The estimation of the sampling variability of any statistic must take into account the sample design. Observations of different students cannot be assumed to be independent of each other (and are generally positively correlated) because of the effects of cluster sampling (students within schools, schools within primary sampling units), nonresponse, and poststratification adjustments. To account for the differential probabilities of selection and the various adjustments, each student has an associated sampling weight, which is used in the computation of any statistic and which is subject to sampling variability. Ignoring the special characteristics of the sample design, and treating the data as if the observations are independent and identically distributed, will generally produce underestimates of the true sampling variability due to the clustering and unequal sampling weights.
The proper estimation of the sampling variability of a statistic based on the NAEP data is complicated and requires techniques beyond those commonly available in standard statistical packages. Fortunately, the jackknife procedure provides good quality estimates of the sampling variability of most statistics, at the expense of increased computation, and can be used in concert with standard statistical packages to obtain a proper estimate of sampling variability.