## 6. TOTAL VARIANCE OF THE LINKING FUNCTION

Table 10 collects together all of the components of the variance of . Shown in the table for the values of Var(x) and x from Table 2, are the overall variance of , the standard error of , and the percentage of the total variance attributable to the naive variance estimate, to the sampling variance component, to the measurement error component, and to the component due to temporal shift. The number of times that the total variance is larger than the naive variance is 100 divided by the percentage of total variance attributable to the naive variance. Consequently, the total variance is at least 5 times larger than the naive variance, and so, the standard error with all components included will more than double that of the naive standard error. The sampling component is by far the most important component of the variance, accounting for 76 to 80 percent of the total.

Table 10.—Total variance of with percentages due to components for grade 8

 Subject Total Standard Error () Percentage of total variance due to: Naive Var Sampling Measurement error Temporal shift Mathematics x = U.S. mean 35.83 5.99 20 % 79 % 1 % 0.0 % x = U.S. 90th percentile 42.52 6.52 17 % 80 % 4 % 0.0 % Science x = U.S. mean 38.71 6.22 19 % 76 % 2 % 2.4 % x = U.S. 90th percentile 47.40 6.88 16 % 77 % 5 % 2.0 %

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