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Procedures for Estimating Group Scale Score Statistics and Their Standard Errors
The objective of the National Assessment of Educational Progress (NAEP) is not to compare the scale scores for individuals, but to estimate population and subpopulation characteristics. When doing this, estimating the variability of the statistics of interest is necessary. When survey variables are observed without error from every respondent, usual variance estimators quantify the uncertainty associated with sample statistics from the only source of uncertainty, namely the sampling of respondents. Item-level statistics for NAEP cognitive items meet this requirement, but scale score values do not. The Item Response Theory (IRT) models used in their construction posit an unobservable scale score variable θ to summarize performance on the items in a scale. The fact that θ values are not observed even for the respondents in the sample requires additional statistical analyses to draw inferences about θ distributions and to quantify the uncertainty associated with those inferences.