1

Puerto Rico Accuracy Document 

Accuracy of the Data (2010) 

INTRODUCTION 

The data contained in these data products are based on the Puerto Rico Community Survey 

(PRCS) sample interviewed from January 1, 2010 through December 31, 2010. The PRCS 

sample is selected from all municipios in Puerto Rico (PR). Data for Puerto Rico was first 

released in 2005. In 2006, the PRCS began collection of data from sampled persons in group 

quarters (GQs) – for example, military barracks, college dormitories, nursing homes, and 

correctional facilities. Persons in group quarters are included with persons in housing units 

(HUs) in all 2010 PRCS estimates that are based on the total population. All PRCS population 

estimates from years prior to 2006 include only persons in housing units. The PRCS, like any 

other statistical activity, is subject to error. The purpose of this documentation is to provide data 

users with a basic understanding of the PRCS sample design, estimation methodology, and 

accuracy of the PRCS data. The PRCS is sponsored by the U.S. Census Bureau, and is part of 

the 2010 Decennial Census Program. 

Additional information on the design and methodology of the PRCS, including data collection 

and processing, can be found at: 

http://www.census.gov/acs/www/methodology/methodology_main/. 

The 2010 Accuracy of the Data for the United States can be found at: 

http://www.census.gov/acs/www/Downloads/data_documentation/Accuracy/ACS_Accuracy_of_

Data_2010.pdf. 
2

TABLE OF CONTENTS

INTRODUCTION .......................................................................................................................... 1

DATA COLLECTION ................................................................................................................... 3

SAMPLING FRAME ..................................................................................................................... 3

SAMPLE DESIGN ......................................................................................................................... 4

WEIGHTING METHODOLOGY .................................................................................................. 8

CONFIDENTIALITY OF THE DATA ........................................................................................ 14

ERRORS IN THE DATA ............................................................................................................. 15

MEASURES OF SAMPLING ERROR ....................................................................................... 16

CALCULATION OF STANDARD ERRORS ............................................................................. 18

Sums and Differences of Direct Standard Errors ...................................................................... 18

Ratios ........................................................................................................................................ 19

Proportions/percents ................................................................................................................. 19

Percent Change ......................................................................................................................... 20

Comparing 1-Year Period Estimates with Overlapping 3-Year Period Estimates: .................. 20

Comparing 1-Year Period Estimates with Overlapping 5-Year Period Estimates: .................. 20

TESTING FOR SIGNIFICANT DIFFERENCES ........................................................................ 21

EXAMPLES OF STANDARD ERROR CALCULATIONS....................................................... 22

CONTROL OF NONSAMPLING ERROR ................................................................................. 24

ISSUES WITH APPROXIMATING THE STANDARD ERROR OF LINEAR 

COMBINATIONS OF MULTIPLE ESTIMATES ...................................................................... 29
3

DATA COLLECTION 

Housing Units 

The PRCS employs three modes of data collection: 

• Mailout/Mailback 

• Computer Assisted Telephone Interview (CATI) 

• Computer Assisted Personal Interview (CAPI) 

The general timing of data collection is: 

Month 1: Addresses in sample that are determined to be mailable are sent a questionnaire via 

the U.S. Postal Service. 

Month 2: All mail non-responding addresses with an available phone number are sent to 

CATI. 

Month 3: A sample of mail non-responses without a phone number, CATI non-responses, and 

unmailable addresses are selected and sent to CAPI. 

Note that mail responses are accepted during all three months of data collection. 

 Group Quarters 

Field representatives have several options available to them for data collection. These include 

completing the questionnaire while speaking to the resident in person or over the telephone, 

conducting a personal interview with a proxy, such as a relative or guardian, or leaving paper 

questionnaires for residents to complete for themselves and then pick them up later. This last 

option is used for data collection in Federal prisons. 

Group Quarters data collection spans six weeks, except for Federal prisons, where the data 

collection time period is four months. All Federal prisons are assigned to September with a four 

month data collection window. 

SAMPLING FRAME 

Housing Units 

The universe for the PRCS consists of all valid, residential housing unit addresses in all 

municipios in Puerto Rico. The Master Address File (MAF) is a database maintained by the 

Census Bureau containing a listing of residential and commercial addresses in the U.S. and 

Puerto Rico. The MAF is updated twice each year with the Delivery Sequence Files provided by 

the U.S. Postal Service, however this update covers only the U.S. The DSF does not provide 
4

changes and updates to the MAF for Puerto Rico. The MAF is also updated with the results 

from various Census Bureau field operations, including the PRCS. 

Group Quarters 

The main source of group quarters (GQ) in the ACS GQ sampling frame is the 2000 Census. 

Updates to the 2000 Census GQ inventory come from various operations such as the GQ 

Incomplete Information Operation (IIO) at the Census Bureau’s National Processing Center in 

Jeffersonville, Indiana, the Census Questionnaire Resolution (CQR) Program and GQs closed on 

Census day. GQs that were closed on Census day were added from a preliminary inventory file 

obtained from Decennial Systems and Contract Management Office (DSCMO) since it was 

possible that while these GQs were closed on Census day, they could be open when the ACS 

contacts them. The GQ frame also underwent an unduplication process. Updates identified 

during various field operations, including ACS GQ data collection operations and the 

Demographic Area Address Listing (DAAL), are also reflected in the GQ frame. Headquarters 

Staff uses the state Department of Corrections websites to update the inventory and attributes of 

state prisons. The inventory of federal prisons in the ACS GQ frame is kept current through a 

partnership with the Bureau of Prisons where they provide ACS with an annual updated list of 

federal prisons. After the frame was put together from these different sources, it was then sorted 

geographically. 

SAMPLE DESIGN 

Housing Units 

The PRCS employs a two-phase, two-stage sample design. The PRCS first-phase sample 

consists of two separate samples, Main and Supplemental, each chosen at different points in 

time. Together, these constitute the first-phase sample. Both the Main and the Supplemental 

samples are chosen in two stages referred to as first- and second-stage sampling. Subsequent to 

second-stage sampling, sample addresses are randomly assigned to one of the twelve months of 

the sample year. The second-phase of sampling occurs when the CAPI sample is selected (see 

Section 2 below). 

The Main sample is selected during the summer preceding the sample year. Approximately 99 

percent of the sample is selected at this time. Each address in sample is randomly assigned to 

one of the 12 months of the sample year. Supplemental sampling occurs in January/February of 

the sample year and accounts for approximately 1 percent of the overall first-phase sample. The 

Supplemental sample is allocated to the last nine months of the sample year. A sub-sample of 

non-responding addresses and of any addresses deemed unmailable is selected for the CAPI data 

collection mode. 

Several of the steps used to select the first-phase sample are common to both Main and 

Supplemental sampling. The descriptions of the steps included in the first-phase sample selection 
5

below indicate which are common to both and which are unique to either Main or Supplemental 

sampling. 

1. First-phase Sample Selection 

• First-stage sampling (performed during both Main and Supplemental sampling) – First 

stage sampling defines the universe for the second stage of sampling through two steps. 

First, all addresses that were in a first-phase sample within the past four years are 

excluded from eligibility. This ensures that no address is in sample more than once in 

any five-year period. The second step is to select a 20 percent systematic sample of 

“new” units, i.e. those units that have never appeared on a previous MAF extract. Each 

new address is systematically assigned to either the current year or to one of four backsamples. This procedure maintains five equal partitions of the universe. 

• Assignment of blocks to a second-stage sampling stratum (performed during Main 

sampling only) – Second-stage sampling uses five sampling strata in PR. The stratum 

level rates used in second-stage sampling account for the first-stage selection 

probabilities. These rates are applied at a block level to addresses in PR by calculating a 

measure of size for Municipios. 

The measure of size is an estimate of the number of occupied HUs in the Municipio. This 

is calculated by multiplying the number of PRCS addresses by the occupancy rate from 

Census 2000 at the block level. A measure of size for each Census Tract is also 

calculated in the same manner. 

 

Each block is then assigned the smallest measure of size from the set of all entities of 

which it is a part. The second-stage sampling strata and the overall first-phase sampling 

rates are shown in Table 1 below. 

• Calculation of the second-stage sampling rates (performed during Main sampling only) – 

The overall first-phase sampling rates given in Table 1 are calculated using the 

distribution of PRCS valid addresses by second-stage sampling stratum in such a way as 

to yield an overall target sample size for the year of approximately 36,000. These rates 

also account for expected growth of the HU inventory between Main and Supplemental 

of roughly 1 percent. The first-phase rates are adjusted for the first-stage sample to yield 

the second-stage selection probabilities. 

• Second-stage sample selection (performed in Main and Supplemental) – After each block 

is assigned to a second-stage sampling stratum, a systematic sample of addresses is 

selected from the second-stage universe (first-stage sample) within each municipio. 

• Sample Month Assignment (performed in Main and Supplemental) – After the second 

stage of sampling, all sample addresses are randomly assigned to a sample month. 
6

Addresses selected during Main sampling are allocated to each of the 12 months. 

Addresses selected during Supplemental sampling are assigned to the months of AprilDecember. 

Table 1. First-phase Sampling Rate Categories for Puerto Rico 

Sampling Rate Category Sampling Rates 

Blocks in smallest Municipios 

(MOS1

 < 200) 10.0% 

Blocks in smaller Municipios 

(200 ≤ MOS < 800) 7.86% 

Blocks in small Municipios 

(800 ≤ MOS ≤ 1200) 3.93% 

Blocks in large Municipios 

(MOS >1200, TRACTMOS2 ≥ 2000) 1.97% 

All other blocks 

(MOS >1200, TRACTMOS < 2000) 2.62% 

 1MOS = Measure of size. 

 2TRACTMOS = Census Tract measure of size. 

2. Second-phase Sample Selection – Subsampling the Unmailable and Non-Responding 

Addresses 

All addresses determined to be unmailable are subsampled for the CAPI phase of data 

collection at a rate of 2-in-3. Unmailable addresses do not go to the CATI phase of data 

collection. Subsequent to CATI, all addresses for which no response has been obtained prior 

to CAPI are subsampled based on the expected rate of completed interviews at the tract level 

using the rates shown in Table 2. 

 Table 2. Second-Phase (CAPI) Subsampling Rates for Puerto Rico 

Address Characteristics CAPI Subsampling Rate

Unmailable addresses 66.7% 

Mailable addresses 50.0% 

Group Quarters

The GQ sampling frame is divided into three strata: one for small GQs (having 15 or fewer 

people according to Census 2000 or updated information), one for GQs that were closed on 

Census Day 2000, and one for large GQs (having more than 15 people according to Census 2000 

or updated information). GQs in the first two strata are sampled using the same procedure, and 

GQs in the large stratum are sampled using different a method. The small GQ stratum and the 
7

 

stratum for GQs closed on Census Day are combined into one sampling stratum and sorted 

geographically 1

. 

1. First-phase Sample Selection for Small GQ Stratum 

• First-stage sampling - Small GQs are only eligible to be selected for the PRCS once 

every five years. To accomplish this, the first stage sampling procedure systematically 

assigned all small GQs to one of five partitions of the universe. Each partition was 

assigned to a particular year (2010-2014) and the one assigned to 2010 became the first 

stage sample. In future years, each new GQ will be systematically assigned to one of the 

five samples. These samples are rotated over five year periods and become the universe 

for selecting the second stage sample. 

 

• Second-stage sampling - A simple 1-in-8 systematic sample of the GQs in the first stage 

sample is selected. Regardless of their actual size, all GQs in the small stratum have the 

same probability of selection. Since the first stage sample is 20% of the universe, this 

yields the targeted sampling rate of 2.5%. 

2. Sample Selection for the Large GQ Stratum 

Unlike housing unit address sampling and the small GQ sample selection, the large GQ 

sampling procedure has no first-stage in which sampling units are randomly assigned to one 

of five years. All large GQs are eligible for sampling each year. The large GQ samples are 

selected using a two-phase design. 

• First-phase Sampling - In the large GQ stratum, GQ hits are selected using a systematic 

PPS (probability proportional to size) sample, with a target sampling rate of 2.5%. A hit 

refers to a grouping of 10 expected interviews. GQs are selected with probability 

proportional to its most current count of persons or capacity. For stratification, and for 

sampling the large GQs, a GQ measure of size (GQMOS) is computed, where GQMOS is 

the expected population of the GQ divided by 10. This reflects that the GQ data is 

collected in groups of 10 GQ persons. People are selected in hits of 10 in a systematic 

sample of 1-in-40 hits. All GQs in this stratum are eligible for sampling every year, 

regardless of their sample status in previous years. For large GQs, hits can be selected 

multiple times in the sample year. For most GQ types, the hits are randomly assigned 

throughout the year. Some GQs may have multiple hits with the same sample date if 

more than 12 hits are selected from the GQ. In these cases, the person sample within that 

month is unduplicated. 

3. Sample Month Assignment 

1

Note that all references to the small GQ stratum include both small GQs and GQs closed on 

Census day. 
8

In order to assign a panel month to each hit, all of the GQ samples from Puerto Rico are 

combined and sorted by small/large stratum and second-phase order of selection. 

Consecutive samples are assigned to the twelve panel months in a predetermined order, 

starting with a randomly determined month, except for Federal prisons. Correctional 

facilities have their sample clustered. All Federal prisons hits are assigned to the September 

panel. In non-Federal correctional facilities, all hits for a given GQ are assigned to the same 

panel month. However, unlike Federal prisons, the hits in state and local correctional 

facilities are assigned to randomly selected panels spread throughout the year. 

4. Second Phase Sample: Selection of Persons in Small and Large GQs 

Small GQs in the second phase sampling are “take all,” i.e., every person in the selected GQ 

is eligible to receive a questionnaire. If the actual number of persons in the GQ exceeds 15, a 

field subsampling operation is performed to reduce the total number of sample persons 

interviewed at the GQ to 10. If the actual number of persons in the GQ is 10 or fewer, then 

the group size will be less than 10. 

For each hit in the large GQs, the automated instrument uses the population count at the time 

of the visit and selects a subsample of 10 people from the roster. The people in this 

subsample receive the questionnaire. 

WEIGHTING METHODOLOGY 

The estimates that appear in this product are obtained from a raking ratio estimation procedure 

that results in the assignment of two sets of weights: a weight to each sample person record and a 

weight to each sample housing unit record. Estimates of person characteristics are based on the 

person weight. Estimates of family, household, and housing unit characteristics are based on the 

housing unit weight. For any given tabulation area, a characteristic total is estimated by 

summing the weights assigned to the persons, households, families or housing units possessing 

the characteristic in the tabulation area. Each sample person or housing unit record is assigned 

exactly one weight to be used to produce estimates of all characteristics. For example, if the 

weight given to a sample person or housing unit has a value 40, all characteristics of that person 

or housing unit are tabulated with the weight of 40. 

The weighting is conducted in two main operations: a group quarters person weighting operation 

which assigns weights to persons in group quarters, and a household person weighting operation 

which assigns weights both to housing units and to persons within housing units. The group 

quarters person weighting is conducted first and the household person weighting second. The 

household person weighting is dependent on the group quarters person weighting because 

estimates for total population, which include both group quarters and household population, are 

controlled to the Census Bureau’s official 2010 total resident population estimates. 
9

Group Quarters Person Weighting 

Each GQ person is first assigned to a Population Estimates Program Major GQ Type Group (the 

type groups used by the Population Estimates Program). The major type groups used are: 

Table 3: Population Estimates Program Major GQ Type Groups 

Major GQ Type Group Definition Institutional / Non-Institutional 

1 Correctional Institutions Institutional 

2 Juvenile Detention Facilities Institutional 

3 Nursing Homes Institutional 

4 Other Long-Term Care Facilities Institutional 

5 College Dormitories Non-Institutional 

6 Military Facilities Non-Institutional 

7 Other Non-Institutional Facilities Non-Institutional 

The procedure used to assign the weights to the GQ persons is performed independently within 

the commonwealth of Puerto Rico. The steps to assign the weights to the GQ persons are as 

follows: 

• Base Weight—The initial base weight after the first phase of sampling is the inverse of 

its first-phase sampling rate, which is equal to 40 for all sample cases in 2010. This 

initial base weight is then adjusted for the second-phase sampling that occurs at the time 

of interview. 

• Non-Interview Factor—This factor adjusts the weight of all responding GQ persons to 

account for the non-responding GQ persons including those persons contained in whole 

non-responding GQs. The non-interview factor is computed and assigned using the 

following groups: 

 Major GQ Type Group × Municipio 

• GQ Person Post-stratification Factor—This factor adjusts the GQ person weights so that 

the weighted sample counts equal independent population estimates from the Population 

Estimates Program by Major Type Group for the commonwealth of Puerto Rico. 

Because of collapsing of groups in applying this factor, only total GQ population is 

assured of agreeing with the Census Bureau’s official 2010 population estimates. The 

GQ person post-stratification factor is computed and assigned using the following 

groups: 

 Major GQ Type Group 

• Rounding—The final GQ person weight is rounded to an integer. Rounding is performed 

so that the sum of the rounded weights is within one person of the sum of the unrounded 
10

weights for any of the groups listed below: 

 Major GQ Type Group 

 Major GQ Type Group × Municipio 

Housing Unit and Household Person Weighting 

The housing unit and household person weighting use weighting areas built from collections of 

whole municipios. The 2010 Census data are used to group municipios of similar demographic 

and social characteristics. The characteristics considered in the formation include: 

• Percent in poverty 

• Percent renting 

• Percent in rural areas 

• Race/ethnicity, age, and sex distribution 

• Distance between the centroids of the municipios 

• Core-based Statistical Area status 

Each weighting area is also required to meet a threshold of 400 expected person interviews in the 

2010 PRCS. The process also tries to preserve as many municipios that met the threshold to 

form their own weighting areas. In total, there are 55 weighting areas formed from the 78 

municipios in Puerto Rico. 

The estimation procedure used to assign the weights is then performed independently within 

each of the PRCS weighting areas. 

1. Initial Housing Unit Weighting Factors—This process produces the following factors: 

• Base Weight (BW)—This initial weight is assigned to every housing unit as the inverse of 

its block’s sampling rate. 

• CAPI Subsampling Factor (SSF)—The weights of the CAPI cases are adjusted to reflect 

the results of CAPI subsampling. This factor is assigned to each record as follows: 

Selected in CAPI subsampling: SSF = 2.0 

Not selected in CAPI subsampling: SSF = 0.0 

Not a CAPI case: SSF = 1.0 

Some sample addresses are unmailable. A two-thirds sample of these is sent directly to 

CAPI and for these cases SSF = 1.5. 

• Variation in Monthly Response by Mode (VMS)—This factor makes the total weight of 

the Mail, CATI, and CAPI records to be tabulated in a month equal to the total base 
11

weight of all cases originally mailed for that month. For all cases, VMS is computed and 

assigned based on the following groups: 

Weighting Area × Month 

• Noninterview Factor (NIF)—This factor adjusts the weight of all responding occupied 

housing units to account for nonresponding housing units. The factor is computed in two 

stages. The first factor, NIF1, is a ratio adjustment that is computed and assigned to 

occupied housings units based on the following groups: 

Weighting Area × Building Type × Tract 

A second factor, NIF2, is a ratio adjustment that is computed and assigned to occupied 

housing units based on the following groups: 

Weighting Area × Building Type × Month 

NIF is then computed by applying NIF1 and NIF2 for each occupied housing unit. 

Vacant housing units are assigned a value of NIF = 1.0. Nonresponding housing units are 

now assigned a weight of 0.0. 

• Noninterview Factor—Mode (NIFM)—This factor adjusts the weight of the responding 

CAPI occupied housing units to account for CAPI nonrespondents. It is computed as if 

NIF had not already been assigned to every occupied housing unit record. This factor is 

not used directly but rather as part of computing the next factor, the Mode Bias Factor. 

NIFM is computed and assigned to occupied CAPI housing units based on the following 

groups: 

Weighting Area × Building Type (single or multi unit) × Month 

Vacant housing units or non-CAPI (mail and CATI) housing units receive a value of 

NIFM = 1.0. 

• Mode Bias Factor (MBF)—This factor makes the total weight of the housing units in the 

groups below the same as if NIFM had been used instead of NIF. MBF is computed and 

assigned to occupied housing units based on the following groups: 

Weighting Area × Tenure (owner or renter) × Month × Marital Status of the Householder 

(married/widowed or single) 

Vacant housing units receive a value of MBF = 1.0. MBF is applied to the weights 

computed through NIF. 
12

2. Person Weighting Factors—Initially the person weight of each person in an occupied housing 

unit is the product of the weighting factors of their associated housing unit 

(BW × ... × MBF). At this point everyone in the household has the same weight. The 

person weighting is done in a series of three steps which are repeated until a stopping 

criterion is met. These three steps form a raking ratio or raking process. These person 

weights are individually adjusted for each person as described below. 

The three steps are as follows: 

• Municipio Controls Raking Factor (SUBEQRF) - This factor is applied to individuals 

based on their geography. It adjusts the person weights so that the weighted sample 

counts equal independent population estimates of total population for the municipio. For 

those municipios which are their own weighting area, this adjustment factor will be 1.0. 

Because of later adjustments to the person weights, total population is not assured of 

agreeing exactly with the official 2010 population estimates for municipios which are not 

their own weighting area. 

• Spouse Equalization/Householder Equalization Raking Factor (SPHHEQRF)—This factor 

is applied to individuals based on the combination of their status of being in a marriedcouple or unmarried-partner household and whether they are the householder. All persons 

are assigned to one of four groups: 

1. Householder in a married-couple or unmarried-partner household 

2. Spouse or unmarried partner in a married-couple or unmarried-partner household 

(non-householder) 

3. Other householder 

4. Other non-householder 

The weights of persons in the first two groups are adjusted so that their sums are each 

equal to the total estimate of married-couple or unmarried-partner households using the 

housing unit weight (BW × ... × HPF). At the same time the weights of persons in the 

first and third groups are adjusted so that their sum is equal to the total estimate of 

occupied housing units using the housing unit weight (BW × ... × HPF). The goal of this 

step is to produce more consistent estimates of spouses or unmarried partners and 

married-couple and unmarried-partner households while simultaneously producing more 

consistent estimates of householders, occupied housing units, and households. 

• Demographic Raking Factor (DEMORF)—This factor is applied to individuals based on 

their age and sex in Puerto Rico (note that there are 13 Age groupings). It adjusts the 

person weights so that the weighted sample counts equal the independent population 

estimates by age and sex at the weighting area level. Because of collapsing of groups in 

applying this factor, only the total population is assured of agreeing with the official 2010 

population estimates at the weighting area level. 
13

These three steps are repeated several times until the estimates for Puerto Rico 

achieve their optimal consistency with regard to the spouse and householder 

equalization. The Person Post-Stratification Factor (PPSF) is then equal to the 

product (SUBEQRF × SPHHEQRF × DEMORF) from all of iterations of these three 

adjustments. The unrounded person weight is then the equal to the product of PPSF 

times the housing unit weight (BW × ... × MBF × PPSF). 

3. Rounding—The final product of all person weights (BW × ... × MBF × PPSF) is rounded to 

an integer. Rounding is performed so that the sum of the rounded weights is within one 

person of the sum of the unrounded weights for any of the groups listed below: 

Municipio 

Municipio × Sex 

Municipio × Sex × Age 

Municipio × Sex × Age × Tract 

Municipio × Sex × Age × Tract × Block 

For example, the number of Males, Age 30 estimated for a municipio using the rounded 

weights is within one of the number produced using the unrounded weights. 

4. Final Housing Unit Weighting Factors—This process produces the following factors: 

• Householder Factor (HHF)—This factor adjusts for differential response depending 

on the sex and age of the householder. The value of HHF for an occupied housing 

unit is the PPSF of the householder. Since there is no householder for vacant units, 

the value of HHF = 1.0 for all vacant units. 

• Rounding—The final product of all housing unit weights (BW × ... × HHF) is 

rounded to an integer. For occupied units, the rounded housing unit weight is the 

same as the rounded person weight of the householder. This ensures that both the 

rounded and unrounded householder weights are equal to the occupied housing unit 

weight. The rounding for vacant housing units is then performed so that total 

rounded weight is within one housing unit of the total unrounded weight for any of 

the groups listed below: 

 Municipio 

Municipio × Tract 

Municipio × Tract × Block 
14

CONFIDENTIALITY OF THE DATA 

The Census Bureau has modified or suppressed some data on this site to protect confidentiality. 

Title 13 United States Code, Section 9, prohibits the Census Bureau from publishing results in 

which an individual's data can be identified. 

The Census Bureau’s internal Disclosure Review Board sets the confidentiality rules for all data 

releases. A checklist approach is used to ensure that all potential risks to the confidentiality of 

the data are considered and addressed. 

• Title 13, United States Code: Title 13 of the United States Code authorizes the Census 

Bureau to conduct censuses and surveys. Section 9 of the same Title requires that any 

information collected from the public under the authority of Title 13 be maintained as 

confidential. Section 214 of Title 13 and Sections 3559 and 3571 of Title 18 of the 

United States Code provide for the imposition of penalties of up to five years in prison 

and up to $250,000 in fines for wrongful disclosure of confidential census information. 

• Disclosure Avoidance: Disclosure avoidance is the process for protecting the 

confidentiality of data. A disclosure of data occurs when someone can use published 

statistical information to identify an individual that has provided information under a 

pledge of confidentiality. For data tabulations, the Census Bureau uses disclosure 

avoidance procedures to modify or remove the characteristics that put confidential 

information at risk for disclosure. Although it may appear that a table shows information 

about a specific individual, the Census Bureau has taken steps to disguise or suppress the 

original data while making sure the results are still useful. The techniques used by the 

Census Bureau to protect confidentiality in tabulations vary, depending on the type of 

data. 

• Data Swapping: Data swapping is a method of disclosure avoidance designed to protect 

confidentiality in tables of frequency data (the number or percent of the population with 

certain characteristics). Data swapping is done by editing the source data or exchanging 

records for a sample of cases when creating a table. A sample of households is selected 

and matched on a set of selected key variables with households in neighboring 

geographic areas that have similar characteristics (such as the same number of adults and 

same number of children). Because the swap often occurs within a neighboring area, 

there is no effect on the marginal totals for the area or for totals that include data from 

multiple areas. Because of data swapping, users should not assume that tables with cells 

having a value of one or two reveal information about specific individuals. Data 

swapping procedures were first used in the 1990 Census, and were used again in Census 

2000 and the 2010 Census. 
15

• Synthetic Data: The goals of using synthetic data are the same as the goals of data 

swapping, namely to protect the confidentiality in tables of frequency data. Persons are 

identified as being at risk for disclosure based on certain characteristics. The synthetic 

data technique then models the values for another collection of characteristics to protect 

the confidentiality of that individual. 

ERRORS IN THE DATA 

• Sampling Error — The data in the PRCS products are estimates of the actual figures that 

would have been obtained by interviewing the entire population using the same 

methodology. The estimates from the chosen sample also differ from other samples of 

housing units and persons within those housing units. Sampling error in data arises due to 

the use of probability sampling, which is necessary to ensure the integrity and 

representativeness of sample survey results. The implementation of statistical sampling 

procedures provides the basis for the statistical analysis of sample data. Measures used to 

estimate the sampling error are provided in the next section. 

• Nonsampling Error — In addition to sampling error, data users should realize that other 

types of errors may be introduced during any of the various complex operations used to 

collect and process survey data. For example, operations such as data entry from 

questionnaires and editing may introduce error into the estimates. Another source is 

through the use of controls in the weighting. The controls are designed to mitigate the 

effects of systematic undercoverage of certain groups who are difficult to enumerate and 

to reduce the variance. The controls are based on the population estimates extrapolated 

from the previous census. Errors can be brought into the data if the extrapolation 

methods do not properly reflect the population. However, the potential risk from using 

the controls in the weighting process is offset by far greater benefits to the PRCS 

estimates. These benefits include reducing the effects of a larger coverage problem found 

in most surveys, including the PRCS, and the reduction of standard errors of PRCS 

estimates. These and other sources of error contribute to the nonsampling error 

component of the total error of survey estimates. Nonsampling errors may affect the data 

in two ways. Errors that are introduced randomly increase the variability of the data. 

Systematic errors which are consistent in one direction introduce bias into the results of a 

sample survey. The Census Bureau protects against the effect of systematic errors on 

survey estimates by conducting extensive research and evaluation programs on sampling 

techniques, questionnaire design, and data collection and processing procedures. In 

addition, an important goal of the PRCS is to minimize the amount of nonsampling error 

introduced through nonresponse for sample housing units. One way of accomplishing this 

is by following up on mail nonrespondents during the CATI and CAPI phases. For more 

information, see the section entitled “Control of Nonsampling Error”. 
16

MEASURES OF SAMPLING ERROR 

Sampling error is the difference between an estimate based on a sample and the corresponding 

value that would be obtained if the estimate were based on the entire population (as from a 

census). Note that sample-based estimates will vary depending on the particular sample selected 

from the population. Measures of the magnitude of sampling error reflect the variation in the 

estimates over all possible samples that could have been selected from the population using the 

same sampling methodology. 

Estimates of the magnitude of sampling errors – in the form of margins of error – are provided 

with all published PRCS data. The Census Bureau recommends that data users incorporate this 

information into their analyses, as sampling error in survey estimates could impact the 

conclusions drawn from the results. 

Confidence Intervals and Margins of Error 

Confidence Intervals – A sample estimate and its estimated standard error may be used to 

construct confidence intervals about the estimate. These intervals are ranges that will contain 

the average value of the estimated characteristic that results over all possible samples, with a 

known probability. 

For example, if all possible samples that could result under the PRCS sample design were 

independently selected and surveyed under the same conditions, and if the estimate and its 

estimated standard error were calculated for each of these samples, then: 

1. Approximately 68 percent of the intervals from one estimated standard error below 

the estimate to one estimated standard error above the estimate would contain the 

average result from all possible samples; 

2. Approximately 90 percent of the intervals from 1.645 times the estimated standard 

error below the estimate to 1.645 times the estimated standard error above the 

estimate would contain the average result from all possible samples. 

3. Approximately 95 percent of the intervals from two estimated standard errors below 

the estimate to two estimated standard errors above the estimate would contain the 

average result from all possible samples. 

 The intervals are referred to as 68 percent, 90 percent, and 95 percent confidence intervals, 

respectively. 

Margin of Error – Instead of providing the upper and lower confidence bounds in published 

PRCS tables, the margin of error is provided instead. The margin of error is the difference 

between an estimate and its upper or lower confidence bound. Both the confidence bounds 
17

and the standard error can easily be computed from the margin of error. All PRCS published 

margins of error are based on a 90 percent confidence level. 

Standard Error = Margin of Error / 1.645 

Lower Confidence Bound = Estimate - Margin of Error 

Upper Confidence Bound = Estimate + Margin of Error 

Note that for 2005, PRCS margins of error and confidence bounds were calculated using a 90 

percent confidence level multiplier of 1.65. Beginning with the 2006 data release, we are 

now employing a more accurate multiplier of 1.645. Margins of error and confidence bounds 

from previously published products will not be updated with the new multiplier. When 

calculating standard errors from margins of error or confidence bounds using published data 

for 2005, use the 1.65 multiplier. 

When constructing confidence bounds from the margin of error, the user should be aware of 

any “natural” limits on the bounds. For example, if a characteristic estimate for the 

population is near zero, the calculated value of the lower confidence bound may be negative. 

However, a negative number of people does not make sense, so the lower confidence bound 

should be reported as zero instead. However, for other estimates such as income, negative 

values do make sense. The context and meaning of the estimate must be kept in mind when 

creating these bounds. Another of these natural limits would be 100 percent for the upper 

bound of a percent estimate. 

If the margin of error is displayed as ‘*****’ (five asterisks), the estimate has been controlled 

to be equal to a fixed value and so it has no sampling error. When using any of the formulas 

in the following section, use a standard error of zero for these controlled estimates. 

Limitations –The user should be careful when computing and interpreting confidence intervals. 

• The estimated standard errors (and thus margins of error) included in these data products 

do not include portions of the variability due to nonsampling error that may be present in 

the data. In particular, the standard errors do not reflect the effect of correlated errors 

introduced by interviewers, coders, or other field or processing personnel. Nor do they 

reflect the error from imputed values due to missing responses. Thus, the standard errors 

calculated represent a lower bound of the total error. As a result, confidence intervals 

formed using these estimated standard errors may not meet the stated levels of confidence 

(i.e., 68, 90, or 95 percent). Thus, some care must be exercised in the interpretation of the 

data in this data product based on the estimated standard errors. 

• Zero or small estimates; very large estimates — The value of almost all PRCS 

characteristics is greater than or equal to zero by definition. For zero or small estimates, 
18

use of the method given previously for calculating confidence intervals relies on large 

sample theory, and may result in negative values which for most characteristics are not 

admissible. In this case the lower limit of the confidence interval is set to zero by default. 

A similar caution holds for estimates of totals close to a control total or estimated 

proportion near one, where the upper limit of the confidence interval is set to its largest 

admissible value. In these situations the level of confidence of the adjusted range of 

values is less than the prescribed confidence level. 

CALCULATION OF STANDARD ERRORS 

Direct estimates of the standard errors were calculated for all estimates reported in this product. 

The standard errors, in most cases, are calculated using a replicate-based methodology that takes 

into account the sample design and estimation procedures. Excluding the base weights, replicate 

weights were allowed to be negative in order to avoid underestimating the standard error. 

Exceptions include: 

1. The estimate of the number or proportion of people, households, families, or housing 

units in a geographic area with a specific characteristic is zero. A special procedure is 

used to estimate the standard error. 

2. There are either no sample observations available to compute an estimate or standard 

error of a median, an aggregate, a proportion, or some other ratio, or there are too few 

sample observations to compute a stable estimate of the standard error. The estimate is 

represented in the tables by “-” and the margin of error by “**” (two asterisks). 

3. The estimate of a median falls in the lower open-ended interval or upper open-ended 

interval of a distribution. If the median occurs in the lowest interval, then a “-” follows 

the estimate, and if the median occurs in the upper interval, then a “+” follows the 

estimate. In both cases the margin of error is represented in the tables by “***” (three 

asterisks). 

Sums and Differences of Direct Standard Errors 

The standard errors estimated from these tables are for individual estimates. Additional 

calculations are required to estimate the standard errors for sums of or the differences between 

two or more sample estimates. 

The standard error of the sum of two sample estimates is the square root of the sum of the two 

individual standard errors squared plus a covariance term. That is, for standard errors ܵܧሺܺ෠

ଵሻ

and ܵܧሺܺ෠

ଶሻ of estimates ܺ෠ଵ and ܺ෠ଶ: 
19

෠ܺ൫ܧܵ

ଵ േ ܺ෠ଶ൯ ൌ ටሺܵܧ൫ܺ෠ଵ൯ሻଶ ൅ ሺܵܧ൫ܺ෠

(1) ݅ܽ݊ܿ݁ݎܽݒ݋ܿ േ ଶ൯ሻଶ

The covariance measures the interaction between two estimates. Currently the covariance terms 

are not available. Data users should use the approximation: 

෠ܺ൫ܧܵ

ଵ േ ܺ෠ଶ൯ ൎ ටሺܵܧ൫ܺ෠ଵ൯ሻଶ ൅ ሺܵܧ൫ܺ෠ଶ൯ሻଶ (2)

This method, however, will underestimate or overestimate the standard error if the two estimates 

interact in either a positive or negative way. 

The approximation formula (2) can be expanded to more than two estimates by adding in the 

individual standard errors squared inside the radical. As the number of estimates involved in the 

sum or difference increases, the results of formula (2) become increasingly different from the 

standard error derived directly from the ACS microdata. Care should be taken to work with the 

fewest number of estimates as possible. If there are estimates involved in the sum that are 

controlled in the weighting then the approximate standard error can be increasingly different. 

Several examples are provided starting on page 29 to demonstrate how issues associated with 

approximating the standard errors when summing large numbers of estimates together. 

Ratios 

The statistic of interest may be the ratio of two estimates. First is the case where the numerator 

is not a subset of the denominator. The standard error of this ratio between two sample estimates 

is approximated as: 

෠ܺቆ ܧܵ

ܻ෠

ቇ ൌ 1

ܻ෠

ඨሾܵܧ൫ܺ෠൯ሿଶ ൅

ܺ෠ଶ

ܻ෠ଶ ሾܵܧሺܻ෠ሻሿଶ (3)

Proportions/percents 

For a proportion (or percent), a ratio where the numerator is a subset of the denominator, a 

slightly different estimator is used. If ܲ෠ ൌ ܺ෠⁄ܻ෠, then the standard error of this proportion is 

approximated as: 

ܵܧ൫ܲ෠൯ ൌ 1

ܻ෠

ඨሾܵܧ൫ܺ෠൯ሿଶ െ ܺ෠ଶ

ܻ෠ଶ ሾܵܧ൫ܻ෠൯ሿଶ (4)

If ܳ෠ ൌ 100% ൈ ܲ෠ (where P is the proportion and Q is its corresponding percent), then ܵܧ൫ܳ෠൯ ൌ

100% ൈ ܵܧሺܲ෠ሻ. Note the difference between the formulas to approximate the standard error for 

proportions and ratios - the plus sign in the ratio formula has been replaced with a minus sign in 

proportions formula. If the value under the square root sign is negative, use the ratio standard 

error formula instead. 
20

Percent Change 

Calculating the percent change from one time period to another. For example, computing the 

percent change of a 2008-2010 estimate to a 2007-2009 estimate. Normally, the current estimate 

is compared to the older estimate. 

Let the current estimate = X and the r estima = Y, then t earlie te he formula for percent change is: 

ܵܧ ቆܺ෠ െ ܻ෠

ܻ෠ ቇ ൌ ܵܧ ቆܺ෠ܻ෠ െ 1ቇ ൌ ܵܧሺܺ෠ܻ෠

ሻ (1)

This reduces to a ratio. The ratio formula above may be used to calculate the standard error. As 

a caveat, this formula does not take into account the correlation when calculating overlapping 

time periods. 

Products – For a product of two estimates - for example if you want to estimate a proportion’s 

numerator by multiplying the proportion by its denominator - the standard error can be 

approximated as: 

ܵܧ൫ܺ෠ ൈ ܻ෠൯ ൌ ටܺ෠ଶ ൈ ሾܵܧ൫ܻ෠൯ሿଶ ൅ ܻ෠ଶ ൈ ሾܵܧ൫ܺ෠൯ሿଶ (5)

Comparing 1-Year Period Estimates with Overlapping 3-Year Period Estimates: 

It should be noted that the 1-year and 3-year estimates represent period estimates. Due to the 

difficulty in interpreting the “difference” in period estimates of different lengths, the Census 

Bureau currently discourages users from making such comparisons. The standard error of this 

difference is approximated as: 

෠ܺ൫ܧܵ

ଵିYear െ ܻ෠ଷିYear൯ ൌ ඨ1

3 ሾܵܧሺܺ෠

ଵିYearሻሿ

ଶ ൅ ሾܵܧሺܻ෠

ଷିYearሻሿଶ (6)

Comparing 1-Year Period Estimates with Overlapping 5-Year Period Estimates: 

It should be noted that the 1-year and 5-year estimates represent period estimates. Due to the 

difficulty in interpreting the “difference” in period estimates of different lengths, the Census 

Bureau currently discourages users from making such comparisons. The standard error of this 

difference is approximated as: 
21

෠ܺ൫ܧܵ

ଵିYear െ ܻ෠ହିYear൯ ൌ ඨ3

5 ሾܵܧ൫ܺ෠

ଵିYear൯ሿଶ ൅ ሾܵܧ൫ܻ෠ହିYear൯ሿଶ (7)

TESTING FOR SIGNIFICANT DIFFERENCES 

Significant differences – Users may conduct a statistical test to see if the difference between a 

PRCS estimate and any other chosen estimates is statistically significant at a given confidence 

level. “Statistically significant” means that the difference is not likely due to random chance 

alone. With the two estimates (Est1 and Est2) and their respective standard errors (SE1 and SE2), 

calculate 

ܼ ൌ

ݐݏܧଵ െ ݐݏܧଶ

ඥሺܵܧଵሻଶ ൅ ሺܵܧଶሻଶ (8)

If Z > 1.645 or Z < -1.645, then the difference can be said to be statistically significant at the 90 

percent confidence level.1 Any estimate can be compared to a PRCS estimate using this method, 

including other PRCS estimates from the current year, the PRCS estimate for the same 

characteristic and geographic area but from a previous year, ACS estimates, 2010 Census counts, 

estimates from other Census Bureau surveys, and estimates from other sources. Not all estimates 

have sampling error — 2010 Census counts do not — but they should be used if they exist to 

give the most accurate result of the test. 

Users are also cautioned to not rely on looking at whether confidence intervals for two estimates 

overlap or not to determine statistical significance, because there are circumstances where that 

method will not give the correct test result. If two confidence intervals do not overlap, then the 

estimates will be significantly different (i.e. the significance test will always agree). However, if 

two confidence intervals do overlap, then the estimates may or may not be significantly different. 

The Z calculation above is recommended in all cases. 

Here is a simple example of why it is not recommended to use the overlapping 

confidence bounds rule of thumb as a substitute for a statistical test. 

Let: X1 = 5.0 with SE1 = 0.2 and X2 = 6.0 with SE2 = 0.5. 

The Upper Bound for X1 = 5.0 + 0.2 * 1.645 = 5.3 while the Lower Bound for X2 = 6.0 - 

0.5 * 1.645 = 5.2. The confidence bounds overlap, so, the rule of thumb would indicate 

that the estimates are not significantly different at the 90% level. 

However, if we apply the statistical significance test we obtain: 

 

1

 The PRCS Accuracy of the Data was not separate from the ACS Accuracy of the Data in 2005, which used a Z 

statistic of +/-1.65. Data users should use +/-1.65 for estimates published in 2005 or earlier. 
22

ܼ ൌ

5െ6

√0.2ଶ ൅ 0.5ଶ

ൌ 1.857

Z = 1.857 > 1.645 which means that the difference is significant (at the 90% level). 

All statistical testing in PRCS data products is based on the 90 percent confidence level. Users 

should understand that all testing was done using unrounded estimates and standard errors, and it 

may not be possible to replicate test results using the rounded estimates and margins of error as 

published. 

EXAMPLES OF STANDARD ERROR CALCULATIONS 

Example 1 - Calculating the Standard Error from the Margin of Error 

The estimated number of males, never married is 566,489 from summary table B12001 

for Puerto Rico for 2010. The margin of error is 8,808. 

 Standard Error = Margin of Error / 1.645 

 Calculating the standard error using the margin of error, we have: 

 SE(566,489) = 8,808/ 1.645 = 5,354. 

Example 2 - Calculating the Standard Error of a Sum or Difference 

We are interested in the number of people who have never been married. From Example 

1, we know the number of males, never married is 566,489. From summary table 

B12001 we have the number of females, never married is 523,558 with a margin of error 

of 8,401. So, the estimated number of people who have never been married is 556,489 + 

523,558 = 1,090,047. To calculate the approximate standard error of this sum, we need 

the standard errors of the two estimates in the sum. We have the standard error for the 

number of males never married from Example 1 as 5,354. The standard error for the 

number of females never married is calculated using the margin of error: 

 SE(523,558) = 8,401 / 1.645 = 5,107. 

 So using formul ( ) for th approxim te st a 2 e a andard error of a sum or difference we have: 

 ܵܧሺ1,090,047ሻ ൌ ඥ5,354ଶ ൅ 5,107ଶ ൌ 7,399

Caution: This method will underestimate or overestimate the standard error if the two 

estimates interact in either a positive or negative way. 
23

To calculate the lower and upper bounds of the 90 percent confidence interval around 

1,090,047 using the standard error, simply multiply 7,399 by 1.645, then add and subtract 

the product from 1,090,047. Thus the 90 percent confidence interval for this estimate is 

[1,090,047 - 1.645(7,399)] to [1,090,047 + 1.645(7,399)] or 1,077,876 to 1,102,218. 

Example 3 - Calculating the Standard Error of a Proportion/Percent 

We are interested in the percentage of females who have never been married to the 

number of people who have never been married. The number of females, never married 

is 523,558 and the number of people who have never been married is 1,090,047. To 

calculate the approximate standard error of this percent, we need the standard errors of 

the two estimates in the percent. We have the approximate standard error for the number 

of females never married from Example 2 as 5,107 and the approximate standard error 

for the number of people never married calculated from example 2 as 7,399. 

 The estimate is (523,558 / 1,090,047) * 100% = 48.0% 

So, using formula (4) for the approximate standard error of a proportion or percent, we 

have: 

 

ܵܧሺ48.0%ሻ ൌ 100% כ

1

1,090,047 ඥ5,107ଶ െ 0.480ଶ כ 7,399ଶ ൌ 0.34%

To calculate the lower and upper bounds of the 90 percent confidence interval around 

48.0 using the standard error, simply multiply 0.34 by 1.645, then add and subtract the 

product from 48.0. Thus the 90 percent confidence interval for this estimate is 

 [48.0 - 1.645(0.34)] to [48.0 + 1.645(0.34)], or 47.4% to 48.6%. 

Example 4 - Calculating the Standard Error of a Ratio 

Now, let us calculate the estimate of the ratio of the number of unmarried males to the 

number of unmarried females and its standard error. From the above examples, the 

estimate for the number of unmarried men is 566,489 with a standard error of 5,354, and 

the estimates for the number of unmarried women is 523,558 with a standard error of 

5,107. 

The estimate of the ratio is 566,489 / 523,558 = 1.082. 

Using formula (3) for the approximate standard error we have: 

ܵܧሺ1.082ሻ ൌ 1

523,558 ඥ5,354ଶ ൅ 1.082ଶ כ 5,107ଶ ൌ 0.015
24

The 90 percent margin of error for this estimate would be 0.015 multiplied by 1.645, or 

about 0.025. The 90 percent lower and upper 90 percent confidence bounds would then 

be [1.082– 0.025] to [1.082 + 0.025], or 1.057 and 1.107. 

Example 5 - Calculating the Standard Error of a Product 

We are interested in the number of 1-unit detached owner-occupied housing units. The 

number of owner-occupied housing units is 929,831 with a margin of error of 9,777 from 

subject table S2504 for 2010, and the percent of 1-unit detached owner-occupied housing 

units is 80.0% (0.800) with a margin of error of 0.6 (0.006). So the number of 1-unit 

detached owner-occupied housing units is 929,831 * 0.800 = 743,865. Calculating the 

standard error for the estimates using the margin of error we have: 

ܵܧሺ929,831ሻ ൌ 9,777 1.645 ൌ 5,943 ⁄

and 

ܵܧሺ0.800ሻ ൌ 0.006 1.645 ൌ 0.0036474 ⁄

The approximate standard error for number of 1-unit detached owner-occupied housing 

units is calculated using formula (5) for products as: 

ܵܧሺ743,865ሻ ൌ ඥ929,831ଶ כ 0.0036474ଶ ൅ 0.800ଶ כ 5,943ଶ ൌ 5,840

To calculate the lower and upper bounds of the 90 percent confidence interval around 

743,865 using the standard error, simply multiply 5,840 by 1.645, then add and subtract 

the product from 743,865 Thus the 90 percent confidence interval for this estimate is 

[743,865 - 1.645(5,840)] to [743,865 + 1.645(5,840)] or 734,258 to 753,472. 

CONTROL OF NONSAMPLING ERROR 

As mentioned earlier, sample data are subject to nonsampling error. This component of error 

could introduce serious bias into the data, and the total error could increase dramatically over 

that which would result purely from sampling. While it is impossible to completely eliminate 

nonsampling error from a survey operation, the Census Bureau attempts to control the sources of 

such error during the collection and processing operations. Described below are the primary 

sources of nonsampling error and the programs instituted for control of this error. The success of 

these programs, however, is contingent upon how well the instructions were carried out during 

the survey. 

• Coverage Error — It is possible for some sample housing units or persons to be missed 

entirely by the survey (undercoverage), but it is also possible for some sample housing 
25

units and persons to be counted more than once (overcoverage). Both the undercoverage 

and overcoverage of persons and housing units can introduce biases into the data, 

increase respondent burden and survey costs. 

A major way to avoid coverage error in a survey is to ensure that its sampling frame, for 

Puerto Rico an address list in each municipio, is as complete and accurate as possible. 

The source of addresses for the PRCS is the MAF, which was created using the address 

list for Census 2000. An attempt is made to assign all appropriate geographic codes to 

each MAF address via an automated procedure using the Census Bureau TIGER 

(Topologically Integrated Geographic Encoding and Referencing) files. A manual coding 

operation based in the appropriate regional offices is attempted for addresses, which 

could not be automatically coded. The MAF was used as the source of addresses for 

selecting sample housing units and mailing questionnaires. TIGER produced the location 

maps for CAPI assignments. Sometimes the MAF has an address that is the duplicate of 

another address already on the MAF. This could occur when there is a slight difference 

in the address such as 123 Calle 1, Bayamon versus URB Hermosillo, 123 Calle 1, 

Bayamon. 

In the CATI and CAPI nonresponse follow-up phases, efforts were made to minimize the 

chances that housing units that were not part of the sample were interviewed in place of 

units in sample by mistake. If a CATI interviewer called a mail nonresponse case and was 

not able to reach the exact address, no interview was conducted and the case was eligible 

for CAPI. During CAPI follow-up, the interviewer had to locate the exact address for 

each sample housing unit. If the interviewer could not locate the exact sample unit in a 

multi-unit structure, or found a different number of units than expected, the interviewers 

were instructed to list the units in the building and follow a specific procedure to select a 

replacement sample unit. Person overcoverage can occur when an individual is included 

as a member of a housing unit but does not meet PRCS residency rules. 

Coverage rates give a measure of undercoverage or overcoverage of persons or housing 

units in a given geographic area. Rates below 100 percent indicate undercoverage, while 

rates above 100 percent indicate overcoverage. Coverage rates are released concurrent 

with the release of estimates on American FactFinder in the B98 series of detailed tables. 

Further information about PRCS coverage rates may be found at 

http://www.census.gov/acs/www/methodology/sample_size_and_data_quality/. 

• Nonresponse Error — Survey nonresponse is a well-known source of nonsampling error. 

There are two types of nonresponse error – unit nonresponse and item nonresponse. 

Nonresponse errors affect survey estimates to varying levels depending on amount of 

nonresponse and the extent to which nonrespondents differ from respondents on the 

characteristics measured by the survey. The exact amount of nonresponse error or bias 

on an estimate is almost never known. Therefore, survey researchers generally rely on 
26

proxy measures, such as the nonresponse rate, to indicate the potential for nonresponse 

error. 

o Unit Nonresponse — Unit nonresponse is the failure to obtain data from housing 

units in the sample. Unit nonresponse may occur because households are unwilling 

or unable to participate, or because an interviewer is unable to make contact with a 

housing unit. Unit nonresponse is problematic when there are systematic or variable 

differences between interviewed and noninterviewed housing units on the 

characteristics measured by the survey. Nonresponse bias is introduced into an 

estimate when differences are systematic, while nonresponse error for an estimate 

evolves from variable differences between interviewed and noninterviewed 

households. 

The PRCS made every effort to minimize unit nonresponse, and thus, the potential for 

nonresponse error. First, the PRCS used a combination of mail, CATI, and CAPI 

data collection modes to maximize response. The mail phase included a series of 

three to four mailings to encourage housing units to return the questionnaire. 

Subsequently, mail nonrespondents (for which phone numbers are available) were 

contacted by CATI for an interview. Finally, a subsample of the mail and telephone 

nonrespondents was contacted for by personal visit to attempt an interview 

PRCS response rates measure the percent of units with a completed interview. The 

higher the response rate, and consequently the lower the nonresponse rate, the less 

chance estimates may be affected by nonresponse bias. Response and nonresponse 

rates, as well as rates for specific types of nonresponse, are released concurrent with 

the release of estimates on American FactFinder in the B98 series of detailed tables. 

Further information about response and nonresponse rates may be found at 

http://www.census.gov/acs/www/methodology/sample_size_and_data_quality/. 

o Item Nonresponse — Nonresponse to particular questions on the survey questionnaire 

and instrument allows for the introduction of error or bias into the data, since the 

characteristics of the nonrespondents have not been observed and may differ from 

those reported by respondents. As a result, any imputation procedure using 

respondent data may not completely reflect this difference either at the elemental 

level (individual person or housing unit) or on average. 

Some protection against the introduction of large errors or biases is afforded by 

minimizing nonresponse. In the PRCS, item nonresponse for the CATI and CAPI 

operations was minimized by the requirement that the automated instrument receive a 

response to each question before the next one could be asked. Questionnaires 

returned by mail were edited for completeness and acceptability. They were reviewed 

by computer for content omissions and population coverage. If necessary, a telephone 
27

follow-up was made to obtain missing information. Potential coverage errors were 

included in this follow-up. 

Allocation tables provide the weighted estimate of persons or housing units for which 

a value was imputed, as well as the total estimate of persons or housing units that 

were eligible to answer the question. The smaller the number of imputed responses, 

the lower the chance that the item nonresponse is contributing a bias to the estimates. 

Allocation tables are released concurrent with the release of estimates on American 

Factfinder in the B99 series of detailed tables with the overall allocation rates across 

all person and housing unit characteristics in the B98 series of detailed tables. 

Additional information on item nonresponse and allocations can be found at 

http://www.census.gov/acs/www/methodology/sample_size_and_data_quality/. 

• Measurement and Processing Error — The person completing the questionnaire or 

responding to the questions posed by an interviewer could serve as a source of error, 

although the questions were cognitively tested for phrasing, and detailed instructions for 

completing the questionnaire were provided to each household. 

o Interviewer monitoring — The interviewer may misinterpret or otherwise incorrectly 

enter information given by a respondent; may fail to collect some of the information 

for a person or household; or may collect data for households that were not 

designated as part of the sample. To control these problems, the work of interviewers 

was monitored carefully. Field staff were prepared for their tasks by using specially 

developed training packages that included hands-on experience in using survey 

materials. A sample of the households interviewed by CAPI interviewers was 

reinterviewed to control for the possibility that interviewers may have fabricated data. 

o Processing Error — The many phases involved in processing the survey data 

represent potential sources for the introduction of nonsampling error. The processing 

of the survey questionnaires includes the keying of data from completed 

questionnaires, automated clerical review, follow-up by telephone, manual coding of 

write-in responses, and automated data processing. The various field, coding and 

computer operations undergo a number of quality control checks to insure their 

accurate application. 

o Content Editing — After data collection was completed, any remaining incomplete or 

inconsistent information was imputed during the final content edit of the collected 

data. Imputations, or computer assignments of acceptable codes in place of 

unacceptable entries or blanks, were needed most often when an entry for a given 

item was missing or when the information reported for a person or housing unit on 

that item was inconsistent with other information for that same person or housing 

unit. As in other surveys and previous censuses, the general procedure for changing 
28

unacceptable entries was to allocate an entry for a person or housing unit that was 

consistent with entries for persons or housing units with similar characteristics. 

Imputing acceptable values in place of blanks or unacceptable entries enhances the 

usefulness of the data. 

 
29

ISSUES WITH APPROXIMATING THE STANDARD ERROR OF LINEAR 

COMBINATIONS OF MULTIPLE ESTIMATES 

Several examples are provided here to demonstrate how different the approximated standard 

errors of sums can be compared to those derived and published with ACS microdata. 

A. Suppose we wish to estimate the total number of males using age and citizenship status. The 

relevant data from table B05003 is displayed in table A below. 

Table A: 2010 Estimates of males from B05003: Sex by Age by Citizenship Status 

Characteristic Estimate MOE 

Male 1,905,314 2,996 

 Under 18 Years 493,406 2,300 

 Native 491,306 2,312 

 Foreign Born 2,100 697 

 Naturalized U.S. Citizen 94 163 

 Not a U.S. Citizen 2,006 687 

 18 Years and Older 1,411,908 1,572 

 Native 1,363,983 4,197 

 Foreign Born 47,925 4,108 

 Naturalized U.S. Citizen 19,549 2,150 

 Not a U.S. Citizen 28,376 3,290 

 2010 American FactFinder 

The estimate and its MOE are actually published. However, if they were not available in the 

tables, one way of obtaining them would be to add together the number of males under 18 and 

over 18 to get: 

ݐݏܧ݅݉ܽݐ݁ሺܯ݈ܽ݁ሻ ൌ 493,406 ൅ 1,411,908 ൌ 1,905,314

And the first approximated 

ܵܧሺܯ݈ܽ݁ሻ ൌ ܵܧሺ1,905,314ሻ ൌ ඨ൬

2,300

1.645൰

ଶ

൅ ൬

1,572

1.645൰

ଶ

ൌ 1,693.6

Another way would be to add up the estimates for the three subcategories (Native, Foreign Born: 

Naturalized U.S. Citizen, and Foreign Born: Not a U.S. Citizen), for males under and over 18 

years of age. From these eight estimates we obtain: 

28,376 ൅ 19,549 ൅ 1,363,983 ൅ 2,006 ൅ 94 ൅ 491,306 ൌ ሻ݈ܽ݁ܯሺ݁ݐ݅݉ܽݐݏܧ

ൌ 1,905,314

With a second approximation to the SE of: 
ܵܧሺܯ݈ܽ݁ሻ ൌ ܵܧሺ1,905,314ሻ

ൌ ඨ൬

2,312

1.645

30

൰

ଶ

൅ ൬ 163

1.645൰

ଶ

൅ ൬ 687

1.645൰

ଶ

൅ ൬

4,197

1.645൰

ଶ

൅ ൬

2,150

1.645൰

ଶ

൅ ൬

3,290

1.645൰

ଶ

ൌ 3,791.7

We do know that the standard error using the published MOE is 2,996 / 1.645 = 1,821.3. With a 

quick glance, we can see that the ratio of the standard error of the first method to the publishedbased standard error yields 0.93; an under-estimate of roughly 7%, whereas the second method 

yields a ratio of 2.08 or an over-estimate of 108%. This is an example of what could happen to 

the approximate SE when the sum involves a controlled estimate. In this case, it is sex by age. 

B. Suppose we are interested in the total number of people aged 65 or older and its standard error. 

Table B shows some of the estimates for the national level from table B01001 (the estimates in 

gray were derived for the purpose of this example only). 

Table B: Some Estimates from AFF Table B01001: Sex by Age for 2010 

Age 

Category 

Estimate, 

Male 

MOE, 

Male 

Estimate, 

Female 

MOE, 

Female Total 

Estimated 

MOE, 

Total 

65 and 66 

years old 

34,387 2,210 44,262 2,719 78,649 3,504

67 to 69 

years old 

47,639 2,418 51,547 2,511 99,186 3,486

70 to 74 

years old 

58,924 2,576 77,508 3,335 136,432 4,214

75 to 79 

years old 

42,091 1,912 58,788 2,639 100,879 3,259

80 to 84 

years old 

31,494 2,237 41,300 2,418 72,794 3,294

85 years 

and older 

26,399 1,790 45,063 2,595 71,462 3,152

Total 240,934 NA 318,468 NA 559,402 8,579

To begin we find the total number of people aged 65 and over by simply adding the totals for 

males and females to get 240,934 + 318,468 = 559,402. One way we could approximate the 

standard error of this estimate is to sum the males and females for each age category and then use 

their MOEs to obtain the approximate standard error for the total number of people over 65. 

ൌ ൌ ଶ ൅ 9 ܯܱܧ ଶ ሺܣ݃݁ 65 ܽ݊݀ 66ሻ ܯܱܧሺ78,649ሻ ඥ2,210 2,71 ൌ 4 3,50

ܯܱܧሺܣ݃݁ 67 ݋ݐ 69ሻ ൌ ܯܱܧሺ99,186ሻ ൌ ඥ2,418ଶ ൅ 2,511ଶ ൌ 3,486
31

... etc. ... 

Now, we calculate the number of people aged 65 or older, 559,402, using the six derived 

estimates and approximate its standard error: 

ܵܧሺ559,402ሻ ൌ ඨሺ

3,504

1.645ሻଶ ൅ ሺ3,4861.645ሻଶ ൅ ሺ4,2141.645ሻଶ ൅ ሺ3,2591.645ሻଶ ൅ ሺ3,2941.645ሻଶ ൅ ሺ3,1531.645ሻଶ

ൌ 5,215 

For this example the estimate and MOE are published in table B09017. The total number of 

people aged 65 or older is 559,402 with a margin of error of 2,595. Therefore the publishedbased standard error is: 

ܵܧሺ559,402ሻ ൌ 2,595 1.645 ൌ 1,578 ⁄

The approximated standard error, using the six derived age group estimates, yields an 

approximated standard error roughly 3.3 times larger than the published-based standard error. 

There are two additional ways to approximate the standard error of people aged 65 and over in 

addition to the way used above. The first is to find the MOEs for the males age 65 and older, 

and females aged 65 and older separately and then combine them to find the approximated 

standard error. 

For males it is: 

ܯܱܧሺܯ݈ܽ݁ሻ ൌ ܯܱܧሺ240,934ሻ ൌ ඥ2,210ଶ ൅ 2,418ଶ ൅ 2,576ଶ ൅ 1,912ଶ ൅ 2,237ଶ ൅ 1,790ଶ

ൌ 5,406.5

And for females: 

ሻ318,468ሺܧܱܯ ൌ ሻ݈݁݉ܽ݁ܨሺܧܱܯ

ൌ ඥ2,719ଶ ൅ 2,511ଶ ൅ 3,335ଶ ൅ 2,639ଶ ൅ 2,418ଶ ൅ 2,595ଶ ൌ 6,660.7

And the total is then: 

ܵܧሺܶݐ݋݈ܽሻ ൌ ܵܧሺ559,402ሻ ൌ ඨሺ

5,406.5

1.645 ሻଶ ൅ ൬

6,660.7

1.645 ൰

ଶ

ൌ 5,215

The second is to use all twelve of the published estimates together, that is, all estimates from the 

male age categories and female age categories, to create the SE for people aged 65 and older. 
32

ܯܱܧሺܶݐ݋݈ܽሻ ൌ ܯܱܧሺ559,402ሻ ൌ ඥ2,210ଶ ൅ 2,418ଶ ൅ 2,576ଶ ൅ ڮ ൅ 2,595ଶ ൌ 8,578.8

And the standard error is: 

ܵܧሺܶݐ݋݈ܽሻ ൌ ܵܧሺ559,402ሻ ൌ 8,578.8 1.645 ⁄ ൌ 5,215. 

 In this particular situation, the results from all three ways are the same. So no matter which way 

you use, you will obtain the same approximation for the standard error. This is not always the 

case. The corresponding section in the ACS Accuracy of the Data document gives an example 

where the three methods yield different answers. It is located at 

http://www.census.gov/acs/www/Downloads/data_documentation/Accuracy/ACS_Accuracy_of_

Data_2010.pdf. 

C. For an alternative to approximating the standard error for people 65 years and older seen in part 

B, we could find the estimate and its SE by summing all of the estimate for the ages less than 65 

years old and subtracting them from the estimate for the total population. Due to the large 

number of estimates, the Table C does not show all of the age groups. In addition, the estimates 

in the part of the table shaded gray were derived for the purposes of this example only and 

cannot be found in base table B01001. 

 Table C: Some Estimates from AFF Table B01001: Sex by Age for 2010: 

Age Category Estimate, 

Male 

MOE, 

Male 

Estimate, 

Female 

MOE, 

Female 

Total Estimated 

MOE, 

Total 

Total 

Population 1,905,314 2,996 2061974 2996 3,967,288 4,237 

Under 5 years 119,417 1,668 114,240 1,654 233,657 2,349 

5 to 9 years 

old 132,539 4,060 118,361 3,987 250,900 5,690 

10 to 14 years 

old 150,027 4,079 150,448 4,085 300,475 5,773 

... ... ... ... ... ... ... 

62 to 64 years 

old 61,327 3,232 73,135 3,368 134,462 4,668 

Total for Age 

0 to 64 years 

old 

1,664,380 11,945 1,743,506 12,948 3,407,886 17,617 

Total for Age 

65 years and 

older 

240,934 12,315 318,468 13,290 559,402 18,119 
33

An estimate for the number of people age 65 and older is equal to the total population minus the 

population between the ages of zero and 64 years old: 

Number of people aged 65 and older: 3,967,288 – 3,407,886 = 559,402. 

The way to approximate the SE is the same as in part B. First we will sum males and females 

estimates across each age category and then approximate the MOEs. We will use that 

information to approximate the standard error for our estim e f inter st: at o e

ܯܱܧ ଶ ଶ ሺܶݐ݋݈ܽ ܲݑ݌݋݈ܽݐ݅݋݊ሻ ൌ ܯܱܧሺ3,967,288ሻ ൌ ඥ2,996 ൅ 2,996 ൌ

ܯܱܧሺܷ݊݀݁ݎ 5 ݕ݁ܽݏݎሻ ൌ ܯܱܧሺ233,657ሻ ൌ ඥ1,668ଶ ൅ 1,654

4,237

ൌ 2,349

... etc. ... 

And the SE for the total number of people aged 65 and older is: 

ሻ559,402ሺܧܵ ൌ ሻݎ݈݀݁݋ ܽ݊݀ 65 ݃݁ܣሺܧܵ

ൌ ඨሺ

4,237

1.645ሻଶ ൅ ሺ2,349

1.645ሻଶ ൅ ሺ5,6901.645ሻଶ ൅ ሺ5,7731.645ሻଶ ൅ڮ൅ሺ3,3681.645ሻଶ ൌ 18,119

Again, as in Example B, the estimate and its MOE are published in table B09017. The total 

number of people aged 65 or older is 559,402 with a margin of error of 2,595. Therefore the 

standard error is: 

SE(559,402) = 2,595 / 1.645 = 1,578. 

The approximated standard error using the thirteen derived age group estimates yields a standard 

error roughly 11.5 times larger than the actual SE. 

Data users can mitigate the problems shown in examples A through C to some extent by using a 

collapsed version of a detailed table if available which will reduce the number of estimates and 

SEs. These issues may also be avoided by creating estimates and SEs using the Public Use 

Microdata Sample (PUMS) or by requesting a custom tabulation, a fee-based service offered 

under certain conditions by the Census Bureau. More information regarding custom tabulations 

may be found at http://www.census.gov/acs/www/data_documentation/custom_tabulations/.