U.S. Census Bureau
APPENDIX

Formulas for School District Estimators and Evaluation Statistics

The generic form of the school district estimators is as follows Let formula be the quantity of interest (poor 5-17, population 5-17, or total population) for school district j in county i. Also define

formulas

For a school district whose boundaries cross county lines, the subscript j corresponding to that district for county i will indicate the piece of the school district in county i. The ij thus can denote either pieces of school districts or complete school districts that are contained in county in county i. When a school district is split across two or more counties, the estimate formula for that district is obtained by summing estimates for all of its pieces obtained from (1) across all the counties i that it intersects. For simplicity of discussion, this complication will be be ignored in what follows.

For the 1995 school district estimates, formula come from the 1990 census, and the other quantities formula refer to income year 1995 (for 5-17 poverty) or to calendar year 1996 (for population). For the evaluation of 80-based school district estimates against the 1990 census results, formula in (1) come from the 1980 census (and will be written formula to distinguish this), and the other quantities formula refer to income year 1989 (for 5-17 poverty) or the calendar year 1990 (for population), as do the 1990 census results. In fact, for purposes of evaluation we make the assumption that

formula

Of course, this assumption is untrue as the census poverty figures are themselves estimates containing nontrivial amounts of sampling error (for small to moderate size school districts), and both poverty and population figures can contain varying amounts of nonsampling error.

Alternative Estimates for Comparison in Evaluations

Two alternative estimates are considered for purposes of evaluation against 1990 census results. The census county-based estimate formula replaces formula in (1) by the 90 census estimate for county i, formula This yields

formulas

This "estimate" in not usable for 1995, of course. It is included in the evaluations merely for purposes of comparison of its performance to that of the model-based estimator (1). Note formula i.e., the estimates from (3) aggregated to counties agree with the 90 census results, which are being used as the standard of comparison. Thus, evaluating the Census county-based estimator (2) shows, to the extent these evaluations can examine this, the conributions to error of change from 1980 to 1990 in the school-district-to-county ratios formula Note that if these ratios don't change from 1980 to 1990, formula comes out to be formula the standard.

The naive estimate formula replaces the county i census growth ratio formula in (3), with the corresponding ratio for the total U.S.:

formula

This estimate changes the previous census school district estimates formula only by the crude measure of growth at the national level formula. This growth ration refers to total population for evaluating estimates of same, or to 5-17 population for evaluating estimates of 5-17 population or poor children 5-17.

Evaluation Statistics

The same evaluation statistics are computed for all three estimates (model-based, Census county-based, naive). Explicit formulas will be given here only for the model-based estimates formula

The mean algebraic percent error is

formula

where formula sums over all districts involved in the particular comparison, and n (=formula) is total number of such school districts. The number n varies over the various evaluations shown, which include groupings of school districts into categories according to various characteristics (e.g., size). The MALP is a measure of bias (relative to the 90 census).

The mean absolute percent error is

formula

where formula and n are defined as for (5). The MAPE is a measure of accuracy (with respect to the 90 census results).

The above two measures are unweighted statistics. Corresponding weighted statistics are

formula

where formula for all ij that formula The following weights were used:

formula children 5-17 in the 1990 census for school district in j in county i.

The same weights are used for all three sets of evaluations (poor 5-17, 5-17 population, and total population). Later evaluations will examine use of different sets of weights, e.g., weights proportional to 5-17 poor for evaluating the 5-17 school district poverty estimates.


Source: U.S. Census Bureau, Population Division,
Administrative Records & Methodology Research Branch
Author: Esther R. Miller

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Created: May 29, 2002
Last Revised: October 31, 2011 at 10:03:09 PM