Evaluating Forecast Error in State Population Projections 

Using Census 2000 Counts 

By Paul R. Campbell 

Population Division

U.S. Bureau of the Census

Washington, D.C. 20233

July 2002

Population Division Working Paper Series No. 57
Evaluating Forecast Error in State Population Projections Using Census 2000 Counts 

Paul R. Campbell 

Population Division

U.S. Bureau of the Census

Washington, D.C. 20233

Abstract

In order to determine if a popular summary statistic, the mean absolute percentage error (MAPE), is a valid measure 

of forecast error for the Census Bureau=s 1995 to 2000 state population projections, statistical tests and graphs were 

used to determine if the error distribution is strongly influenced by outliers. It was found that the absolute 

percentage error distribution is skewed and asymmetrical. Since the MAPE understates accuracy, MAPE-R, a 

variant of MAPE derived from the transformed absolute percentage error distribution was accepted as more 

accurate. Using simple extrapolated projections as a standard to compare forecast error, the findings suggest that 

the Census Bureau=s projections are fairly accurate over a short projection horizon. 

Paper Presented at the Federal Forecasters Conference in Washington, D.C. April 18, 2002. 

Disclaimer: 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. 
EVALUATING FORECAST ERROR IN STATE POPULATION PROJECTIONS 

USING CENSUS 2000 COUNTS 

Paul R. Campbell, U.S. Bureau of the Census 

Introduction. Users of the Census Bureau’s 1990 

census-based state population projections for 1995 to 

2025 are interested in the Bureau’s forecast error. Even 

counties or smaller geographic areas are dependent 

upon state level accuracy, since projections for these 

areas often are prorated to the state figures. In this 

study, Census 2000 counts are used to measure forecast 

error in projections for April 1, 2000. This is the first 

opportunity to evaluate the 1995 to 2025 projections 

with the ‘truth’ assuming that the Census 2000 results 

are correct. This evaluation specifically examines the 

forecast error of the 2000 state population projection 

totals (including the District of Columbia) for Series A 

and B, and a simple extrapolated projection to identify 

the more accurate set of projections and the state 

outliers. The basic statistical method used to detect 

state population projection error is to examine the 

Percentage Error (PE) and the Absolute Percentage 

Error (APE), while the overall accuracy of the set of 

projections is measured by the Mean Absolute 

Percentage Error (MAPE). Additionally, this study 

examines the skewness and asymmetry of the APE 

distribution for states to determine if there is a need to 

use MAPE-R, derived from the transformed APEs, as 

recommended in the literature in order to correct for the 

influence of outliers on the mean. 

Several questions are addressed in this state projections 

evaluation. How accurate are the Census Bureau’s state 

population projections for the year 2000? Which 

Census Bureau state population projection series is the 

most accurate? Are the Census Bureau’s projections as 

good as or better than the results obtained from simple 

extrapolated projections? To answer these questions, a 

discussion on evaluating the Census Bureau’s state 

population projections is presented in several sections. 

First, the “Prior Research” section reviews recent 

literature on the evaluation of subnational population 

projections. The “State Projections for 2000” section 

discusses the methodology of the Census Bureau’s state 

population projections for 1995 to 2025. The 

“Extrapolated Projections” section identifies the 

procedures used to produce a simple extrapolated 

projection from the enumerated 1990 census counts and 

the post-1990 census estimates for 1995. Next, the 

“Calculation of MAPE” section explains how the basic 

statistical summary measures are derived. The 

“Transformed APEs and Test for Symmetry” section (1) 

describes how the APE distribution is examined to see 

if it is asymmetrical and needs correcting, and (2) 

presents the necessary transformation and conversion 

formula used to correct for the influence of extreme 

outliers. Finally, the results are discussed in the 

“Findings and Conclusions” section. 

Prior Research. Since literature on the evaluation of 

state-level population forecasts is not extensive, a 

broader review of research on the evaluation of 

subnational population estimates and projections 

provide useful guidelines for measuring forecast errors. 

Most frequently subnational population estimates or 

projections were evaluated using the mean absolute 

percentage error (MAPE), as indicated in evaluations by 

Swanson, Tayman, and Barr (2000), Tayman and 

Swanson (1999), Campbell (1997), Davis (1994), Smith 

and Sincich (1992). The MAPE is a measure of the 

central tendency of errors calculated by averaging the 

sum of the percentage differences between the 

projections and the census for states, ignoring the plus 

or minus sign. 

There are other alternative statistical measures used to 

evaluate errors (or differences) in subnational 

population estimates and projections. Davis (1994), 

evaluating post-1980 county populations estimates with 

the 1990 census used several summary measures. In 

addition to the MAPE, he used the mean algebraic 

percentage error (MALPE), weighted mean absolute 

percentage error (WMAPE), root mean square error 

(RMSE), index of dissimilarity (INDISS), median 

percent difference, and 90th percentile (or percent error 

at which 90 percent of the observations are lower). A 

complete description of these and other statistical 

measures is available in his study and in Armstrong 

(1977). Davis (1994) limits his discussion of findings 

to the “familiar” MAPE, since this summary statistic is 

highly correlated with the other summary measures for 

most of his tabulations, except for the MALPE and the 

90th percentile. 

In questioning the validity of the MAPE measure, 

Tayman and Swanson (1999) use the MAPE, the 

Symmetrical MAPE (SMAPE), and a class of measures 

known as Minimization-Estimators (M-estimators) to 

evaluate county forecasts for selected states. The Mestimators are described by Tayman and Swanson 

(1998:303) as “minimizing a more general objective 

function using maximum likelihood procedures rather 

than the sum of squared residuals associated with the 

sample mean, a sum that is highly sensitive to outliers.” 
Their findings suggest that a robust M-estimator like the 

Tukey-M statistic is a suitable alternative summary 

measure of forecast error. 

Swanson, et al., (2000) in evaluating subnational 

estimates, argues that the MAPE is reliable, easy to 

interpret, and clear in its presentation. On the other 

hand, they acknowledge that the MAPE in many cases 

lacks validity, since the APEs used in its calculation are 

right-skewed, such that extreme values can unduly 

influence the MAPE. To reduce the effects of outliers 

and asymmetrical distributions on the arithmetic mean, 

the alternatives are the median, the geometric mean, the 

weighted mean, the M-estimator, and data 

transformation. 

In their evaluation of county estimates, Swanson, et al., 

(2000) and Tayman and Swanson (1999) suggest that 

some extreme outliers may influence the MAPE by 

pulling its value upward so that it is not valid for a data 

set. They suggest validating the MAPE with tests for 

skewness and symmetry. If the MAPE is not valid, then 

a variant of the MAPE is calculated using a data 

transformation that corrects for the error inflation due to 

the outliers. 

This paper follows guidelines found in evaluation 

literature that recommend performing a data 

transformation to obtain a variant of the MAPE 

whenever the original distribution of APE is not 

symmetrical. In order to test for the effect that outliers 

have on the summary measure, Swanson, et al., used a 

modified Box-Cox (1964) transformation to obtain a 

symmetrical distribution of the original APEs, such that 

very large errors (or outliers) are compressed. The 

original distribution can be tested for asymmetry using 

graphic devices like histograms and boxplots; or 

statistical measures like the skewness coefficients 

(Snedecor and Cochran 1980:78); and the D’Agostino 

skewness test (D’Agostino, et al., 1990). When the 

original APEs are biased upward, then Swanson, et al., 

(2000) recommend calculating a summary measure 

from the transformed APEs that they refer to as MAPET. Since the MAPE-T, the average of the transformed 

APEs, is not represented in a familiar scale, they 

recommend using a nonlinear power function to 

statistically map the scales of the original error 

observations to the transformed observations. Swanson, 

et al., (2000) suggest calculating a re-expressed average 

(MAPE-R) in metrics, which is solved using linear 

regression results and the logarithm of MAPE-T. They 

argue that MAPE-R is a measure of central tendency of 

the error that is not influenced by the asymmetry and 

outliers that characterize the untransformed absolute 

percentage error distribution. Evaluating county 

estimates, Swanson, et.al. (2000:199) validated the 

MAPE-R statistic by finding consistent results using 

Tukey-M. 

Swanson, et al., (2000) identified some of the 

shortcomings of the alternative summary measures that 

may more accurately describe APEs as follows: Mestimators are not easy to explain, the median is 

“influenced by changes in the centermost observations 

resulting from grouping,” the geometric mean is 

affected by the logarithmic transformation which 

sometimes fails to yield a distribution that has optimal 

symmetry, and the loss function lacks a standard 

weighting scheme.

1

 Additionally, they acknowledge that 

MAPE-R was cumbersome to calculate and required the 

use of different statistical software packages. 

From a different perspective, economists Kolb and 

Stekler (1993) recommend using mean square error 

(MSE) to test whether a set of economic forecasts are 

statistically significant or better than “naïve” or “no 

change” forecasts. Using the simple extrapolated 

population difference as a standard, they used Theil’s U 

statistic to determine if the more complex projection 

model performs at least as well as the simplest model. 

Furthermore, evaluating state population projection 

models that ranged from simple to complex, Smith and 

Sincich (1992) concluded that the simple trend 

projections derived from linear extrapolation are just as 

accurate as the more complex models like the Census 

Bureau’s cohort-component projections. This analysis 

goes a step further than Smith and Sincich (1992) or 

Campbell (1997), the MAPE or MAPE-R from the 

Census Bureau’s 2000 state projection and the census 

population is compared to the same summary statistics 

from a standard, the simple extrapolated population 

projection and the census population. 

State Projections for 2000. The Census Bureau’s state 

population projections use detailed demographic 

accounting procedures and professional judgement in 

developing projection assumptions. The Census 

Bureau’s state projections were prepared for July 1 of 

each year from 1995 to 2025 using the cohortcomponent projection method. The cohort-component 

method is based on the traditional demographic 

accounting system: 

P1 = P0 + B - D + DIM - DOM + IM - EM 

where: P1 = population at the end of the period, P0 = 

population at the beginning of the period, and the 

1

For a discussion on the merits of the loss function versus MAPE / 

MAPE-R and other summary measures see Coleman (2000 and 2002). 
-- 

following events during the period: B = births, D = 

deaths, DIM = domestic in-migration, DOM = domestic 

out-migration, (both DIM and DOM are aggregations of 

the state-to-state migration flows), IM = immigration, 

and EM = emigration. 

Each component of population change -- births, deaths, 

internal migration (domestic or state-to-state migration 

flows), and international migration (immigration and 

emigration) utilizes separate projection assumptions 

for each birth cohort by single year of age, sex, race, 

and Hispanic origin. The race and Hispanic origin 

groups projected separately were: non-Hispanic White; 

non-Hispanic Black; non-Hispanic American Indian, 

Eskimo, and Aleut; non-Hispanic Asian and Pacific 

Islander; Hispanic White, Hispanic Black, Hispanic 

American Indian, Eskimo, and Aleut; Hispanic Asian 

and Pacific Islander. The detailed components used in 

the state population projections were derived from vital 

statistics, administrative records, 1990 census data, state 

population estimates (U.S. Bureau of the Census, 

1996c), and the middle series of the national population 

projections (Day, 1996). Detailed assumptions and 

procedures by which these data were generated by 

single year of age, sex, race, and Hispanic origin are 

described in detail in the report, “Population Projections 

for States, by Age, Sex, Race, and Hispanic Origin: 

1995 to 2025,” (Campbell, 1996). Overall, the 

assumptions concerning the future levels of fertility, 

mortality, and international migration are consistent 

with the assumptions developed for the national 

population projections (Day, 1996). 

Once separate data components were developed, the 

cohort-component method was applied, producing the 

detailed demographic projections. For the start of each 

projection year, the beginning population for each state 

was disaggregated into race and Hispanic origin 

categories (the eight groups previously identified) by 

sex and single year of age (0 to 84, and 85 plus). 

Components of change were individually applied to 

each group to project the next year's population. For 

the mortality component, each age-sex-race/ethnic 

group was survived forward one year using the 

pertinent survival rate. The internal redistribution of the 

population was accomplished by applying the 

appropriate state-to-state migration rates to the survived 

population in each state. The projected out-migrations 

were subtracted from the state of origin and added to 

the state of destination (as in-migrants). Next, the 

number of immigrants from abroad was added to each 

state, while the number of emigrants leaving each state 

was subtracted. Applying the appropriate agerace/ethnic-specific birth rates to females of 

childbearing age created the populations less than one 

year of age. The number of births by sex and 

race/ethnicity were survived forward and exposed to the 

appropriate migration rate to yield the population less 

than one year of age. The results were adjusted to be 

consistent with the national population projections by 

single years of age, sex, and race/ethnicity. Both the 

state and national population projection reports indicate 

that 1994 was the base year or the most recent year 

estimates were used to begin the forecast. However, the 

first year of the states projection horizon, 1995, was 

also adjusted to be consistent with a set of preliminary 

1995 state estimates only available by age and sex. The 

entire process was then repeated for each year of the 

projection. 

Two sets of state population projections were prepared 

and the only component specified differently in each 

projection model was the domestic migration 

component. The dynamic possibilities of change in 

state-to-state migration make it the most difficult 

component to forecast. Migration trends in the Census 

Bureau’s state projections are based on matched 

Internal Revenue Service (IRS) individual income tax 

return data sets containing 19 annual observations (from 

1975-76 to 1993-94) on each of 2,550 state-to-state 

migration flows. The two projection series provide 

users with different domestic migration scenarios since 

one set includes the rate of change in employment. 

Both sets of state projections were summed and 

adjusted by age, sex, race and Hispanic origin to agree 

with the middle series of the national population 

projection. 

The Census Bureau refers to Series A, which uses a 

time series model, as the “preferred series.” The first 

five years of the projection horizon (1995 to 2000) use 

the time series projection exclusively. The next ten 

years on the projection horizon (2000 to 2010) are 

interpolated toward the mean of the series, while the 

final 15 years (2010 to 2025) use the mean of the series 

exclusively. Series B is the economics model. State-tostate migration flows are derived from the Bureau of 

Economic Analysis projected rate of change in 

employment in the origin and the destination states. 

The “preferred series” was accepted as the projection 

model most likely to be the more accurate series based 

on results from ex-post facto evaluations.2 

The current sets of state population projections were 

previously evaluated in the Census Bureau using post2

 Ideally, the “preferred series” should be based on the evaluation of 

preliminary projections where the most recent launch year estimates 

are withheld from the projection so that estimates can be compared 

against the preliminary projections and prior evaluation results. 
1990 census estimates for July 1, 1996. The MAPEs 

calculated for 1996 were found to be fairly accurate (at 

or below the U.S. total of 0.40 percent for Series A and 

0.30 percent for Series B, for all regions except the 

West). That earlier evaluation also looked separately at 

the components of change and found that the birth 

component was the most accurate followed by the 

mortality component. The study concluded that both 

domestic and international migration components were 

more difficult to forecast accurately, and domestic 

migration was the least accurate component in the 

projections (Campbell, 1997). 

An important first step in this evaluation was to obtain 

projections that are consistent with the Census 2000 

reference date. While projections are available annually 

for July 1 of each year, there are none readily available 

for the target date centered on April 1, 2000. To match 

the projections to the census date, the solution was to 

linearly interpolate between the July 1, 1999 and July 1, 

2000 state population projection totals to obtain 

projections for the census date April 1, 2000 using 

Waring’s formula (see Shryock and Siegel, 1976:533).3 

Extrapolated Projections. A simple extrapolated total 

population projection for each state was used as a 

standard to evaluate the forecast error in the state’s total 

population projections. The extrapolated state 

population projections were derived by linearly 

extrapolating from the enumerated April 1, 1990 census 

population and the July 1, 1995 populations4 to April 1, 

2000 for every state. Smith and Sincich (1992), using 

several techniques in an evaluation which ranged from 

extrapolating growth rates and ratio shares to time series 

models, concluded that the linear extrapolation and ratio 

share models performed the best. Based on their 

recommendation, the following formula was used to 

extrapolate to April 1, 2000: 

Pt = P0 + X / Y(P0 – Pb) 

where the Pt is the state population projection for the 

target year (April 1, 2000), P0 is the state population 

size on July 1, 1995, Pb is the state population size in 

the base year (April 1, 1990), X is the number of years 

3

 Waring’s two-point interpolation formula was used for each state: 

f(x) = [f(a)*(x-b)/(a-b)]+[f(b)*(x-a)/(b-a)], where f(x) = April 1, 2000 

population; f(a) = July 1, 1999 population; f(b) = July 1, 2000 

population; and the proportions of the year were x = 2000+92/366; a = 

1999+182/365; and b = 2000+183/366. Wang (2002) using 

geometric interpolation to calculate the April 1, 2000 projections 

reported slightly different results. 

4

 The July 1, 1995 totals rather than July 1, 1994 totals were used 

since the state projections for the first target year 1995 were 

inflated/deflated to the preliminary 1995 state estimates. 

in the base period (5.25 years between April 1, 1990 to 

July 1, 1995) and Y is the number of years in the 

projection horizon (4.75 years between July 1, 1995 to 

April 1, 2000). No attempt was made to control the 

extrapolated state totals to independently derived 

national projection totals, which results in simple 

extrapolations for states not affected by 

inflation/deflation errors (from states being forced to 

sum prorata to the nation).5 

Calculation of MAPE. Forecasters tend to treat the 

terms projection, extrapolation, prediction, and forecast 

as synonymous (Armstrong, 2001:39). In this study, the 

initial forecast error refers to the percentage difference 

or error between a state’s total population projection 

and the Census 2000 population enumerated for the 

same date.6 Calculating the percentage error (PE) is 

useful in examining the error magnitude, direction of 

error, and identifying outliers in the evaluation of the 

state projections with census counts (see Table 1). The 

absolute percentage error (APE) is calculated without 

regard to the direction of error. The statistical measure 

that summarizes the APE distribution is the mean 

absolute percentage error (MAPE). The formula for the 

APEs and MAPE, ignoring the ± sign, are as follows: 

APEi = |Pi – Ci| ÷ Ci *100 

MAPE = Σ(APEi) ÷ N 

where N refers to the number of states (in the U.S., a 

region, or a division), P is the projected or extrapolated 

population, C is the census population, and i refers to 

the state. 

MAPEs were developed for the United States (the states 

and the District of Columbia), where N equals 51; and 

for each census region or division, where N equals the 

number of states in each region or division. All data 

evaluated are from unrounded state population 

projection figures reported in U.S. Bureau of the Census 

(2000, 1996a, 1996b, and 1996c) and Campbell (1996). 

Transformed APEs and Test for Symmetry. After 

APE and MAPE are calculated, Swanson, et al., 

(2000:194) suggest using transformed APEs that are 

5

 Comparison of the April 1, 2000 U.S. extrapolated state population 

totals with the Census 2000 total indicated that the extrapolated 

projections underprojected the U.S. total population by 2.12 percent. 

This is more accurate than the Series A and B projections, which were 

underprojected by 2.62 percent. 

6

 Projection evaluations based on estimates are ephemeral, since 

estimates may be corrected several times during the intercensal 

decade, only to be finalized after incorporating the latest census 

results. 
Table 1. Percentage Error in Census Bureau's Projections Series A and B, and Extrapolated Projections, 

2000 

Region, Division Projection Series A Projection Series B Extrapolated Projections 

and State Percentage Percentage Percentage 

Census Number Error Number Error Number Error 

NORTHEAST 

New England 

Maine...............................................1,2.....74, .....92..... 3 ......... 1,258,277 -1.31 1,249,996 -1.96 1,253,593 -1.67 

New Hampshire................................. 1,2.... 35,..... 786 1,220,918 -1.20 1,213,932 -1.77 1,183,513 -4.23 

Vermont.............................................608 ......... ,82..... 7 ... 615,398 1.08 606,141 -0.44 604,642 -0.69 

Massachusetts................................6,3.....49, .....097 6,192,883 -2.46 6,216,274 -2.09 6,125,219 -3.53 

Rhode Island..................................1,0.....48, ......... 319.. 997,150 -4.88 988,592 -5.70 977,458 -6.76 

Connecticut.....................................3,4.....05, .....56....5 3,283,684 -3.58 3,285,888 -3.51 3,263,436 -4.17 

Middle Atlantic 

New York......................................18, .....976 .....,4....57......... 18,144,490 -4.38 18,171,587 -4.24 18,267,830 -3.73 

New Jersey....................................... 8,4....14, .....35.... 0 8,167,064 -2.94 8,174,604 -2.85 8,139,921 -3.26 

Pennsylvania................................12, .....281 .......... ,054... 12,196,852 -0.69 12,213,689 -0.55 12,243,940 -0.30 

MIDWEST 

East North Central 

Ohio..............................................11, .....353 ......... ,140............ 11,311,801 -0.36 11,343,222 -0.09 11,425,043 0.63 

Indiana.............................................6,0.....80, .....48....5... 6,033,717 -0.77 6,047,746 -0.54 6,038,106 -0.70 

Illinois.............................................. 12,..... 419.....,2..... 93 12,040,250 -3.05 12,057,487 -2.91 12,191,274 -1.84 

Michigan.............................................. 9,938, ........ 44..... 4 .. 9,673,709 -2.66 9,703,959 -2.36 9,779,185 -1.60 

Wisconsin............................................ 5,363, ......... 675....... 5,317,128 -0.87 5,314,560 -0.92 5,331,994 -0.59 

West North Central 

Minnesota......................................... 4,9..... 19,..... 47..... 9 .... 4,819,724 -2.03 4,812,141 -2.18 4,821,686 -1.99 

Iowa................................................... 2,9.... 26,..... 32..... 4 ....... 2,897,210 -0.99 2,888,920 -1.28 2,900,595 -0.88 

Missouri............................................... 5,595, ......... 211..... 5,530,291 -1.16 5,535,993 -1.06 5,510,330 -1.52 

North Dakota....................................... 642.....,20....0.. 660,765 2.89 656,275 2.19 643,632 0.22 

South Dakota........................................ 754..... ,84....4. 774,908 2.66 768,379 1.79 758,890 0.54 

Nebraska............................................. 1,711, ........ 26..... 3 ... 1,702,280 -0.52 1,697,153 -0.82 1,690,255 -1.23 

Kansas............................................2,6.....88, .....41.....8........... 2,663,372 -0.93 2,669,488 -0.70 2,644,715 -1.63 

SOUTH 

South Atlantic 

Delaware............................................. 783.....,60.....0...... 765,340 -2.33 756,486 -3.46 763,341 -2.59 

Maryland..........................................5,2.....96, .....48..... 6 .... 5,264,225 -0.61 5,251,268 -0.85 5,278,573 -0.34 

District of Columbia..................... 572,059 524,104 -8.38 530,118 -7.33 506,624 -11.44 

Virginia............................................7,0.....78, .....51.....5.. 6,979,511 -1.40 6,949,653 -1.82 7,008,299 -0.99 

West Virginia..................................1,8.....08, ... 344 1,840,407 1.77 1,832,857 1.36 1,859,488 2.83 

North Carolina................................8,0.....49, ....313 7,750,347 -3.71 7,761,108 -3.58 7,707,711 -4.24 

South Carolina.................................4,0.....12, ...012 3,849,116 -4.06 3,843,811 -4.19 3,842,137 -4.23 

Georgia............................................8,1.....86, .....45..... 3 ... 7,843,520 -4.19 7,859,752 -3.99 7,854,730 -4.05 

Florida............................................. 15,..... 982.....,3..... 78.. 15,181,072 -5.01 15,197,736 -4.91 15,276,275 -4.42 

East South Central 

Kentucky..........................................4,0.....41, .....76..... 9 .... 3,988,406 -1.32 3,983,394 -1.44 4,018,483 -0.58 

Tennessee.......................................5,6.....89, .....28....3....... 5,638,679 -0.89 5,648,707 -0.71 5,598,854 -1.59 

Alabama.........................................4,4.....47, .....10....0....... 4,441,087 -0.14 4,426,738 -0.46 4,445,124 -0.04 

Mississippi........................................... 2,844, ........ 658 2,810,204 -1.21 2,819,891 -0.87 2,809,485 -1.24 

West South Central 

Arkansas.........................................2,6.....73, .....40.....0..... 2,624,492 -1.83 2,615,811 -2.15 2,604,110 -2.59 

Louisiana........................................... 4,4..... 68,..... 97..... 6 .. 4,420,298 -1.09 4,439,675 -0.66 4,453,055 -0.36 

Oklahoma............................................ 3,450, ........ 65..... 4 .... 3,367,555 -2.41 3,365,485 -2.47 3,397,212 -1.55 

Texas...........................................20, .....851 ......... ,820..............20,050,906 -3.84 20,106,138 -3.58 20,296,038 -2.67 

WEST 

Mountain 

Montana............................................... 902.....,19....5...... 946,058 4.86 933,518 3.47 934,667 3.60 

Idaho...............................................1,2.....93, .....95....3.......... 1,338,081 3.41 1,324,245 2.34 1,304,869 0.84 

Wyoming.............................................. 493.....,78.....2...... 522,521 5.82 517,023 4.71 504,236 2.12 

Colorado........................................... 4,3..... 01,..... 26..... 1 ... 4,149,286 -3.53 4,135,073 -3.86 4,155,703 -3.38 

New Mexico....................................... 1,8..... 19,.... 04.... 6 1,852,019 1.81 1,849,867 1.69 1,839,492 1.12 

Arizona...........................................5,1.....30, .....63.....2.... 4,770,861 -7.01 4,807,423 -6.30 4,717,977 -8.04 

Utah................................................... 2,2.... 33,..... 16..... 9 ....... 2,194,955 -1.71 2,202,945 -1.35 2,158,202 -3.36 

Nevada................................................ 1,998, ........ 25..... 7 ......... 1,856,460 -7.10 1,847,723 -7.53 1,827,109 -8.56 

Pacific 

Washington.....................................5,8.....94, .....12....1. 5,838,090 -0.95 5,810,813 -1.41 5,941,423 0.80 

Oregon............................................3,4.....21, .....39.....9........... 3,385,241 -1.06 3,384,972 -1.06 3,410,454 -0.32 

California........................................ 33,..... 871.....,6..48 32,462,610 -4.16 32,377,783 -4.41 33,244,071 -1.85 

Alaska................................................... 626..... ,93..... 2 ......... 650,925 3.83 630,985 0.65 652,081 4.01 

Hawaii............................................... 1,2....11, .....53..... 7 ......... 1,253,667 3.48 1,234,930 1.93 1,257,884 3.83 

UNITED STATES 281,421,906 274,061,914 -2.62 274,061,954 -2.62 275,462,964 -2.12 

Resident Population for April 1, 2000. Series A and B reflect different interstate migration assumptions and do not sum to the same total 

due to rounding. See text for explanations. Source: Population Division, U.S. Bureau of the Census. 
symmetrically distributed to produce an average that are graphic devices used to identify location, spread,

reflects more accurately the error

represented by most of the

observations. The MAPE in most

instances is based on a right-skewed,

asymmetrical distribution of absolute

percentage errors where outliers are

likely to pull the summary measure

of error upward, thereby overstating

the error represented by most of the

observations.

Examining data spread. Emerson

and Stoto (1983) and Swanson, et

al., (2000:194) recommend first

looking at the spread of the data to

determine if a distribution of APEs

appears to be unduly dominated by

Figure 1. Box-Cox Maximum-Likelihood Values for 

Series A, B, and Extrapolated Projections, 2000 

-300 

-250 

-200 

-150 

-100 

-50 

0 

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 

Power 

Source:PopulationDivision,U.S.BureauoftheCensus 

Box-Cox F 

SeriesA 

SeriesB 

Extrapolated 

outliers. They suggest applying a

transformation when the ratio of the largest value to the

smallest value exceeds 20.

Power transformation. Swanson, et al., (2000) have 

described a modified Box-Cox power transformation 

procedure to determine the most symmetrical 

transformed distribution of APEs, which was defined 

as: 

Y = (Xλ - λ) / λ, for X ≠ 0; or Y = LN(X), for X = 0, 

where X is the untransformed APE, Y is the 

transformed APE, and λ (lambda) is the power 

transformation constant. Lambda is determined by 

finding its value that maximizes the function: 

ML(λ)=-(N/2)(LN[(1/N)Σ(Yi –Y)

2

]+(λ – 1) (ΣLN(Xi))), 

where N is the number of states, Yi is the transformed 

APE, Y is the mean of the transformed APEs, X is the 

untransformed APE, and Σ represents the sum over all 

observations. A “coarse grid” search, set up in a 

Microsoft Excel spreadsheet, was used to solve for 

values of λ from –2 to 2 inclusive, using increments of 

.10. 

Figure 1 shows the nonlinear relationship of the BoxCox maximum-likelihood values associated with λ for 

each set of projection APEs. The optimal value of λ

(0.3) corresponds to the largest maximum-likelihood 

value (the smallest negative value in the graph) for 

Series A, Series B, and extrapolated projection APEs. 

Test for skewness. The original APEs and transformed 

APEs can be compared for skewness using graphic 

devices and the D’Agostino skewness test. Boxplots 

skewness, tail length, and outliers. The spread of the 

box represents 50 percent of the values between the first 

and third quartiles (qt). The boxplot for the 

untransformed APEs indicates a right-skewed 

distribution whenever the median (the crossbar in the 

box) is closer to the lower end of the box with a long 

upper tail. Similarly, the histogram can be used to 

visually identify asymmetrical and right skewed 

distributions (see Figures 2 to 4). 

Identifying extreme outliers. Swanson, et al., 

(2000:196) and Emerson and Strenio (1983:59-60) 

suggest that extreme population outliers for the original 

APEs should be mathematically identified using 

information from the boxplot. They suggest calculating 

extreme outlier cutoff points by multiplying the fourth 

spread or width of the middle half of the data by 1.5, 

adding that product to the third quartile value, and 

subtracting the resulting sum from the first quartile 

value. 

Calculating MAPE-T and MAPE-R. Once it is 

established that the transformation of APEs was 

necessary to correct for skewness and asymmetry, the 

MAPE-T is calculated from the transformed APE 

distribution using the APE and MAPE formula 

discussed above. The next step is to calculate MAPER, the re-expressed average that matches the original 

metric distribution, since it is not easy to interpret 

MAPE-T, the average of the transformed observations, 

which is in a different unit of measurement. Swanson, 

et al., (2000:199) recommended using a nonlinear 

power function to map the scales of the transformed and 

original observations such as: 

Xi = A * Yi

b

, 
where X is the original APE, Y is the transformed APE, 

and A and B are estimated parameters. The estimated 

parameters from the linear regression expressed in 

logarithm form: 

LN(X) = LN(A) + B [LN(Y)] 

can be used with MAPE-T to estimate: 

MAPE-R = e{A + (B * [LN(MAPE-T)])}. 

The resulting MAPE-R is reported to be a better 

measure of the central tendency of the error that is not 

influenced by the asymmetry and outliers that are found 

in the original absolute percentage error distribution. 

Figu r e 2. Dis t r ibu tio n of 

A b s o lu te Pe r ce n tag e Er r o r , 

Pr o je ctio n Se r ie s A , 2000 

0 

5 

10 

15 

20 

25 

30 

1 3 5 6 7 

APE 

Source:U.S.B ureau ofthe C ensus 

2 4 + 

Findings and Conclusions. The findings below 

suggest the need for a more robust summary measure of 

forecast error, such as MAPE-R, to evaluate the Census 

Bureau’s state population projections. Additionally, the 

use of 1) statistical cutoffs to identify extreme outliers, 

and 2) simple extrapolation as a standard to evaluate 

state population projections, provides statistical 

guidelines, rather than subjective conclusions for 

identifying forecast errors. Data issues associated with 

comparing the 1990-based projections with the 2000 

census are also presented below. 

Descriptive analysis. Clearly, the first step in this 

evaluation was to identify outliers by looking at the 

percent error for the magnitude and direction of forecast 

error (see Table 1). There appears to be some overall 

consistency in the direction of forecast error for states. 

All three sets of projections underprojected nearly fourfifths of the same states. The few states that were 

consistently overprojected for all three sets of 

projections were mostly in the West; i.e., Montana, 

Idaho, Wyoming, New Mexico, Alaska, and Hawaii, 

while the remainder were West Virginia in the South, 

and South Dakota and North Dakota in the Midwest. 

For three states, the direction of error was not consistent 

across all of the projections. Vermont was 

overprojected only on Series A, while Ohio and 

Washington were only overprojected on the 

extrapolated series. The most accurate state projections 

were those for Alabama on Series A (-0.14 percent) and 

the extrapolated projections (-0.04 percent), and Ohio 

on Series B (-0.09 percent). 

The range of error was smallest for Series B. The

Series B projections ranged from an underprojected

population of –7.5 percent for Nevada to an

overprojected population of 4.7 percent for Wyoming.

In comparison,

error in the Series A projections ranged from –8.4

Figu r e 3. Dis tr ib u tion o f 

Tran s form e d A b s o lu te Pe r ce n t ag e 

Er r o r , Pr o je ctio n Se r ie s A , 2000 

0 

5 

10 

15 

20 

25 

30 

1 2 5 7 

T ransA P E 

Source:U.S.B ureau ofthe C ensus 

3 4 6 + 

percent for the District of Columbia to 5.8 percent for 

Wyoming. The range of variation for the extrapolated 

projections was much wider than either Series A or B, 

ranging from –11.4 percent for the District of Columbia 

to 4.0 percent for Alaska. 

Among the three sets of population projections, two 

states, Nevada and Arizona, plus the District of 

Columbia, consistently stand out as extremely low 

outliers (see Table 1). Both sets of the Census Bureau 

projections were more accurate than the extrapolated 

projection for these three outliers. 

Clearly, the simple descriptive review so far suggests 

that Series B appears to be the most accurate. All three 

sets of projections had trouble accurately forecasting the 

up and down swings in population growth that occurred 

in the West during the 1990’s. Additionally, the quality 

of the 1990 census and post census estimates probably 

contributes to error in the projections; however, these 

issues were not the focus of the current study. 
MAPEs. A comparison of the MAPEs in Table 2 

suggests that Series B is the most accurate for the U.S. 

and the regions. The MAPE for Series B (2.44 percent) 

is slightly lower than the extrapolated projections (2.54 

percent) and Series A (2.63 percent). Furthermore, 

Series B tends to underproject the actual population (42 

of the 51 states which include the District of Columbia 

were too low; see Table 1). Nearly half (25 out of 51 

APEs) were within 2.5 points of the MAPE for Series 

B. 

Forecast error for the regions and divisions varied 

greatly, but tend to be consistent across the three sets of 

projections. The MAPEs for regions were highest for 

the West and lowest for the Midwest. Two-thirds of the 

division MAPEs were lower for the extrapolated 

projections than for the Series A and B projections. 

Due to the smaller number of observations for MAPEs 

at the region and division level, no attempt was made to 

validate the region or division results. The next step is 

to measure the variation in the APEs and determine if 

they are asymmetrically distributed. 

Spread and Asymmetry. Review of the data found a 

wide range of variation in the APEs which warrants the 

application of the Emerson-Stoto spread ratio to the 

original distribution. The transformation of the original 

distribution of errors was performed since the spread 

ratio of the highest original APE to the lowest original 

APE exceeds 20 for each of the three projections. In 

Series A, the highest APE, 8.38, is for the District of 

Columbia, while the lowest APE, 0.14, is for Alabama 

which results in a spread ratio of 60 (8.38/0.14). For 

Series B, the highest APE, 7.53 in Nevada, and the 

lowest APE, 0.09 in Ohio, results in a ratio of 84. The 

widest range occurs in the extrapolated projections 

where the highest APE, 11.44 for the District of 

Columbia, and the lowest APE, 0.04, for Alabama, 

results in spread ratio of 286. 

In contrast, transformed APEs for Series A and B, and 

the extrapolated projections had Emerson-Stoto spread 

ratios below 20. For example, the log-percentage errors 

for the transformed APEs (not shown) for Series A 

ranged from a high of 5.31 percent for the District of 

Columbia to a low of 0.83 percent for Alabama, which 

results in a spread ratio of 6. Similarly, the transformed 

APEs for Series B ranged from 5.11 percent for Nevada 

to 0.60 percent for Ohio, with a spread ratio of 9. The 

transformed APEs for the extrapolated projection 

ranged from a high of 5.92 percent for the District of 

Columbia to a low of 0.31 for Alabama, with a spread 

ratio of 19. 

After calculating transformed MAPEs (MAPE-T) using 

the modified Box-Cox method, histograms and boxplots 

are created to evaluate the shape of both the original 

and the transformed distributions. Histograms of the 

original APEs show data that are asymmetrical and 

slightly right-skewed (see Series A in Figure 2), while 

Table 2. Mean Absolute Percentage Error in State Population Projections, By 

Region And Division From Series A and B, And Extrapolated Projections, 2000 

Region and 

division Series A Series B Extrapolation 

United States 2.63 2.44 2.54 

Northeast 2.50 2.57 3.15 

New England 2.42 2.58 3.51 

Middle Atlantic 2.67 2.55 2.43 

Midwest 1.58 1.40 1.11 

East North Central 1.54 1.36 1.07 

West North Central 1.60 1.43 1.14 

South 2.60 2.58 2.69 

South Atlantic 3.50 3.50 3.90 

East South Central 0.89 0.87 0.86 

West South Central 2.29 2.21 1.79 

West 3.75 3.13 3.22 

Mountain 4.41 3.91 3.88 

Pacific 2.69 1.89 2.16 

Mean absolute percentage error (MAPEs) are results for 5 years-out from the 

1995 population. Based on the enumerated 2000 census counts, the 2000 

population for Series A, B, and Extrapolated Projections derived from the 

Absolute Percentage Errors calculated for the states and the District of Columbia, 

see text for detailed explanation. Source: U.S. Bureau of the Census. 

the histograms for the transformed APEs (TransAPEs) 

are symmetrical (see Series A in Figure 3). The same 

patterns were found for Series B and the extrapolated 

projections, but the histograms are not shown here. The 

histograms for the three sets of projections validated the 

need for the data transformations.

7 

Similarly, the boxplots in Figure 4 confirm that original 

APEs are asymmetrical and right-skewed for all three 

sets of projections. The median (the crossbar in the 

box) appears between the middle and bottom of the box, 

with a long upper tail for the three original APE 

distributions. The box spread is narrower for the 

transformed APEs (TransAPE or T-APE in the graph) 

and symmetrical with the median in the middle of the 

box (the same location as the mean), with a lower and 

upper tail of equal length. All of the calculations and 

graphs were derived using Microsoft Excel, which does 

not easily facilitate showing the asterisks for extreme 

outliers in the boxplot graphs. 

7

 The histogram sorts the APE and TransAPE distributions in a 

Microsoft Excel spreadsheet using bin range values of 1 to 7. The 

lowest bin values include APE or TransAPE values less than one, 

while the highest values were included with 7. A rule of thumb for 

choosing the APE bin range is one minus the highest value and one 

plus the lowest value. 
The skewness coefficients of 1.02 for Series A, 1.05 for comparison to the MAPE-R. The ratio of the MAPE to

Series B, and 1.74 for the extrapolated projections the median (absolute percent error) is another useful

imply that each set of projections was asymmetrical and descriptive tool that shows the overstated forecast error

right-skewed. A symmetrical distribution would have a (Tayman and Swanson 1999:307). In Table 3, the

skewness coefficient of

zero. The D’Agostino

skewness test suggests that

the null hypothesis (the data

are not skewed) should be

rejected (p = 0.000) for all

three of the original APE

distributions.

Extreme outliers. The

Emerson-Strenio “fourth

spread” procedure was used

to identify extreme outliers

among the original APE

distributions. The “fourth

spread” upper cutoff values

were 7.80 for Series A, 7.38

FIGURE 4. BOXPL OT S OF A PE & T r an s A PE FOR SERIES A , 

SERIES B, A ND EXT RA P OL A T ED P ROJEC T IONS, 2 00 0 

0 

2 

4 

6 

8 

10 

12 

14 

Source: Population Division, U.S. Bureau of the Census 

A PE & Tr ans A PE 

A PE Trans A PE A PE TransA PE A PE TransA PE 

Series A Series B Extrapolated 

for Series B, and 7.93 for

the extrapolated projections. The District of Columbia

and Nevada were identified as the only extreme outlier

in Series A and B, respectively. In the extrapolated

projections, three extreme outliers with APEs above the

cutoff value were Arizona, the District of Columbia,

and Nevada.

MAPE-R results. While the individual transformed

APEs are of no interest in the evaluation of the state

projections, the summary statistic for the transformed

APEs is useful. Additionally, the MAPE-T which

equals 3.21, 3.10, and 3.04 for Series A, B, and the

extrapolated projections, respectively, is difficult to

explain, since the results are log-based. In order to

explain the transformed summary statistic in the original

metric format, the next step is to re-express MAPE-T as

MAPE-R using the logarithm regression results (see

formula discussed earlier).8 The MAPE-R derived for

Series B projections at 2.06 percent is slightly more

accurate than Series A at 2.24 percent. Additionally,

the MAPE-R for the extrapolated projection at 2.00

percent was about the same as the Series B projections

and slightly more accurate than the Series A

projections.

Table 3 shows MAPE overstating the forecast error in

8

 The logarithm regression results used to derive MAPE-R were (1) 

Series A: A = -1.950, B = 2.366, R2

 = 0.992, standard error (SE) = 

0.074; (2) Series B: A = -1.850, B = 2.275, R2

 = 0.986, and SE = 

0.104 and (3) the extrapolated projections: A = -1.649, B = 2.104, R2 

= 0.968, and SE = 0.194. 

MAPE-to-median ratios confirm that MAPE overstates 

forecast error, since the ratios are greater than 1.0 for all 

three projections. A different conclusion would have 

been drawn if the original error distribution were not 

corrected for skewness and asymmetry. 

Initially, Theil’s U was considered as a potential 

summary measure to determine if the Census Bureau’s 

forecast models were more accurate than the 

extrapolated projections. However, it was not accepted 

as a valid measure since the distributions of APEs were 

found to be skewed and asymmetrical. Armstrong and 

Collopy (1992:77) reported that RMSE (used to derive 

Theil’s U) is unreliable due to its poor protection 

against outliers.9 Additional issues related to the 

guidelines used for choosing appropriate forecast error 

measures are discussed by Ahlburg (1992). 

To summarize, MAPE-R was used to replace the 

summary measure MAPE in the evaluation of the 

Census Bureau’s projections since the data distributions 

were skewed and asymmetrical. The results show that 

the Census Bureau’s state population projections for 

April 2000 (Series B - the economic model) had the 

least forecast error, with an average absolute percentage 

9

 Theil’s U, interpreted as the RMSE of the projections model divided 

by the RMSE of the extrapolated or no-change model, is derived from 

the formula: U = [Σ (Pi – Ei)

2

]

1⁄2 / [Σ Ei

2

]

1⁄2, where Pi refers to the 

projection error for each state, and Ei is the corresponding 

extrapolation error used as the standard for each state (see Kolb and 

Stekler, 1993). Using either absolute or percent change or the 

transformed percent change, Theil’s U coefficients found both Series 

A and B with fewer errors than the extrapolated projections. 
error of 2.06 percent. This is slightly better than Series 

A, with an average absolute percentage error of 2.24 

percent. The forecast error in the Census Bureau’s 

Series B projections was the same as that found in the 

extrapolated projections (2.00 percent), while the 

extrapolated projections slightly out-performed Series A 

- the preferred series. All three projections consistently 

underprojected approximately four-fifths of the states 

(40 states in the extrapolated projections, 41 states in 

Series A, and 42 states in Series B) out of a total of 51 

states (including the District of Columbia). The widest 

range of variation and the most extreme outliers were 

found for the extrapolated projections. 

Table 3. Comparison of the MAPE, Ratio to Median, and MAPE-R 

for Series A, Series B, and Extrapolated Projections, 2000. 

Ratio of MAPE 

MAPE to Median MAPE-R 

Series A 2.63 1.13 2.24 

Series B 2.44 1.24 2.06 

Extrapolated 2.54 1.38 2.00 

Summary Statistics for projections evaluation using enumerated

Census 2000 results, see text for detailed explanation.

Source: Population Division, U.S. Bureau of Census.

An added feature of the extrapolated projection is that 

base period (1990-1995) growth trends are held 

constant over the projection horizon (1995-2000). This 

information is useful for identifying changes in trends 

(or error) between the base period and the projection 

horizon. Ideally, when the extrapolated projection error 

is zero, there is no evidence of change in the pattern of 

growth between the base period and the projection 

horizon. In this study, nearly a third of the states (16 

states with error ranging from 1.0 percent to -1.0 

percent) showed little change in the 1990-95 pattern of 

population growth extrapolated to 2000. 

Several issues or differences between the 1990 and 

2000 censuses not examined in this study probably 

affect the accuracy of the state projections. First, 

adjusted 1990 census counts were not used as the base 

year and any undercoverage in the 1990 census is 

carried throughout the post-1990 estimates and 

projections.10 Second, this evaluation only examines 

the aggregated population totals and does not evaluate 

the separate component totals, such as births, deaths, 

10 An evaluation of the factors affecting the accuracy of the state 

projections, such as census undercount, estimates error, and error in 

the projected components of change have been addressed by Wang 

(2002). 

state-to-state, and international migration, by age, sex, 

and race/Hispanic origin. The domestic migration and 

international migration components are the most 

difficult to adequately baseline or project. Additionally, 

retrospective census information on place of residence 

during 1985-90 used in the projections may not reflect 

changes in the age pattern of migrants during the 

1990’s. Third, the race/Hispanic origin categories are 

quite differently defined in each of the censuses, the 

vital statistics, and administrative records. Fourth, the 

state projections use national data as a proxy in the 

absence of detailed demographic components. Mulder 

(2001) evaluating the Census Bureau’s national 

population projections produced between 1947 and 

1994 has documented the inability of past projections to 

accurately forecast turning points, particularly for the 

immigration and fertility components of the projections. 

Finally, there is the issue of the multi-dimensional 

raking, in other words the state projection results are 

aggregated pro-rata to the national estimates and 

projections for consistency at the national level by age, 

sex, and race/Hispanic origin. 

The 2000 state population projections appear to be 

slightly more accurate than vintage projections 

produced decades earlier. Wetrogan and Campbell 

(1990) calculated MAPEs ranging from 3.0 percent to 

5.2 percent for a 5-year projection horizon in their 

evaluations of 1970’s and 1980’s Census Bureau 

projections using corresponding 1970’s and post-1980 

census estimates.11 They reported U.S. MAPEs from 

the Census Bureau’s 1987 projection at 0.5 percent, 1.1 

percent, and 1.6 percent for one-, two-, and three-year 

projection horizons, respectively. MAPEs of 0.5 

percent per year appear to be a reasonable level of 

accuracy to expect for state population projections over 

a short term or 5-year projection horizon. 

This study found that the Census Bureau’s 2000 state 

population projections are as accurate as simple 

extrapolated projections and have fewer extreme 

outliers. Further evaluation of the detailed demographic 

components should aid in identifying areas of the 

projection model that needs to be improved. It appears 

that tests for skewness and asymmetry are necessary to 

validate the use of the popular summary measure, such 

as the MAPE or its variant MAPE-R. 

The advantage of using MAPE-R in conjunction with 

the original absolute percentage error is that users are 

more familiar with interpreting this summary measure 

and MAPE-R resolves the central tendency issues 

11 An evaluation of the 1970’s and 1980’s state population projections 

using the 1990 post census estimates, final intercensal estimates, and 

MAPE-R would probably yield lower forecast errors. 
whenever MAPE is found to be invalid. Clearly, a 

drawback to its widespread use is the cumbersome 

statistical calculation needed to carry out its application; 

nevertheless, all of the results for this evaluation were 

carried out in Microsoft Excel spreadsheets.12 With a 

few modifications, the spreadsheets can be used to 

evaluate error in other small subnational estimates or 

projection data sets. 

Acknowledgments. I would like to thank Greg 

Spencer and Edwin Byerly for inspiring this research, 

and Ching-Li Wang for his comments and sharing 

evaluation results. 

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