order to meet the target student sample, and achieve a reasonable compromise among the other four objectives above, NAEP sample design staff used the following algorithm to assign a measure of size to each school based on its enrollment per grade as indicated on the sampling frame.
The measures of size vary by grade, enrollment size, and within school target sample size. In general, very small schools will be sampled at one quarter the rate of larger "take-all-student" schools. The initial measures of size (MOS) were set as follows:
Eighth and twelfth grades:
where xjs is the estimated grade enrollment for grade j in school s, and PSU_WTs is the PSU weight for school s.
A school with more than 15 percent Black and Hispanic students and at least 10 Black or Hispanic students in the sample grade is in the high Black/Hispanic stratum for NAEP. The measures of size for schools in the high Black/Hispanic stratum are doubled to increase their chances of selection:
The next task in this development is to describe bj, the constant of proportionality for each grade. It is a sampling parameter that, when multiplied with a school’s preliminary measure of size (Mjs), yields the school’s final measure of size. It is computed in such a way that, when used with the systematic sampling procedure, the target student sample size is achieved. For public schools, bj is 0.000039192 for fourth grade, 0.000069935 for eighth grade, and 0.000132007 for twelfth grade.
The final measure of size, Ejs, is defined as:
The quantity uj (the maximum number of “hits” allowed) in this formula is designed to put an upper bound on the burden for the sampled schools. For public schools, uj is 1.
Schools were ordered within each jurisdiction using the serpentine sort described under the stratification of public schools. A systematic sample was then drawn using this serpentine sorted list and the measures of size. The numbers of schools selected were about 360, 460, and 670 for fourth, eighth, and twelfth grades, respectively.