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NAEP Technical DocumentationComputation of Measures of Size for the 2007 State Assessment

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 any one 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 certain jurisdictions, a census of students was taken to meet, as nearly as possible, the target student sample size. Elsewhere, to meet the target student sample and achieve a reasonable compromise among the other four objectives above, the 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 differ between grades and between jurisdictions that participated in all three assessments (reading, mathematics, and writing) and jurisdictions that participated in two assessments (reading and mathematics). The former set of jurisdictions are called "reading-mathematics-writing jurisdictions" below (abbreviated in some cases as "rmw"), and the latter set are called "reading-mathematics-only jurisdictions" below (abbreviated in some places as "rmo").

For fourth grade, 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 eighth grade reading-mathematics-writing 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 eighth grade reading-mathematics-only jurisdictions, the measures of size were as follows:

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

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

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

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 (MOS_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 Puerto Rico, u_j was set to 1.

In addition, new- and newly-eligible-schools were sampled from the new-school frame. The assigned measures of size, E_js, 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 Common Core of Data schools within each jurisdiction). The variable  is the probability of selection of the district into the new-school district (d) sample.

After applying this procedure, there was one jurisdiction in fourth grade (Department of Defense schools) and two jurisdictions in eighth grade (Hawaii and Rhode Island) where a very high proportion of schools were included, but not all. 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.

For three jurisdictions in fourth grade (Bureau of Indian Education [BIE], District of Columbia, and Wyoming) and 10 jurisdictions in eighth grade (BIE, District of Columbia, Delaware, Department of Defense schools, North Dakota, Vermont, Wyoming, Atlanta, Boston, and Cleveland), the number of students on the sampling frame was smaller than the target sample. Also, in the eighth-grade sample for Austin, the target sample size accounted for more than 90 percent of all students on the frame. For these jurisdictions, all students in all schools were sampled.

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