 NCES 2007-017July 2007

## Appendix A.1. Sampling Errors

The samples used in surveys are selected from large numbers of possible samples of the same size that could have been selected using the same sample design. Estimates derived from the different samples would differ from each other. The difference between a sample estimate and the average of all possible samples is called the sampling deviation. The standard, or sampling, error of a survey estimate is a measure of the variation among the estimates from all possible samples and thus is a measure of the precision with which an estimate from a particular sample approximates the average result of all possible samples.

The sample estimate and an estimate of its standard error permit us to construct interval estimates with prescribed confidence that the interval includes the average result of all possible samples. If all possible samples were selected under essentially the same conditions and an estimate and its estimated standard error were calculated from each sample, then (1) approximately 66.7 percent of the intervals from one standard error below the estimate to one standard error above the estimate would include the average value of all possible samples; and (2) approximately 95.0 percent of the intervals from two standard errors below the estimate to two standard errors above the estimate would include the average value of all possible samples. We call an interval from two standard errors below the estimate to two standard errors above the estimate a 95 percent confidence interval.

To illustrate this concept, consider the data and standard errors appearing in table 104. For the 2005 estimate that 9.4 percent of 16- to 24-year-olds were high school dropouts, the table shows that the standard error is 0.22 percent. The sampling error above and below the stated figure is approximately double (1.96) the standard error, or about 0.44 percentage points. Therefore, we can create a 95 percent confidence interval, which is approximately 8.97 to 9.83 (9.4 percent ± 1.96 x 0.22 percent).

Analysis of standard errors can help assess how valid a comparison between two estimates might be. The standard error of a difference between two independent sample estimates is equal to the square root of the sum of the squared standard errors of the estimates. The standard error (se) of the difference between independent sample estimates "a" and "b" is:

sea,b = (sea2+seb2)1/2

It should be noted that most of the standard error estimates presented in the Digest and in the original documents are approximations. That is, to derive estimates of standard errors that would be applicable to a wide variety of items and could be prepared at a moderate cost, a number of approximations were required. As a result, the standard error estimates provide a general order of magnitude rather than the exact standard error for any specific item. The preceding discussion on sampling variability was directed toward a situation concerning one or two estimates. Determining the accuracy of statistical projections is more difficult. In general, the further away the projection date is from the date of the actual data being used for the projection, the greater the probable error in the projections. If, for instance, annual data from 1970 to 2004 are being used to project enrollment in institutions of higher education, the further beyond 2004 one projects, the more variability there is in the projection. One will be less sure of the 2015 enrollment projection than of the 2007 projection. A detailed discussion of the projections methodology is contained in Projections of Education Statistics to 2015 (National Center for Education Statistics [NCES] 2006-084). Guide to Sources Top