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:
to meet the target student sample size for each grade,
to select an equal-probability sample of students,
to limit the number of students that are selected from a school,
to ensure that the sample within a school does not include a very high percentage of the students in the school, unless all students are included, and
to reduce the rate of sampling of small schools, in recognition of the greater cost and burden per student of conducting assessments in such schools.
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. Although in some years sampling rates differ between private and public schools, in 2006, student sampling was the same for both school sample types. The preliminary measures of size (MOS) were set as follows:
Eighth and twelfth grades:
xjs is the estimated grade enrollment for grade j in school s,
PSCHWTs= the Private School Universe Survey (PSS) area frame weight for school s, computed by the Census Bureau, and
PSU_WTs = the Primary Sampling Unit (PSU) weight for school s.
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 (MOSjs), 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 private schools, bj is 0.000084541 for fourth grade, 0.000147658 for eighth grade, and 0.000304864 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 private schools, uj is 1.
Schools were ordered within each jurisdiction using the serpentine sort described under the stratification of private schools. A systematic sample was then drawn using this serpentine sorted list and the measures of size. The numbers of schools selected were about 140, 160, and 170 for fourth, eighth, and twelfth grades, respectively.