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NAEP Technical DocumentationComputation of Measures of Size

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.

These five principles are not fully compatible, conflicting with one another to a greater or lesser extent, depending on the size and structure of the student and school populations.

In certain jurisdictions, a census of students was taken so as to meet, as nearly as possible, the target student sample size. These were:

• Delaware,
• District of Columbia,
• Department of Defense (DoD) schools,
• Atlanta, Georgia,
• Boston, Massachusetts (fourth grade only),
• North Dakota,
• South Dakota (fourth grade only),
• Vermont, and
• Wyoming.

For these jurisdictions, one can view the measure of size for each school as equal to the expected number of hits in the school, and the upper bound for the burden is unlimited in the sense that all eligible students are selected.

Elsewhere, 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 measure of size differs between jurisdictions that participated in all three assessments (reading, mathematics, and science) and jurisdictions that participated in two assessments (reading and mathematics). The former set of jurisdictions were called "reading-mathematics-science jurisdictions" below (abbreviated in some cases as "rms"), and the latter set are called "reading-mathematics-only jurisdictions" below (abbreviated in some places as "rmo"). For reading-mathematics-science jurisdictions the preliminary measures of size (MOS) were set as follows:

where x_js is the estimated grade enrollment for jurisdiction j and school s.

For reading-mathematics-only jurisdictions the measures of size were as follows:

For Puerto Rico the measures of size were set as follows:

The next task in this development is to describe b_j, the constant of proportionality for a specified jurisdiction. It is a sampling parameter that, when multiplied with a school’s preliminary measure of size (M_js), 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.

The final measure of size, E_js, is defined as:

The quantity u_j (the maximum number of “hits” allowed) in this formula is designed to put an upper bound on the burden for the sampled schools. In most jurisdictions, u_j was set to 3. In Alaska, u_j was set to 8, and in Austin for eighth grade only, u_j was set to 4.

In addition, new and newly eligible schools were sampled from the new-school frame. The assigned measures of size for these schools

used the b_j and u_j values from the main school sample for the jurisdiction (i.e., the same sampling rates as  for CCD schools within each jurisdiction).  is the probability of selection of the district into the new-school district sample.

After applying this procedure, there were five jurisdictions at grade 8 where a very high proportion of schools were included, but not all. These jurisdictions were:

• Hawaii,
• Rhode Island,
• Boston, Massachusetts,
• Charlotte, North Carolina, and
• Austin, Texas.

To achieve a more robust (and readily understood) design, it was determined that in these cases all eighth-grade schools in these jurisdictions were included in the sample with certainty, and the number of hits per school was adjusted accordingly so that the target student sample size would be achieved.

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