Formulas for School District Estimators and Evaluation Statistics
The generic form of the school district estimators is as follows Let
be the quantity of
interest (poor 5-17, population 5-17, or total population) for school
district j in county i. Also define
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
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, 
come from the 1990 census, and the other
quantities 
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, in (1) come from the 1980 census
(and will be written 
to distinguish this), and the other quantities 
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
Alternative Estimates for Comparison in Evaluations
Two alternative estimates are considered for purposes of evaluation
against 1990 census results. The census county-based estimate
replaces
in (1) by
the 90 census estimate for county i, 
This yields
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 
Note that if these ratios don't change from 1980 to 1990,
comes out to be
the standard.
The naive estimate 
replaces the county i census growth ratio
in (3),
with the corresponding ratio for the total U.S.:
only by the crude
measure of growth at the national level 
. 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
The mean algebraic percent error is
sums over all
districts involved in the particular comparison, and n (=) 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
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
for all ij that
The following
weights were used:
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.
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