1 

American Community Survey 

Accuracy of the Data (2010) 

INTRODUCTION 

The data contained in these data products are based on the American Community Survey (ACS) 

sample interviewed from January 1, 2010 through December 31, 2010. The ACS sample is 

selected from all counties and county-equivalents in the United States. In 2006, the ACS 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 ACS estimates that are 

based on the total population. All ACS population estimates from years prior to 2006 include 

only persons in housing units. The ACS, 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 

ACS sample design, estimation methodology, and accuracy of the ACS data. The ACS 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 ACS, including data collection and 

processing, can be found at: http://www.census.gov/acs/www/methodology/methodology_main/. 

The 2010 Accuracy of the Data from the Puerto Rico Community Survey can be found at 

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

_Data_2010.pdf. 

 
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TABLE OF CONTENTS

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

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

SAMPLING FRAME ..................................................................................................................... 4

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

WEIGHTING METHODOLOGY ................................................................................................ 10

CONFIDENTIALITY OF THE DATA ........................................................................................ 16

ERRORS IN THE DATA ............................................................................................................. 17

CALCULATION OF STANDARD ERRORS ............................................................................. 20

Sums and Differences of Direct Standard Errors.................................................................................... 21

Ratios ......................................................................................................................................................22

Proportions/Percents ............................................................................................................................... 22

Percent Change .......................................................................................................................................22

Products...................................................................................................................................................23

Comparing 1-Year Period Estimates with Overlapping 3-Year Period Estimates..................................23

Comparing 1-Year Period Estimates with Overlapping 5-Year Period Estimates..................................23

TESTING FOR SIGNIFICANT DIFFERENCES ........................................................................ 23

EXAMPLES OF STANDARD ERROR CALCULATIONS....................................................... 24

CONTROL OF NONSAMPLING ERROR ................................................................................. 27

ISSUES WITH APPROXIMATING THE STANDARD ERROR OF LINEAR 

COMBINATIONS OF MULTIPLE ESTIMATES ...................................................................... 31
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DATA COLLECTION 

Housing Units 

The ACS employs three modes of data collection: 

• Mailout/Mailback 

• Computer Assisted Telephone Interview (CATI) 

• Computer Assisted Personal Interview (CAPI) 

With the exception of addresses in Remote Alaska, 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. 

All Remote Alaska addresses are assigned to one of two data collection periods, January-April, 

or September-December and are sampled for CAPI at a rate of 2-in-3. Data for these addresses 

are collected using CAPI only and up to four months are given to complete the interviews in 

Remote Alaska for each data collection period. 

Group Quarters 

Group Quarters data collection spans six weeks, except in Remote Alaska and for Federal 

prisons, where the data collection time period is four months. As is done for HUs, Group 

Quarters in Remote Alaska are assigned to one of two data collection periods, January-April, or 

September-December and up to four months is allowed to complete the interviews. Similarly, all 

Federal prisons are assigned to September with a four month data collection window. 

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. 
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SAMPLING FRAME 

Housing Units 

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

and county equivalents in the 50 states, including the District of Columbia. 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 (DSF) provided by the U.S. Postal Service. The DSF covers only 

the U.S. These files identify mail drop points and provide the best available source of changes 

and updates to the housing unit inventory. The MAF is also updated with the results from 

various Census Bureau field operations, including the ACS. 

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 ACS employs a two-phase, two-stage sample design. The ACS 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 secondstage 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). 
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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 

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 seven sampling strata in the U.S. 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 the U.S. by 

calculating a measure of size for each of the following entities: 

ο Counties 

ο Places 

ο School Districts (elementary, secondary, and unified) 

ο American Indian Areas 

ο Tribal Subdivisions 

ο Alaska Native Village Statistical Areas 

ο Hawaiian Homelands 

ο Minor Civil Divisions – in Connecticut, Maine, Massachusetts, Michigan, 

Minnesota, New Hampshire, New Jersey, New York, Pennsylvania, Rhode 

Island, Vermont, and Wisconsin (these are the states where MCDs are active, 

functioning governmental units) 

ο Census Designated Places – in Hawaii only 
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The measure of size for all areas except American Indian Areas, Tribal Subdivisions, and 

Alaska Native Village Statistical Areas is an estimate of the number of occupied HUs in 

the area. This is calculated by multiplying the number of ACS addresses by an estimated 

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

calculated in the same manner. 

For American Indian, Tribal Subdivisions, and Alaska Native Village Statistical Areas, 

the measure of size is the estimated number of occupied HUs multiplied by the 

proportion of people reporting American Indian or Alaska Native (alone or in 

combination) in Census 2000. 

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 ACS valid addresses by second-stage sampling stratum in such a way as to 

yield an overall target sample size for the year of approximately 3,000,000 in the U.S. 

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 county and 

county equivalent. 

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

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

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. 

 
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 Table 1. First-phase Sampling Rate Categories for the United States

Sampling Rate Category Sampling 

Rates 

Blocks in smallest governmental units 

(MOS1 < 200) 10.00% 

Blocks in smaller governmental units 

(200 ≤ MOS < 800) 6.67% 

Blocks in small governmental units 

(800 ≤ MOS ≤ 1200) 3.33% 

Blocks in large tracts 

(MOS >1200, TRACTMOS2 ≥ 2000) where Mailable addresses3 ≥ 75% 

and predicted levels of completed mail and CATI interviews prior to 

second-stage sampling > 60% 

1.50% 

Other Blocks in large tracts 

(MOS >1200, TRACTMOS ≥ 2000) 1.63% 

All other blocks 

(MOS >1200, TRACTMOS < 2000) 

where Mailable addresses ≥ 75% and predicted levels of completed mail 

and CATI interviews prior to second-stage sampling > 60% 

2.04% 

All other blocks (MOS >1200, TRACTMOS < 2000) 2.22% 

1

MOS = Measure of size. 

2

TRACTMOS = Census Tract measure of size. 

3

Mailable addresses: Addresses that have sufficient information to be delivered by the U.S. 

Postal Service (as determined by ACS). 

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, which include Remote Alaska 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. 

 
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Table 2. Second-Phase (CAPI) Subsampling Rates for the United States 

Address and Tract Characteristics 

CAPI 

Subsampling 

Rate

United States 

Unmailable addresses and addresses in Remote Alaska 66.7% 

Mailable addresses in tracts with predicted levels of completed 

mail and CATI interviews prior to CAPI subsampling between 0% 

and less than 36% 

50.0% 

Mailable addresses in tracts with predicted levels of completed 

mail and CATI interviews prior to CAPI subsampling greater than 

35% and less than 51% 

40.0% 

Mailable addresses in other tracts 33.3% 

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, while 

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

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 ACS 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 – During the second stage, GQs are selected from the first stage 

sample in a systematic sample of 1-in-x where x is dependent upon the state’s target 

sampling rate. Since the first stage sample is one fifth of the universe, x can be 

calculated as x = (1 /5) x (1 / rate) where rate is the state’s target sampling rate. For 

example, suppose a state had a target sampling rate of 2.5%. The systematic sample 

 

1

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

Census day. 
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would then be 1-in-8 since (1 / 5) x (1 / 0.025) = 8. Regardless of their actual size, all 

GQs in the small stratum have the same probability of selection. 

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 that varies 

according to state. 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-x where x = 1 / rate (one divided by the state’s 

target sampling rate). For example, suppose a state had a target sampling rate of 2.5%. 

The hits would then be selected in a systematic sample of 1-in-40, since 1 / 0.025 = 40. 

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. 

The following table summarizes the 2010 state target sampling rates for the U.S.

Table 3. 2010 State Targeted Sampling Rates for the U.S. 

Alaska 5.53% New Hampshire 3.17% 

Delaware 4.86% New Mexico 3.06% 

District of Columbia 3.08% North Dakota 4.59% 

Hawaii 3.24% Rhode Island 2.75% 

Idaho 3.48% South Dakota 3.91% 

Maine 3.14% Utah 2.79% 

Montana 4.38% Vermont 4.95% 

Nevada 3.36% Wyoming 7.11% 

All Other States 2.5% 
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3. Sample Month Assignment 

In order to assign each hit to a panel month, all of the GQ samples from a state 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 and remote Alaska. Remote Alaska GQs are 

assigned to January and September based on where the GQ is located. 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 people is 15 or less, all people in the 

GQ will receive the questionnaire. 

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 
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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. 

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 4: 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 

state. The steps are as follows: 

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

its first-phase sampling rate. The initial base weight is equal to 40 for sample cases in 

most states in 2010, though in 15 states and the District of Columbia the initial base 

weights are smaller. 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: 

 State × Major GQ Type Group × County 

• 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 at the state level in the U.S. 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 at the state level. 

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

groups: 

 State × Major GQ Type Group 
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• 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 

weights for any of the groups listed below: 

 Major GQ Type Group 

 Major GQ Type Group × County 

Housing Unit and Household Person Weighting 

The housing unit and household person weighting uses two types of geographic areas for 

adjustments: weighting areas and subcounty areas. Weighting areas are county-based and have 

been used since the first year of the ACS. Subcounty areas are based on incorporated place and 

minor civil divisions (MCD). Their use was introduced into the ACS in 2010. 

 

Weighting areas are built from collections of whole counties. Census 2000 data are used to group 

counties 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 counties 

• Core-based Statistical Area status 

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

2005 ACS. The process also tries to preserve as many counties that meet the threshold to form 

their own weighting areas. In total, there are 1,951 weighting areas formed from the 3,141 

counties and county equivalents. 

Subcounty areas are built from incorporated places and MCDs, with MCDs only being used in 

the 20 states where MCDs serve as functioning governmental units. Each subcounty area formed 

has a total population of at least 24,000, as determined by the July 1, 2010 Population Estimates 

data, which are based on the 2010 Census estimates of the population on April 1, 2010, 

extrapolated forward. The subcounty areas can be incorporated places, MCDs, place/MCD 

intersections (in counties where places and MCDs are not coexistent), ‘balance of MCD,’ and 

‘balance of county.’ The latter two types group together unincorporated areas and places/MCDs 

that do not meet the population threshold. If two or more subcounty areas cannot be formed 

within a county, then the entire county is treated as a single area. Thus all counties whose total 

population is less than 48,000 will be treated as a single area since it is not possible to form two 

areas that satisfy the minimum size threshold. 
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The estimation procedure used to assign the weights is then performed independently within 

each of the ACS weighting areas. 

1. Initial Housing Unit Weighting Factors - This process produced 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, 2.5, or 3.0 according to Table 2 

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 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 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 
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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. 

• Housing unit Post-stratification Factor (HPF)—This factor makes the total weight of 

all housing units agree with the 2010 independent housing unit estimates at the 

subcounty level. 

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 × ... × HPF). 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: 

• Subcounty Area 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 subcounty area. Because of later adjustment to the person weights, total 

population is not assured of agreeing exactly with the official 2010 population 

estimates at the subcounty level. 

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

factor is applied to individuals based on the combination of their status of being in a 

married-couple or unmarried-partner household and whether they are the 
15

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, race, sex and Hispanic origin. It adjusts the person weights so 

that the weighted sample counts equal independent population estimates by age, 

race, sex, and Hispanic origin at the weighting area. Because of collapsing of 

groups in applying this factor, only total population is assured of agreeing with the 

official 2010 population estimates at the weighting area level. 

 This uses the following groups (note that there are 13 Age groupings): 

 Weighting Area × Race / Ethnicity (non-Hispanic White, non-Hispanic Black, nonHispanic American Indian or Alaskan Native, non-Hispanic Asian, non-Hispanic 

Native Hawaiian or Pacific Islander, and Hispanic (any race)) × Sex × Age Groups. 

 These three steps are repeated several times until the estimates at the national level 

achieve their optimal consistency with regard to the spouse and householder equalization. 

The effect 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 × ... × HPF × PPSF). 

3. Rounding—The final product of all person weights (BW × ... × HPF × 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: 

County 

County × Race 
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County × Race × Hispanic Origin 

County × Race × Hispanic Origin × Sex 

County × Race × Hispanic Origin × Sex × Age 

County × Race × Hispanic Origin × Sex × Age × Tract 

County × Race × Hispanic Origin × Sex × Age × Tract × Block 

For example, the number of White, Hispanic, Males, Age 30 estimated for a county 

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 race, Hispanic origin, 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: 

County 

County × Tract 

County × Tract × Block 

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 
17

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. 

• 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 ACS 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 
18

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 as 

well as 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 

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

found in most surveys, including the ACS, and the reduction of standard errors of ACS 

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 ACS 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, please see the section entitled “Control of Nonsampling Error”. 

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 ACS 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. 
19

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 ACS 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 

ACS 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 

and the standard error can easily be computed from the margin of error. All ACS 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 and earlier estimates, ACS 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 
20

multiplier. When calculating standard errors from margins of error or confidence bounds 

using published data for 2005 and earlier, 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 ACS 

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

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 
21

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: 

෠ܺ൫ܧܵ

ଵ േ ܺ෠ଶ൯ ൌ ටሺܵܧ൫ܺ෠ଵ൯ሻଶ ൅ ሺܵܧ൫ܺ෠

(1) ݅ܽ݊ܿ݁ݎܽݒ݋ܿ േ ଶ൯ሻଶ

The covariance measures the interactions 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)

However, this method 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 
22

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 31 to demonstrate 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% ൈ ܲ෠ (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 (4) and ratios (3) - the plus sign in the previous formula has been replaced with a 

minus sign. If the value under the radical is negative, use the ratio standard error formula above, 

instead. 

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 earlie te r estima = Y, then the ormula for percent change is: f

ܵܧ ቆܺ෠ െ ܻ෠

ܻ෠ ቇ ൌ ܵܧ ቆܺ෠ܻ෠ െ 1ቇ ൌ ܵܧሺܺ෠ܻ෠

ሻ (5)
23

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 (6)

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 (7)

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: 

෠ܺ൫ܧܵ

ଵିYear െ ܻ෠ହି௒௘௔௥൯ ൌ ඨ3

5 ሾܵܧ൫ܺ෠

ଵିYear൯ሿଶ ൅ ሾܵܧ൫ܻ෠ହିYear൯ሿଶ (8)

TESTING FOR SIGNIFICANT DIFFERENCES 

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

ACS 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 a Z statistic: 

ܼ ൌ

ݐݏܧଵ െ ݐݏܧଶ

ඥሺܵܧଵሻଶ ൅ ሺܵܧଶሻଶ (9)
24

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 an ACS estimate using this method, 

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

characteristic and geographic area but from a previous year, 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, for example), 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: 

ܼ ൌ 

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 ACS 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 

We will present some examples based on the real data to demonstrate the use of the formulas. 

 

1

 The ACS Accuracy of the Data document in 2005 used a Z statistic of +/-1.65. Data users should use +/-1.65 for 

estimates published in 2005 or earlier. 
25

Example 1 - Calculating the Standard Error from the Confidence Interval 

The estimated number of males, never married is 42,741,179 from summary table 

B12001 for the United States for 2010. The margin of error is 100,944. 

 Standard Error = Margin of Error / 1.645 

 Calculating the stand us ard error ing the margin of error, we have: 

 

ܵܧሺ42,741,179ሻ ൌ 100,944

1.645 ൌ 61,364.

 

Example 2 - Calculating the Standard Error of a Sum or a 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 42,741,179. From summary table 

B12001 we have the number of females, never married is 36,898,838 with a margin of 

error of 85,983. So, the estimated number of people who have never been married is 

42,741,179 + 36,898,838 = 79,640,017. 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 61,364. The 

standard error for the number of females never married is calculated using the margin of 

error: 

 

SEሺ36,898,838ሻ   ൌ  85,983 / 1.645  ൌ  52,269.

 So using formula (2) for the approximate standard error of a sum or difference we have: 

 

ܵܧሺ79,640,017ሻ ൌ ඥ61,364ଶ ൅ 52,269ଶ ൌ 80,608

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

estimates interact in either a positive or negative way. 

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

79,640,017 using the standard error, simply multiply 80,608 by 1.645, then add and 

subtract the product from 79,640,017. Thus the 90 percent confidence interval for this 

estimate is [79,640,017 - 1.645(80,608)] to [79,640,017 + 1.645(80,608)] or 79,507,417 

to 79,772,617. 
26

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 36,898,838 and the number of people who have never been married is 79,640,017. 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 52,269 and the approximate standard error 

for the number of people never married calculated from example 2 as 80,608. 

 The estimate is ሺ36,898,838 / 79,640,017ሻ כ  100%  ൌ  46.33%

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

have: 

ܵܧሺ46.33%ሻ ൌ 100% כ ൬ 1

79,640,017 ඥ52,269ଶ െ 0.4633ଶ כ 80,608ଶ൰ ൌ 0.05

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

46.33 using the standard error, simply multiply 0.08 by 1.645, then add and subtract the 

product from 46.33. Thus the 90 percent confidence interval for this estimate is 

 [46.33 - 1.645(0.05)] to [46.33 + 1.645(0.05)], or 46.25% to 46.41%. 

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 approximate standard error. From the above 

examples, the estimate for the number of unmarried men is 42,741,179 with a standard 

error of 61,364, and the estimate for the number of unmarried women is 36,898,838 with 

a standard error of 52,269. 

The estimate of the ratio is 42,741,179 / 36,898,838 = 1.158. 

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

ܵܧሺ1.158ሻ ൌ 1

36,898,838 ඥ61,364ଶ ൅ 1.158ଶ כ 52,269ଶ ൌ 0.00234

The 90 percent margin of error for this estimate would be 0.00234 multiplied by 1.645, or 

about 0.004. The 90 percent lower and upper 90 percent confidence bounds would then 

be [1.158 – 0.004] to [1.158 + 0.004], or 1.154 and 1.162. 
27

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 74,873,372 with a margin of error of 216,091 

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

housing units is 82.0% (0.820) with a margin of error of 0.1 (0.001). So the number of 1-

unit detached owner-occupied housing units is 74,873,372 * 0.820 = 61,396,165. 

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

ܵܧሺ74,873,372ሻ ൌ 216,091 1.645 ⁄ ൌ 131,362

and 

ܵܧሺ0.820ሻ ൌ 0.001 1.645 ൌ 0.0006079 ⁄

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

units is calculated u ing s formula (5) f r products as: o

ܵܧሺ61,396,165ሻ ൌ ඥ74,873,372ଶ כ 0.0006079ଶ ൅ 0.820ଶ כ 131,362ଶ ൌ 116,938

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

61,396,165 using the standard error, simply multiply 116,938 by 1.645, then add and 

subtract the product from 61,396,165. Thus the 90 percent confidence interval for this 

estimate is [61,396,165 - 1.645(116,938)] to [61,396,165 + 1.645(116,938)] or 

61,203,802 to 61,588,528. 

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 

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. 
28

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

ACS an address list in each state, is as complete and accurate as possible. The source of 

addresses for the ACS is the MAF. 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 Main Street versus 123 Maine Street. 

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 ACS 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 ACS coverage rates may be found at 

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

• 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 

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 
29

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 ACS makes every effort to minimize unit nonresponse, and thus, the potential for 

nonresponse error. First, the ACS 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 by personal visit to attempt an interview. Combined, 

these three efforts resulted in a very high overall response rate for the ACS. 

ACS 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/methodology_main/. 

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 ACS, 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 

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 
30

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/methodology_main/. 

• 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 

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. 

 
31

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 with income below the poverty level in 

the past 12 months using both state and PUMA level estimates for the state of Wyoming. Part of 

the collapsed table C17001 is displayed below with estimates and their margins of error in 

parentheses. 

Table A: 2010 Estimates of Males with Income Below Poverty from table C17001: Poverty 

Status in the Past 12 Months by Sex by Age 

Characteristic Wyoming PUMA 

00100 

PUMA 

00200 

PUMA 

00300 

PUMA 

00400 

Male 23,001 

(3,309) 

5,264 

(1,624)

6,508 

(1,395)

4,364 

(1,026) 

6,865 

(1,909)

Under 18 

Years Old 

8,479 

(1,874) 

2,041 

(920)

2,222 

(778)

1,999 

(750) 

2,217 

(1,192)

18 to 64 

Years Old 

12,976 

(2,076) 

3,004 

(1,049)

3,725 

(935)

2,050 

(635) 

4,197 

(1,134)

65 Years and 

Older 

1,546 

(500) 

219

 (237)

561 

(286)

315 

 (173) 

451 

(302)

2010 American FactFinder 

The first way is to sum the three age groups for Wyoming: 

Estimate(Male) = 8,479 + 12,976 + 1,546 = 23,001. 

The first approximation for the standard error in this case gives us: 

ܵܧሺܯ݈ܽ݁ሻ ൌ ඨሺ

1,874

1.645ሻଶ ൅ ሺ2,076

1.645ሻଶ ൅ ሺ

500

1.645ሻଶ ൌ 1,727.1

A second way is to sum the four PUMA estimates for Male to obtain: 

Estimate(Male) = 5,264 + 6,508 + 4,364 + 6,865 = 23,001 as before. 

The second approximation for the standard error yields: 
32

ܵܧሺܯ݈ܽ݁ሻ ൌ ඨሺ

1,624

1.645ሻଶ ൅ ሺ1,3951.645ሻଶ ൅ ሺ1,0261.645ሻଶ ൅ ሺ1,9091.645ሻଶ ൌ 1,851.9

Finally, we can sum up all three age groups for all four PUMAs to obtain an estimate based on a 

total of twelve estim tes: a

23,001 ൌ 451 ൅ ڮ ൅ 2,222 ൅ 2,041 ൌ ሻ݈ܽ݁ܯሺ݁ݐ݅݉ܽݐݏܧ

And the third approximated standard error is 

ܵܧሺܯ݈ܽ݁ሻ ൌ ඨሺ

920

1.645ሻଶ ൅ ሺ

778

1.645ሻଶ ൅ڮ൅ሺ

302

1.645ሻଶ ൌ 1,649.0

However, we do know that the standard error using the published MOE is 3,309 /1.645 = 

2,011.6. In this instance, all of the approximations under-estimate the published standard error 

and should be used with caution. 

B. Suppose we wish to estimate the total number of males at the national level using age and 

citizenship status. The relevant data from table B05003 is displayed in table B below. 

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

Characteristic Estimate MOE 

Male 151,375,321 27,279 

 Under 18 Years 38,146,514 24,365 

 Native 36,747,407 31,397 

 Foreign Born 1,399,107 20,177 

 Naturalized U.S. Citizen 268,445 10,289 

 Not a U.S. Citizen 1,130,662 20,228 

 18 Years and Older 113,228,807 23,525 

 Native 95,384,433 70,210 

 Foreign Born 17,844,374 59,750 

 Naturalized U.S. Citizen 7,507,308 39,658 

 Not a U.S. Citizen 10,337,066 65,533 

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 ൌ 38,146,514 ൅ 113,228,807 ൌ 151,375,321

And the first approximated standard error is 
33

ܵܧሺܯ݈ܽ݁ሻ ൌ ܵܧሺ151,375,321ሻ ൌ ඨ൬

24,365

1.645 ൰

ଶ

൅ ൬

23,525

1.645 ൰

ଶ

ൌ 20,588.8

Another way would be to add up the estimates for the three subcategories (Native, and the two 

subcategories for Foreign Born: Naturalized U.S. Citizen, and Not a U.S. Citizen), for males 

under and over 18 years of age. From these six estimates we obtain: 

ሻ݈ܽ݁ܯሺ݁ݐ݅݉ܽݐݏܧ

ൌ 36,747,407 ൅ 268,445 ൅ 1,130,662 ൅ 95,384,433 ൅ 7,507,308

൅ 10,337,066 ൌ 151,375,321.

With a second approximated stan

ܵܧሺܯ݈ܽ݁ሻ ൌ ܵܧሺ151,375,321ሻ

ൌ ඨ൬

31,397

1.645

dard error of: 

൰

ଶ

൅ ൬

10,289

1.645 ൰

ଶ

൅ ൬

20,228

1.645 ൰

ଶ

൅ ൬

70,210

1.645 ൰

ଶ

൅ ൬

39,658

1.645 ൰

ଶ

൅ ൬

65,533

1.645 ൰

ଶ

ൌ 67,413.0

We do know that the standard error using the published margin of error is 27,279 / 1.645 = 

16,583.0. With a quick glance, we can see that the ratio of the standard error of the first method 

to the published-based standard error yields 1.24; an over-estimate of roughly 24%, whereas the 

second method yields a ratio of 4.07 or an over-estimate of 307%. 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. 

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

Table C 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). 

 
34

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 

65 and 66 

years old 

2,492,871 20,194 2,803,516 23,327 5,296,387 30,854

67 to 69 

years old 

3,029,709 18,280 3,483,447 24,287 6,513,225 30,398

70 to 74 

years old 

4,088,428 21,588 4,927,666 26,867 9,016,094 34,466

75 to 79 

years old 

3,168,175 19,097 4,204,401 23,024 7,372,576 29,913

80 to 84 

years old 

2,258,021 17,716 3,538,869 25,423 5,796,890 30,987

85 years 

and older 

1,743,971 17,991 3,767,574 19,294 5,511,545 26,381

Total 16,781,175 NA 22,725,473 NA 39,506,648 74,932

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

males and females to get 16,781,175 + 22,725,542 = 39,506,717. One way we could use is 

summing males and female for each age category and then using their MOEs to approximate the 

standard error for the total number of people over 65. 

ܯܱܧሺܣ݃݁ 65 ܽ݊݀ 66ሻ ൌ ܯܱܧሺ5,296,387ሻ ൌ ඥ20,194 ൅ 7 ଶ 23,32 ଶ ൌ 4 30,85

ܯܱܧሺܣ݃݁ 67 ݋ݐ 69ሻ ൌ ܯܱܧሺ6,513,225ሻ ൌ ඥ18,280ଶ ൅ 24,287ଶ ൌ 30,398

... etc. ... 

Now, we calculate for the number of people aged 65 or older to be 39,506,648 using the six 

derived estimates and approximate the standard error: 

ܵܧሺ39,506,717ሻ

ൌ  ඨሺ

30,854

1.645 ሻଶ ൅ ሺ30,398

1.645 ሻଶ ൅ ሺ34,4661.645 ሻଶ ൅ ሺ29,9131.645 ሻଶ ൅ ሺ30,9871.645 ሻଶ ൅ ሺ26,3811.645 ሻଶ ൌ 45,552 

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

people aged 65 or older is 39,506,648 with a margin of error of 20,689. Therefore the publishedbased standard error is: 
ܵܧሺ39,506,648ሻ ൌ 20,689 1.645 ൌ 12,577. ⁄

35

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

standard error roughly 3.6 times larger than the published-based standard error. 

As a note, 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 published MOEs for the males 

age 65 and older and of females aged 65 and older separately and then combine to find the 

approximate standard error for the total. 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. However, in this particular example, the results 

from all three ways are the same. So no matter which way you use, you will obtain the same 

approximation for the SE. This is different from the results seen in example A. 

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

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

part of the table shaded gray were derived for the purposes of this example only and cannot be 

found in base table B01001. 

 
36

Table D: 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 151,375,321 27,279 155,631,235 27,280 307,006,556 38,579 

Under 5 

years 10,853,263 15,661 10,355,944 14,707 21,209,207 21,484 

5 to 9 

years old 10,273,948 43,555 9,850,065 42,194 20,124,013 60,641 

10 to 14 

years old 10,532,166 40,051 9,985,327 39,921 20,517,493 56,549 

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

62 to 64 

years old 4,282,178 25,636 4,669,376 28,769 8,951,554 38,534 

Total for 

Age 0 to 

64 years 

old 

134,594,146 117,166 132,905,762 117,637 267,499,908 166,031 

Total for 

Age 65 

years and 

older 

16,781,175 120,300 22,725,473 120,758 39,506,648 170,454 

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: 307,006,556 – 267,499,908 = 39,506,648. 

The way to approximate the SE is the same as in part C. First we will sum male and female 

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

information to approximate the standard error for our estimate of interest: 

ܯܱܧሺܶݐ݋݈ܽ ܲݑ݌݋݈ܽݐ݅݋݊ሻ ൌ ܯܱܧሺ307,006,556ሻ ൌ ඥ27,279ଶ ൅ 27,280ଶ ൌ 38

ܯܱܧሺܷ݊݀݁ݎ 5 ݕ݁ܽݏݎሻ ൌ ܯܱܧሺ21,209,207ሻ ൌ ඥ15,661ଶ ൅ 14,707

,579

ൌ 21,484

... etc. ... 

And the SE for the total number of people aged 65 and older is: 
37

ሻ39,506,648ሺܧܵ ൌ ሻݎ݈݀݁݋݃݁ 65 ܽ݊݀ ܣሺܧܵ

ൌ ඨሺ

38,579

1.645 ሻଶ ൅ ሺ21,4841.645 ሻଶ ൅ ሺ60,6411.645 ሻଶ ൅ ሺ56,5491.645 ሻଶ ൅ڮ൅ሺ38,5341.645 ሻଶ

ൌ 103,620

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

number of people aged 65 or older is 39,506,648 with a margin of error of 20,689. Therefore the 

standard error is: 

SE(39,506,648) = 20,689 / 1.645 = 12,577. 

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

error roughly 8.2 times larger than the actual SE. 

Data users can mitigate the problems shown in examples A through D to some extent by utilizing 

a collapsed version of a detailed table (if it is available) which will reduce the number of 

estimates used in the approximation. 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 feebased 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/.