POPULATION

DIVISION

Evaluation of the 1990 School District

Level Population Estimates Based on the

Synthetic Ratio Approach

by

Esther R. Miller

Working Paper Series: No. 54

September 2001
WORKING PAPER NO. 54

Evaluation of the 1990 School District

Level Population Estimates Based on the

Synthetic Ratio Approach

by 

Esther R. Miller

Population Division

U.S. Census Bureau 

Washington, DC 20233-8800

This paper reports the results of research and analysis undertaken by

Census Bureau staff. It has undergone a more limited review than

official Census Bureau publications. This report is released to inform

interested parties of research and to encourage discussion.
ABSTRACT

The Census Bureau was tasked with conducting research and evaluation and developing a

methodology to produce updated estimates of the total population and the total number of

school-age children in each school district. This paper provides an overview of the methodology

and limitations, the steps necessary to create the synthetic population estimates, problems we

encountered, and results from our evaluation of the data.
Acknowledgments: 

The author would like to thank Paul Siegel (Small Area Income and Poverty Estimates), Bashir

Ahmed and Signe Wetrogan (Population Division) for their invaluable comments and

contributions to this research. 

1

Evaluation of the 1990 School District Level Population Estimates

Based on the Synthetic Ratio Approach

I. INTRODUCTION

The elementary and secondary schools in the United States depend on federal dollars to

supplement programs for disadvantaged children. Title 1 of the Elementary and Secondary

Education Act provides a means for the Department of Education (DOE) to distribute federal

funds to school districts. 

Prior to School Year (SY) 1997/1998, the distribution of federal dollars to school districts was

carried out in a two-step process. First, the DOE allocated federal dollars to counties. States

then had the responsibility to distribute the federal dollars to school districts. In order to

determine the amount of money to allocate to a state, the DOE used the most recent decennial

data on the number of school-age children in poverty in each county within the state. States then

used a variety of data sources to allocate the monies down to the school districts including

special decennial census tabulations of the number of school-age children in poverty in each

school district.

In 1994, Congress enacted a law authorizing the Department of Education to allocate Title 1

funds directly to school districts, beginning with school year 1997/1998. In doing so, Congress

also specified that the DOE use updated estimates of the number of school-age children in

poverty in each school district rather than the once-a-decade measures from the decennial

census. 

The Census Bureau was tasked with conducting research and evaluation and developing a

methodology to produce updated estimates of the number of school-age children in poverty. 

Because the distribution of the funds also requires updated estimates of the total population and

the total number of school-age children in each school district, the Census Bureau also had to

develop methodologies for these data requirements.

This paper focuses on the development and evaluation of the methodologies to produce updated

estimates of the total population and the total number of school-age children in each school

district. It is divided into five sections. Section I is the introduction. Section II describes the

methodology developed to produce the updated population estimates for school districts and

issues that affect the production and subsequent accuracy of the estimates; Section III describes

the methodology used to evaluate the school-district estimates; Section IV presents the results of

the evaluation; and Section V presents conclusions and discusses plans to improve the

population estimates for school districts. A discussion of the development and evaluation of the 
1

See http://www.census.gov/hhes/www/saipe/schooltoc.html for the documentation.

2

methodology to produce updated estimates of the number of school-age children in poverty is

presented in a separate paper.1

II. DEVELOPMENT OF METHODOLOGY

This section presents an overview of the methodology used to produce the estimates of the total

population and the school-age population in each school district.

As noted in the prior section, the Census Bureau was tasked with developing the methodology to

produce updated estimates of the total population and the school-age population in each school

district. To comply with the legislation, the methodology had to be developed and implemented

for the allocation of funds for school year 1997/1998. 

Although the Census Bureau did have a program to develop and produce annual estimates of the

population of functioning governmental units, the methodologies developed for those estimates

could not be used to produce updated estimates of school districts. Therefore, it was necessary

for the Census Bureau to construct a new methodology to produce the population estimates for

school districts.

Factors Affecting Development of Methodology

In developing the methodology, we encountered a number of factors which complicate the

development of estimates for school districts.

School Districts are Small with Unique Boundaries

School districts are small with unique boundaries. As such, little Census or other data are

available as input to an estimation methodology. In 1990, there were 15,226 school districts in

the United States. Table 1 shows that approximately 50 percent of these school districts have a

total population of less than 5,000 people. Approximately 82 percent of all school districts have

an estimated total population of less than 20,000 people (U.S. Census Bureau, 1997). 

[Table 1 about here]

In most parts of the United States, school district boundaries are unique in that they do not

coincide with other governmental units for which data are regularly tabulated. There are only

seven states where school district boundaries coincide with county boundaries, accounting for 

only 928 of the 15,226 school districts in the United States. Although most school districts are

confined to a single county, some cross county boundaries, further adding to the complications in

developing an estimation methodology.

School Districts are Defined by Relevant Grades
2

Special tabulation by the U.S. Census Bureau.

3

School districts are defined according to the grade levels served by the school district. Therefore

the estimates of the number of school-age children in each school district had to be calculated

according to the grade level served by the school district. In 1990, about 74 percent of the

school districts across the United States served grade levels kindergarten through 12th grade. The

remainder of the school districts served only specific grades such as kindergarten through 6th

grade (22 percent) or 9th through 12th grade (4 percent).2

For those school districts which served only partial grade levels, it was necessary to translate the

grade levels served back to relevant ages. The 1990 census data on highest grade completed

together with data from the October supplements of the 1988, 1989, and 1990 Current

Population Surveys provided the necessary information to develop a grade to age relationship. 

The translation of grade to age was done so that each school-age child could be assigned to one

and only one school district. Thus, the sum of school-age children across school districts would

equal the total number of school-age children in the United States. However, this is not true for

the sum of the total population across school districts. Because a school district may provide

elementary grade service on the same piece of land as a district that provides education for

middle school grades, the estimates of the total population for these overlapping school districts

will be double counted. Thus, the sum of the estimates of the total population for all school

districts cannot be compared with the total population of the United States.

School District Boundaries Change Over Time

Several changes may occur to school districts over time. School districts can annex new territory

over time; school districts can close; and new school districts can be created. In order to

maintain correct and up to date boundaries, the Census Bureau must periodically survey school

districts to obtain current boundary information. Additionally, the changes to boundaries

complicate the complete evaluation of any methodology.

Choosing the Ratio Methodology

The complexities outlined above and the scarcity of data available for school districts led the

Census Bureau to choose a ratio or synthetic approach to produce the school district estimates. 

In choosing the ratio approach, the Census Bureau decided to rely upon the 1990 census to

provide a starting point and the annual estimates of the county population to provide the basis for

change. The annual estimates of the total population for counties would provide the basis for

change in the total population for school districts. The annual estimates of the population by age

for counties would provide the basis for change in the school-age population for school districts. 

This approach assumes that all school districts within a county change at the county rate. The

formula for developing the estimates for the post 1990 period is:

P(sd t) = P(sd 1990) / P(county 1990) * P(county t)

where:
4

P(sd t) = Estimated school district population in current boundaries for time t

P(sd 1990) = School district population in current boundaries from 1990 census

P(county 1990) = County population from 1990 census

P(county t) = County population for time t

While most school districts are confined to a single county, some do cross county boundaries. 

For those cases where the school district crosses county boundaries, it is necessary to construct a

separate ratio and separate estimates for the school district piece in each county. In these cases,

as a final step, the separate school district county pieces are summed to produce the school

district estimate.

Assumptions Associated with Ratio Approach

The ratio approach assumes that the ratio of the school district population to the county

population will remain constant over time. In other words, it assumes that the population in each

school district county piece changes at the same rate as that of the county. However, in reality

this may not be the case. If the county population is estimated to decline, but the school district

population in that county increases or vice versa, the resulting estimates of the school district

population will be biased.

The estimate is further complicated when a school district crosses county boundaries. In that

case, the ratio method assumes that each school district-county piece grows at the rate of that

county. In a school district that crosses county boundaries, one of the counties it comprises may

see a population spurt whereas the other county may experience a decline in population. When

the two county pieces are summed together, the school district population may be

underestimated or overestimated, depending upon the size of the school district pieces.

III. EVALUATING THE RATIO APPROACH

Development of Ratio Estimates for Evaluation

To do a complete evaluation of the school district methodology, we need to have school district

data at two points in time. The data for the 1980 and 1990 censuses provide us with that

opportunity. To evaluate the ratio methodology, we used the 1980 census as the base, developed

an estimate for 1990, and compared the estimate to the 1990 census data. The estimates were

produced for both the total population and the school-age population aged 5-17 years. For this

evaluation, we developed four sets of synthetic population estimates. 

Set 1: County Estimates-Based Model

To evaluate the ratio approach applied to an estimate of the county population (as would be the

case in the post 1990 period), we must develop a 1990 estimate for the county. For this test, we
3

 See http://www.census.gov/population/methods/stco99.txt for the methodology.

5

used the 1990 estimate of the county population that had been developed using our standard

county estimates approaches and based on the 1980 census.3

To produce these estimates, we first compute the ratio of the school district population to county

population using the 1980 census data. Then we apply the ratio to the 1980-based estimate of

the 1990 county population developed by the Census Bureau. This evaluation measures the

effect of the ratio approach as well as any error caused by the estimate of the county population. 

P (sd 1990) = P( sd 1980)/P(county 1980)* P(county 1990)

where:

P (sd 1990) = Estimated school district population in 1990

P(sd 1980) = School district population from 1980 census

P(county 1980) = County population from 1980 census

P(county 1990) = Estimated county population in 1990

Set 2: County Count-Based Model

This approach is very similar to Set 1 except that the ratios are multiplied by the 1990 census

data for the county population rather than the 1980-based estimate. We are assuming that all

school districts within the county change at the same rate as the county. Although for the post

1990 period we would only have estimates data available, this estimate is a good benchmark

against which to judge all other model-based estimates.

In this approach, we multiply the ratio of the 1980 school district population to 1980 county

population by the 1990 census county population.

P (sd 1990) = P( sd 1980)/P(county 1980)* P(county 1990)

where:

P (sd 1990) = Estimated school district population in 1990

P(sd 1980) = School district population from 1980 census

P(county 1980) = County population from 1980 census

P(county 1990) = County population from 1990 census

Set 3: State Growth-Based Estimates

This approach is similar to Set 2 except that it assumes that the school districts all change at the

same rate as that of the state. To develop the estimates, we multiply the ratio of the 1990 state

population to 1980 state population by the 1980 school district population.

P (sd 1990) = P(State 1990)/P(State 1980)* P(sd 1980)
6

where:

P (sd 1990) = Estimated school district population in 1990

P(State 1990) = State population from 1990 census

P(State 1980) = State population from 1980 census

P (sd 1980) = School district population from 1980 census

Set 4: National Growth-Based Estimates

This approach is also similar to Sets 2 and 3 except that it assumes that the school districts all

change at the same rate as that of the entire United States. To develop this estimate, we multiply

the ratio of the 1990 national population to 1980 national population by the 1980 school district

population.

P (sd 1990) = P( National 1990)/P(National 1980)* P(sd 1980)

where:

P (sd 1990) = Estimated School district population in 1990

P(National 1990) = National population from 1990 census

P(National 1980) = National population from 1980 census

P (sd 1980) = School district population from 1980 census

Note that the assumptions underlying the models may not be realistic. For example, the

population growth in a school district does not correspond to the growth in a county or state.

Similarly, it is not reasonable to assume that each and every school district will grow at the same

rate as the nation. 

Creating a Comparable Universe of School Districts Across the Decade

To do a complete evaluation of the methodology, we need a comparable universe of school

districts over the 1980 to 1990 time period. Optimally, for our analysis we would use a matched

1980 and 1990 file, geocoded to identical school district boundaries. The advantage of this type

of file is that we would not need to make assumptions about school district boundaries across the

decade. 

If the Census Bureau had a 1980 data file geocoded to the 1990 school district geography we

could simply apply synthetic ratios to 1990 census data and compare the expected value to the

“truth” in 1990. If we were able to geocode 1990 data into 1980 school district geography, we

could administer the same approach. However, neither data set is available.

Considering we do not have files geocoded to the same boundaries, we concluded we needed to

prepare a universe of school districts that are “equivalent” across the decade. The starting point

for our universe is the total number of school districts in 1990 (15,226). (See Table 2). We first

excluded 928 school districts that were coterminous with county boundaries as the stable shares

approach perfectly predicts the population for the 1990 school district for this set of school

districts. 
7

[Table 2 about here]
4

 Most school districts cover the grade range of K-12. These are known as unified

school districts. A non-unified school district does not cover grades K-12 but instead covers

elementary, middle, or high school grades. If a school district is not unified across the decade, it

is not possible to determine whether the grades the district includes are the same across time

(U.S. Department of Education, 1999).

5

 We assumed the school district boundaries did not change if the identification number

did not change over the decade. This assumption may not always be correct because the state

did not always assign new IDs when land was annexed over the decade, political boundaries

changed, etc. (U.S. Department of Education, 1999).

6

 See Appendix for the formulas for school district estimators and evaluation statistics for

the models. The appendix includes references to both population and poverty estimates. There

are some slight differences in the terminology. Our text refers to MALPE whereas the appendix

refers to MALP. Additionally, Model-based refers to our Set 1, census county-based refers to

our Set 2, and the naive-based refers to our growth-based estimates. Thanks to William R. Bell

for providing the statistical explanation for the computations (U.S. Census Bureau, 1998).

8

Essentially, we could apply the synthetic ratio approach to the remaining 14,298 districts. 

However, in order to have an “equivalent” universe file over the decade, we also removed:

1. School districts with limited grade ranges4

 (4,018); 

2. School districts which were newly formed between 1980 and 1990 (416);

3. School districts in counties where the county boundaries changed between 1980

and 1990 (12)5

; 

4. School district county pieces did not match up across the decade (609); and

5. School districts with a population size of less than 31 people (42).

The final universe for the 1980-1990 evaluation file contained 9,201 matched school district

identification numbers.

Evaluation Measures

To compare and evaluate the estimates, we used two standard statistical measures: (1) the Mean

Absolute Percent Error (MAPE), and (2) the Mean Algebraic Percent Error (MALPE).6

 The

MAPE is computed as the sum for all school district pieces of the absolute difference between

the estimate and the 1990 census figure divided by the number of school districts. The MAPE

measures the accuracy of the estimates. The MALPE is computed in a similar manner, except

that we take the sign of the difference into consideration. Positive mean algebraic percent errors

indicate overestimation of a population and negative errors indicate an underestimate of a

population. 

We also examined weighted MAPEs. The unweighted statistics treat each school district with

equal importance, regardless of size. The weighted MAPEs, on the other hand, take into
9

consideration the size of a school district, measured by the total population or the school-age

population in that school district. Weighting by the total population in each school district

addresses the size of the school district population affected. Weighting by the number of schoolage children indicates how accurate the estimates are for the districts containing the average

child.

IV. RESULTS OF THE EVALUATION

For purposes of this evaluation, we developed four sets of synthetic population estimates. Set 1

uses the ratio approach and the 1980 based county population estimate. Set 2 is similar except

that it uses the 1990 census data for the county rather than the 1980 based estimate. The

differences between Set 1 and Set 2 represent the additional error in the ratio approach

introduced by using an estimate of the population rather than the census counts. Sets 3 and 4

represent alternatives to a county-based approach. Set 3 assumes that the school district grows at

the same rate as that of the state, while Set 4 assumes that the school districts all grow at the

national rate.

Overall Quality

As shown in Table 3 and Figure 1, the county count-based estimates have the smallest

unweighted MAPEs (12.6 and 16.0), followed by the county estimates-based (13.3 and 16.9), the

state growth-based (16.4 and 18.9), and national growth-based estimates (18.9 and 20.6). This

pattern holds both for total population and school-age population aged 5-17, whether the MAPEs

are weighted or unweighted.

[Table 3 and Figure 1 about here]

Table 4 presents the results of comparing the MAPEs across each set of estimates. As show in

the first row of Table 4, we lose only a minor amount of accuracy when we use an estimate

rather than the census count as the base for the 1990 county data. Comparing Set 1 to Set 3 and

Set 4 indicate that the use of the ratio approach at the county is superior to one that uses state or

national growth rate assumptions.

[Table 4 about here]

Using the MAPEs as our unit of analysis, we would conclude that the Set 2 approach is the most

accurate for estimating the school district population. However, the Set 2 (county count-based

approach) can be produced only at the census year. Therefore, if we must rely on the synthetic

approach, we need to employ a set of estimates. And as shown by the comparison to Sets 3 and

4, the use of the county estimate is superior to a method that uses state or national growth rate

assumptions. For this reason, the remainder of this section reports results from the county

estimates-based MAPEs and MALPEs.

Quality of the Estimates by Demographic and Economic Characteristics
7

 See National Research Council, 1998.

8

 We will not discuss weighted MALPEs because the sum of the MALPEs for each

economic or demographic characteristic would be equivalent to zero if all of the school districts

in each county were represented in our sample, thus the weighted MALPEs are meaningless to

the analysis.

9

The unweighted number of school districts in each category of the demographic and

economic characteristics remain the same across Table 5 and Table 6. This is because the

demographic and economic categories (e.g., Size of the School District in 1980 or Percent Poor

School-age Children in 1980) were defined based on the characteristics of the total population in

a school district. For example, if the total population in a school district is 9,000 and the schoolage population in a school district is 4,500 the school district falls into the school district

population of 5,000 - 9,999. In Table 5, the total population is determined by weighting the

number of school districts by the total population in each school district. In Table 6, we

determined the school-age population by weighting the number of school districts by the number

of school-age children in each school district. 

10

To evaluate the amount of “bias” or other patterns in the county estimates-based school district

estimates, we selected ten economic and demographic characteristics. These characteristics are a

subset of those the National Academy of Sciences used to evaluate poverty estimates at the

county level.7

 The ten characteristics are: 

1. Size of the School District in 1980; 

2. Size of the School District in 1990;

3. Population Growth, 1980-1990;

4. Percent Poor School-age Children in 1980;

5. Percent Poor School-age Children in 1990;

6. Numerical Change in Poverty Rate for Children, 1980-1990;

7. Census Division;

8. Percent Hispanic in 1980;

9. Percent Black in 1980; and 

10. Percent Group Quarters in 1980.

Table 5 shows both the unweighted and weighted MAPEs and unweighted MALPEs8

 for total

population, by the selected characteristics. Similarly, Table 6 shows the unweighted and

weighted MAPEs and unweighted MALPEs by characteristics for school-age population 

aged 5-17. Additionally, the two tables present the total population (or school-age population)

and the percent of the population in each category.9

 

[Table 5 and Table 6 about here]
10 The findings in the last two bullets above are consistent with findings shown in Table 2

in that about one half of all school districts are made up of less than 5,000 people. The

difference is that Table 2 is based on the total number of school districts as of 1989-1990 (15,226

school districts); whereas the evaluation universe is based on 9,201 districts. 

11

Figures 2 through 11 are pictorial representations of the weighted and unweighted MAPEs for

both the total and school-age population, by demographic and economic characteristics.

Size of the School District in 1980 (See Figure 2 and Tables 5 and 6)

< The unweighted MAPEs for school districts with fewer than 5,000 people are

almost two times as high as the MAPEs for all other population categories (the

MAPE for total population is 18.1 and the MAPE for the school-age population is

22.8). Whereas the weighted MAPEs for the same set of school districts are one

and half time as high as the other categories.

< For small school districts (those with a population less than 5,000), we

overestimated the total population and the school-age population by 8.8 percent

and 6.9 percent, respectively (see MALPEs).

< For larger school districts (those with a population more than 40,000), we

overestimated the total population and the school-age population by 3.2 percent

and 5.0 percent, respectively.

< Almost half of all districts (48.2 percent) have a total population of less than

5,000 people. However, these districts account for only 6.9 percent of the total

population and 7.7 percent of the school-age population.10

< School districts that are populated by 20,000 or more people represent 16.5

percent of all school districts, but are populated by two-thirds (66.6 percent) of

the total population and about two-thirds (64.5 percent) of the school-age

children.

[Figure 2 about here]

Size of the School District in 1990 (See Figure 3 and Tables 5 and 6)

< The relationship between the size of the district and the size of the MAPEs in

1990 show the same patterns as what we see in 1980.
11 Special tabulation by the U.S. Census Bureau.

12 Special tabulation by the U.S. Census Bureau.

13 When there were no related poor school-age children in 1980, then the shares

methodology predicted that the percentage of children in poverty in 1990 will be zero as well. 

12

< With the exception of the smallest school districts, the MALPEs based on

population size in 1990, are lower than those based on the school district

population size in 1980.

[Figure 3 about here]

Population Growth, 1980-1990 (See Figure 4 and Tables 5 and 6)

< When the size of the school district increases or decreases over the decade by 10

percent or more, the MAPEs are much higher as compared with school districts

with a stable amount of growth.

< For school districts that lost 10 percent or more of their total population during

the 1980 to 1990 period, the ratio approach tended to overestimate their total

population by an average of 28.9 percent.

< Conversely, for school districts that grew by 10 percent or more during the 1980

to 1990 period, the ratio approach tended to underestimate their total population

by an average of 10.9 percent.

< One out of five school districts (21.1 percent) is located in areas where the

population declined by 10 percent or more, representing 8.3 percent of the total

population and 8.6 of the school-age population.

< Over one-third (36.5 percent) of the total population and 38.2 percent of the

school-age population live in a district which had a major increase (10 percent or

more) in population throughout the decade.11

< Overall, larger school districts tend to be growing whereas the smaller school

districts appear to be declining in population.12

[Figure 4 about here]

Percent Poor School-age Children in 1980 (See Figure 5 and Tables 5 and 6)

< With the exception of school districts with zero percent poverty and 24 percent or

more children in poverty, there is little difference in the weighted MAPEs.13
Obviously, these situations occur in very, very small school districts. As a result, the predictions

are not accurate and there is a high degree of error between the predictions and the truth.

14 When there are no children in poverty, the percent difference (for that school district) is

undefined and excluded from our tabulations. Even with the missing values removed, smaller

school districts continue to contribute disproportionately to the high MAPEs.

13

< The unweighted MAPE in areas with a poverty rate of 24 percent or more is 16.5

percent for the total population and 21.3 percent for the school-age population.

< The bias in the estimates in school districts with a poverty rate of 24 percent or

more approximates 9.3 percent for the total population and 8.4 percent for the

school-age population.

< Approximately 16.5 percent of all school districts and 20.9 percent of all schoolage children are in areas where the poverty rate is 24 percent or more.

[Figure 5 about here]

Percent Poor School-age Children in 1990 (See Figure 6 and Tables 5 and 6)

< The MAPEs for the percent poor school-age children in 1990 are similar to the

MAPEs for the percent poor school-age children in 1980.14

< For school districts that have a poverty rate of 24 percent or more in 1990, we

tend to overestimate both the total and school-age populations (MALPEs of 10.1

percent and 8.6 percent, respectively).

[Figure 6 about here]

Numerical Change in Poverty Rate for Children, 1980-1990 (See Figure 7 and Tables 5 and 6)

< The pattern for the MAPEs and MALPEs are similar to the MAPEs and MALPEs

for the demographic characteristic representing population growth between 1980-

1990.

< Like the growth rate for the population, there are higher MAPEs and MALPEs for

both the total and school-age populations in school districts with large increases

in the poverty rate and school districts with large decreases in the poverty rate. 
15 Special tabulation by the U.S. Census Bureau.

16 Special tabulation by the U.S. Census Bureau.

14

< Nearly 97 percent of the school districts which experienced a decline in poverty

of 10 percent or more were located in school districts with a population size of

5,000 or less.15

< Eighty percent of the school districts which experienced an increase in poverty of

10 percent or more were located in school districts with less than 20,000 people.16

[Figure 7 about here]

Census Division (See Figure 8 and Tables 5 and 6)

< The largest unweighted MAPE for both the total and school-age population is in

the Mountain region. When weighted, the MAPE for the Mountain Region is

aligned with the remaining census regions.

[Figure 8 about here]

Percent Hispanic in 1980 (See Figure 9 and Tables 5 and 6)

< MAPEs for the Hispanic population may be correlated with the percent of the

population which is Hispanic.

[Figure 9 about here]

Percent Black in 1980 (See Figure 10 and Tables 5 and 6)

< The MAPEs for the total and school-age population are monotonic.

[Figure 10 about here]

Percent Group Quarters in 1980 (See Figure 11 and Tables 5 and 6)

< The MAPEs are higher for school districts where the GQ population comprises

more than 10 percent of the total population. However, the percentage of GQ

population is difficult to estimate over time because GQ facilities are built or

closed over the decade.

[Figure 11 about here]
15

V. CONCLUSIONS AND PLANS TO IMPROVE SCHOOL DISTRICT ESTIMATES

This paper attempted to evaluate the 1990 school district level population estimates which were

developed by the synthetic ratio approach. For both the total population and the school-age

population age 5-17, four sets of synthetic estimates were produced: (1) the 1990 county

estimates-based estimates; (2) the 1990 county count-based estimates; (3) state growth-based

estimates; and (4) national growth-based estimates. To evaluate the estimates, we used both the

Mean Absolute Percent Error (MAPE) and the Mean Algebraic Percent Error (MALPE). We

examined the variations in the MAPEs and MALPEs by selected demographic and economic

characteristics.

To summarize, the state growth and national growth-based models produced the least accurate

estimates. They are feasible alternatives, but the school district growth rate is least likely to be

the same as the state’s or the nation’s. The county count-based and the county estimates-based

models were close to each other although the former provided more accurate estimates than the

latter. We found that the differences were especially apparent for small school districts, districts

with high and low poverty rates; and districts with high and low growth rates. However, the

county count-based estimates can be produced only at the census year. Therefore, if we must

rely on the synthetic estimates, we do need to use the county estimates-based model.

What are our plans to improve the school district estimates?

The Census Bureau is required to produce school district level population estimates for SY

1995/1996 and every two years thereafter. For SY 1995/1996 and SY 1997/1998, the synthetic

estimates were based on data from the 1990 census and updated county estimates thereafter. For

SY 1999/2000, we will use the Census 2000 data. 

However, for post 2000 school district estimates, we plan to conduct further research to improve

the estimates. These research plans include:

< Examine the use of updated TIGER/MAF files to more adequately define school

district boundaries and aggregate blocks to the school district level.

< Evaluate the geocoding of addresses extracted from IRS tax returns to the school

district level.

< Expand the use of administrative records, such as the extract of IRS tax returns,

Common Core of Data, or Free Lunch data to directly and indirectly estimate total

and school-age population estimates.

< Explore the use of the American Community Survey (ACS) data to estimate

school district population.
16

REFERENCES

National Research Council, 1998. Small-Area Estimates of Children in Poverty, Interim Report

2, Evaluation of Revised 1993 County Estimates for Title I Allocations. Panel on Estimates of

Poverty for Small Geographic Areas, C.F. Citro, M.L. Cohen, and G. Kalton, eds., Committee on

National Statistics. Washington, D.C.: National Academy of Press.

U.S. Census Bureau, 1997. Table presented to the National Academy of Sciences, Panel on

Estimates of Poverty for Small Geographic Areas, Sixth Plenary Meeting, November 4, 1997.

U.S. Census Bureau, 1998. Appendix provided to the National Academy of Sciences, Panel on

Estimates of Poverty for Small Geographic Areas, Ninth Plenary Meeting, October 2-3, 1998.

U.S. Department of Education, 1999. National Center for Education Statistics. Classification

Evaluation of the 1994-95 Common core of Data: Public/Elementary/Secondary Education

Agency Universe Survey, NCES 1999-316 by Stephen Owens. Project Officer: Beth Young. 

Washington, DC.

Source: U.S. Census Bureau





































Total Population

13.3 12.6

16.4

18.9

9.6 9.2

11.8

13.9

0

10

20

30

40

50

60

County Estimates-based County Count-based State Growth-based National Growth-based

Mean Absolute Percent Error

 Unweighted

 Weighted



































School-Age Children

16.9 16.0

18.9

20.6

12.0 10.4

13.3

16.6

0

10

20

30

40

50

60

County Estimates-based County Count-based State Growth-based National Growth-based

Mean Absolute Percent Error

Unweighted



Weighted

Figure 1

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Method to Estimate 1990 School Districts
Source: U.S. Census Bureau

.











































Total Population

18.1

8.6 8.5 9.3 9.4

13.3

9.3 9.2 8.9 9.5

0

10

20

30

40

50

60

 Under 5,000 5,000 - 9,999 10,000 - 19,999 20,000 - 39,999 40,000 or more

Mean Absolute Percent Error

 Unweighted

 Weighted



















































School-Age Children

22.8

10.9 10.8 12.4 12.4

15.7

11.6 11.4 11.9 11.8

0

10

20

30

40

50

60

 Under 5,000 5,000 - 9,999 10,000 - 19,999 20,000 - 39,999 40,000 or more

Mean Absolute Percent Error



Unweighted

 Weighted

Figure 2

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Size of School District: 1980
Source: U.S. Census Bureau

.



































Total Population

18.3

8.6 8.6 9.1 9.7

12.3

8.6 8.5 8.9 10.0

0

10

20

30

40

50

60

 Under 5,000 5,000 - 9,999 10,000 - 19,999 20,000 - 39,999 40,000 or more

Mean Absolute Percent Error

 Unweighted

 Weighted











































School-Age Children

23.0

11.1 10.9 12.3 12.4

14.8

10.9 10.8 12.1 12.3

0

10

20

30

40

50

60

 Under 5,000 5,000 - 9,999 10,000 - 19,999 20,000 - 39,999 40,000 or more

Mean Absolute Percent Error

Unweighted

 Weighted

Figure 3

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Size of School District: 1990
Source: U.S. Census Bureau

.















































Total Population

29.2

8.0 6.4 5.5 5.8

13.7

17.3

9.5

6.5 5.5 6.6

11.9

0

10

20

30

40

50

60

Decrease of 10%

or more

Decrease of 5.0 -

9.9%

Decrease of 0.1 -

4.9% 

Increase of 0.0 -

4.9%

Increase of 5.0 -

9.9%

Increase of 10%

or more 

Mean Absolute Percent Error

Unweighted

 Weighted













































School-Age Children

32.8

11.5 11.0 9.7 10.3

16.2 17.1

11.7 10.7 9.5 9.9

13.2

0

10

20

30

40

50

60

Decrease of 10%

or more

Decrease of 5.0 -

9.9%

Decrease of 0.1 -

4.9% 

Increase of 0.0 -

4.9%

Increase of 5.0 -

9.9%

Increase of 10%

or more 

Mean Absolute Percent Error

Unweighted

 Weighted

Figure 4

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Population Growth: 1980-1990
Source: U.S. Census Bureau

.



























































Total Population

33.9

13.7

11.6 11.8 12.1 12.0

16.5

25.4

11.3

8.6 8.6 7.9

10.2 10.0

0

10

20

30

40

50

60

Zero 0.1 - 5.9% 6.0 - 8.9% 9.0 - 12.4% 12.5 - 16.4% 16.5 - 23.9% 24.0% or more

Mean Absolute Percent Error

Unweighted

 Weighted





































































School-Age Children

48.4

18.2

14.9 14.4 14.7 14.4

21.3

28.3

15.0

11.6 11.5

9.6

11.7 11.3

0

10

20

30

40

50

60

Zero 0.1 - 5.9% 6.0 - 8.9% 9.0 - 12.4% 12.5 - 16.4% 16.5 - 23.9% 24.0% or more

Mean Absolute Percent Error

Unweighted

 Weighted

Figure 5

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Percent Poor School-age Children: 1980
Source: U.S. Census Bureau

.

































































Total Population

45.1

10.9 10.5 10.1 10.1 11.6

16.5

18.9

11.3

8.8 8.2 8.7 8.1

10.7

0

10

20

30

40

50

60

Zero 0.1 - 5.9% 6.0 - 8.9% 9.0 - 12.4% 12.5 - 16.4% 16.5 - 23.9% 24.0% or more

Mean Absolute Percent Error

Unweighted

 Weighted































































School-Age Children

59.9

14.5

12.6 12.9 12.7 14.1

20.9 22.3

15.0

11.8 11.0 10.8 10.3 12.0

0

10

20

30

40

50

60

Zero 0.1 - 5.9% 6.0 - 8.9% 9.0 - 12.4% 12.5 - 16.4% 16.5 - 23.9% 24.0% or more

Mean Absolute Percent Error



Unweighted

 Weighted

Figure 6

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Percent Poor School-age Children: 1990
Source: U.S. Census Bureau

.

















































Total Population

24.1

14.0

11.1 10.9 10.8

19.8

12.4

9.5 9.5 9.2 8.8

13.0

0

10

20

30

40

50

60

Decrease of 10.0+

% points

Decrease of 5.0 -

9.9% points

Decrease of 0.1 -

4.9% points

Increase of 0.0 -

4.9% points 

Increase of 5.0 -

9.9% points 

Increase of 10.0 +

% points

Mean Absolute Percent Error

Unweighted



Weighted























































School-Age Children

31.8

16.7

13.7 14.2 13.6

25.3

14.4

12.4 12.2 11.9 10.7

14.1

0

10

20

30

40

50

60

Decrease of 10.0+

% points

Decrease of 5.0 -

9.9% points

Decrease of 0.1 -

4.9% points

Increase of 0.0 -

4.9% points 

Increase of 5.0 -

9.9% points 

Increase of 10.0 +

% points

Mean Absolute Percent Error

Unweighted

 Weighted

Figure 7

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Change in Poverty Rate for Children: 1980-1990
Source: U.S. Census Bureau











































































Total Population

9.0 7.8

13.4

9.0

11.1

13.2

33.9

8.1

17.2

21.2

9.5

16.8

9.8 9.2

5.3

6.7

13.3

9.1

0

10

20

30

40

50

60

 New

England

 Middle

Atlantic

 South

Atlantic

 East North

Central

 East South

Central

 West

North

Central

 West

South

Central

 Mountain Pacific

Mean Absolute Percent Error

Unweighted

 Weighted















































































School-Age Children

15.3

11.8 11.2

13.7

17.4

38.9

24.1

13.5

20.1

13.0

15.6

11.7

17.9

10.3 11.0 10.6 9.9 9.5

0

10

20

30

40

50

60

 New

England

 Middle

Atlantic

 South

Atlantic

 East North

Central

 East South

Central

 West

North

Central

 West

South

Central

 Mountain Pacific

Mean Absolute Percent Error

Unweighted

 Weighted

Figure 8

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Census Division: 1980
Source: U.S. Census Bureau

.





















































Total Population

22.7

10.0

15.7 17.3

22.8

19.9

14.9

8.0 9.6

11.9 12.5 14.1

0

10

20

30

40

50

60

 0.0% 0.1 - 0.9% 1.0 - 4.9% 5.0 - 9.9% 10.0 - 25.0% More than 25%

Mean Absolute Percent Error

Unweighted

 Weighted

































































School-Age Children

35.1

12.8

19.5 18.5

26.1

23.9

20.4

10.2

12.4

14.2 14.4

16.6

0

10

20

30

40

50

60

 0.0% 0.1 - 0.9% 1.0 - 4.9% 5.0 - 9.9% 10.0 - 25.0% More than 25%

Mean Absolute Percent Error

Unweighted

 Weighted

Figure 9

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Percent Hispanic: 1980
Source: U.S. Census Bureau

.

















































Total Population

19.0

11.4 11.0

12.9 13.9 14.0

10.1 9.2 9.5 9.3 9.9 10.7

0

10

20

30

40

50

60

 0.0% 0.1 - 0.9% 1.0 - 4.9% 5.0 - 9.9% 10.0 - 25.0% More than 25%

Mean Absolute Percent Error

 Unweighted

 Weighted





















































School-Age Children

25.4

14.2 15.3 14.4 16.0 15.0

12.8 12.4 12.7

10.9 11.6 10.7

0

10

20

30

40

50

60

 0.0% 0.1 - 0.9% 1.0 - 4.9% 5.0 - 9.9% 10.0 - 25.0% More than 25%

Mean Absolute Percent Error



Unweighted

 Weighted

Figure 10

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Percent Black: 1980
Source: U.S. Census Bureau





































Total Population

16.9

11.1 10.4 9.1

21.4

11.1 10.8

8.2 8.3 9.5

0

10

20

30

40

50

60

 0.0 - 0.18% 0.19 - 1.3% 1.4 - 2.4% 2.5 - 10.0% More than 10.0%

Mean Absolute Percent Error

Unweighted



 Weighted









































School-Age Children

21.3

13.9 12.9 13.3

22.7

13.7 13.2

10.1 11.1 12.5

0

10

20

30

40

50

60

 0.0 - 0.18% 0.19 - 1.3% 1.4 - 2.4% 2.5 - 10.0% More than 10.0%

Mean Absolute Percent Error



Unweighted

 Weighted

Figure 11

Mean Absolute Percent Errors (MAPEs) for Estimates of Total Population and School-age

Population by Percent Group Quarters (GQ): 1980
16

August 15,2001

POPULATION DIVISION WORKING PAPER SERIES

For copies of these working papers, please contact the Statistical

Information Staff, Population Division, Bureau of the Census,

Washington, DC 20233-8800 (301)457-2422/pop@census.gov

NO. 1 - "The Census Bureau Approach for Allocating 

 International Migration to States, Counties, and

Places: 1981-1991." David L. Word. October 1992.

NO. 2 - "Geographic Coding of Administrative Records--

Past Experience and Current Research." 

Douglas K.Sater. April 1993.

NO. 3 - "Postcensal Population Estimates: States, Counties,

and Places." John F. Long. August 1993.

NO. 4 - "Evaluating the Passel-Word Spanish Surname List: 

 1990 Decennial Census Post Enumeration Survey Results."

 R. Colby Perkins. August 1993.

NO. 5 - "Evaluation of Postcensal County Estimates for the

 1980s." Sam T. Davis. March 1994.

NO. 6 - "Metropolitan Growth and Expansion in the 1980s."

 Richard L. Forstall and James D. Fitzsimmons.

 April 1993.

NO. 7 - "Geographic Coding of Administrative Records -- Current

 Research in ZIP/Sector-to-County Coding Process."

 Douglas K. Sater. June 1994.

NO. 8 - "Illustrative Ranges of the Distribution of 

 Undocumented Immigrants by State." Edward W. Fernandez 

 and J. Gregory Robinson. October 1994.

NO. 9 - "Estimates of Emigration of the Foreign-Born 

 Population: 1980-l990." Bashir Ahmed and

 J. Gregory Robinson. December 1994.

NO. 10 - "Estimation of the Annual Emigration of U.S. Born 

 Persons by Using Foreign Censuses and Selected

 Administrative Data: Circa 1980."

 Edward W. Fernandez. January 1995.

NO. ll - "Using Analytic Techniques to Evaluate the 1990 Census

 Coverage of Young Hispanics." Edward Fernandez. 

 May 1995.

NO. 12 - "Metropolitan and Nonmetropolitan Areas:

New Approaches to Geographical Definition."

Donald C. Dahmann and James D. Fitzsimmons.

October 1995.
17

NO. 13 - "Building a Spanish Surname List for the 1990's--

A New Approach to An Old Problem."

David L. Word and R. Colby Perkins, Jr.

February 1996.

NO. 14 - "Fertility of American Men."

Amara Bachu. March 1996.

NO. 15 - "Comparisons of Selected Social and Economic 

 Characteristics Between Asians, Hawaiians, Pacific 

 Islanders, and American Indians (Including Alaskan 

 Natives)." Edward W. Fernandez. June 1996.

NO. 16 - "Findings on Questions on Race and Hispanic Origin

 Tested in the 1996 National Content Survey." 

Prepared in the Population Division by the Racial

 Statistics Branch and the Ethnic and Hispanic

 Statistics Branch. December 1996.

NO. 17 - "Race and Ethnicity Classification Consistency Between

 the Census Bureau and the National Center for Health

 Statistics." Larry Sink. February 1997. 

NO. 18 - "Results of the 1996 Race and Ethnic Targeted Test." 

May 1997.

NO. 19 - "Who Responds/Who Doesn’t? Analyzing Variation in Mail

 Response Rates During the 1990 Census." David L. Word.

 July 1997. 

No. 20 - "Trends in Marital Status of U.S. Women at First Birth:

 1930 to 1994." Amara Bachu. March 1998.

No. 21 - "State Estimates of Organized Child Care Facilities."

Lynne Casper and Martin O’Connell. March 1998. 

No. 22 - "How Well Does the Current Population Survey Measure 

 the Foreign-Born Population in the United States"

Dianne Schmidley and J. Gregory Robinson. April 1998 

No. 23 - "Poverty, Family Structure, and Child Well-Being:

 Indicators from the SIPP." Jason Fields and 

 Kristin Smith. April 1998

No. 24 - "Child Well-Being Indicators from the SIPP." 

 Kristin Smith, Loretta Bass, and Jason Fields.

 April 1998

No. 25 - "Timing of First Births: 1930-34, 1990-94."

 Amara Bachu. May 1998

No. 26 - "Co-Resident Grandparents and Grandchildren:

 Grandparent Maintained Families." 

Lynne Casper and Ken Bryson

 May 1998
18

No. 27 - "Population of the 100 Largest Cities and Other Urban

 Places in the United States: 1790 to 1990." 

 Campbell Gibson. June 1998

No. 28 - "Are There Differences in Voting Behavior Between

 Naturalized and Native-born Americans?"

 Loretta E. Bass and Lynne M. Casper. March 1999.

No. 29 - "Historical Census Statistics on the Foreign-born 

 Population of the United States: 1850-1990." 

 Campbell J. Gibson. February 1999

No. 30 - "Direct Measures of Poverty as Indicators of Economic

 Need: Evidence from the Survey of Income and Program

 Participation." Kurt J. Bauman. November 1998

No. 31 - "American Community Survey and Intercensal Population 

 Estimates: Where Are the Cross-roads?"

 Amy Symens. December 1998

No. 32 - "Women’s Labor Force Attachment Patterns and Maternity 

 Leave: A Review of the Literature."

 Kristen Smith and Amara Bachu. January 1999 

No. 33 - "Evaluation of Relationship, Marital Status, and 

 Grandparents Items on the Census 2000 Dress Rehearsal"

 Charles Clark and Jason Fields. April 1999.

No. 34 - "Unbinding the Ties: Edit Effects of Marital Status on 

 Same Gender Couples." Charles Clark and Jason Fields. 

 April 1999.

No. 35 - "Racial-Ethnic and Gender Differences in Returns to 

 Cohabitation and Marriage: Evidence from the Current 

 Population Survey." Philip N. Cohen. May 1999.

No. 36 - "How Does POSSLQ Measure Up? Historical Estimates of 

 Cohabitation." Lynne Casper, Philip N. Cohen, and Tavia 

 Simmons. May 1999.

No. 37 - "Childlessness Among American Women On the Rise?".

 Amara Bachu. May 1999.

 

No. 38 - "Methodology and Assumptions for the Population 

 Projections of the United States: 1999 to 2100" 

 Frederick Hollman, Tammany Mulder, and

 Jeffrey Kallan. October 1999.

No. 39 - "What Do We Know About the Undercount of Children?"

 Kirsten K. West and J. Gregory Robinson. 

 August 1999.

No. 40 - Canceled 

No. 41 - Canceled

No. 42 - "Measures of Help Available to Households in Need: 

 Their Relationship to Well-Being, Welfare, and Work." 
19

 Kurt Bauman and Barbara Downs. May 2000

No. 43 - "Have We Reached the Top? Educational Attainment 

 Projections of the U.S. Population."

 Jennifer Cheeseman Day and Kurt Bauman

 May 2000

No. 44 - "The Emerging American Voter: An Examination of the 

 Increase in the Black Vote in November 1998."

 Avalaura L. Gaither and Eric C. Newburger. June 2000

No. 45 - "An Analysis of State and County Population Changes by 

 Characteristics: 1990-1999." Amy Symens Smith, Bashir Ahmed, 

 and Larry Sink. November 2000.

No. 46 - "The Effect of Work and Welfare on Living Conditions in Single

Parent Households." Kurt Bauman. August 2000.

No. 47 - "Projections of the Number of Households and Families in the 

 United States: 1999 to 2025." Monique Oosse and Ching li Wang. 

 Release Pending.

No. 48 - "Fertility in the United States: Is State-Level

 Fertility Converging to the National Level?" 

 Laura Heaton. Release Pending. 

No. 49 – “Variation in State Morality over Time.”

Monica Oosse. Release Pending.

No. 50 - “Accuracy of the U.S. Census Bureau National Population

 Projection and Their Respective Components of Change.”

 Tammany Mulder. Release Pending.

No. 51 - “U.S. Census Bureau Measurement of Net International 

 Migration to the United States: 1990 to 2000.”

 Tammany Mulder, Frederick Hollmann, Lisa Lollock, 

 Rachel Cassidy, Joseph Costanzo, and Josephine Baker.

 Release Pending.

No. 52 - “At-Risk Conditions of U.S. School-Age Children.”

 Robert Kominski, Amie Jamieson, and Gladys Martinez. 

 Release Pending.

No. 53 - “Home Schooling in the United States: Trends and 

 Characteristics.” K. J. Bauman. August 2001.

No. 54 - “Evaluation of the 1990 School District Level 

 Population Estimates Based on the Synthetic Ratio

 Approach.” Esther R. Miller Release Pending.

No. 55 - “State Estimates of Child Care Establishments

 1977-1997.” Grace O’Neil and Martin O’Connell