Public Use Microdata Sample (PUMS)
Accuracy of the Data (2013-2017)
INTRODUCTION
This 5-year public use microdata sample (PUMS) for 2013-2017 is a subset of the 2013-2017
American Community Survey (ACS) and Puerto Rico Community Survey (PRCS) samples. It
contains the same sample as the combined PUMS 1-year files for 2013, 2014, 2015, 2016, and
2017. Unless otherwise specified, the term “ACS” in this document will refer to both the ACS
and PRCS.
This 2013-2017 ACS 5-year PUMS contains five years of data for housing units (HUs) and the
population from households and the group quarters (GQ) population. The GQ population,
housing units and population from households are all weighted to agree with the ACS counts,
which are an average over the five year period (2013-2017). The ACS sample was selected from
all counties across the nation, and all municipios in Puerto Rico.
Estimates from the PUMS file are expected to be different from the previously released ACS
estimates because they are subject to additional sampling error and further data processing
operations. The additional sampling error is a result of selecting the PUMS housing and person
records through an additional stage of sampling. In the public use file, the basic unit is an
individual housing unit, except for the sample from GQs. For the GQ sample, the basic unit is
the person. The population sample is defined as all persons living in households selected in the
housing unit sample, plus the persons selected from the GQ sample. Note that microdata records
in this sample do not contain names, addresses, or any information that can identify a specific
housing unit, GQ or person.
Users of the 2013-2017 ACS 5-year PUMS file can find detailed information on differences
between the 2013-2017 files and previous PUMS files in the PUMS ReadMe document. The
PUMS ReadMe document for this PUMS file can be found at:
https://www.census.gov/programs-surveys/acs/technicaldocumentation/pums/documentation.html/.

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Table of Contents
INTRODUCTION......................................................................................................................... 1
CONFIDENTIALITY OF THE DATA ...................................................................................... 3
TITLE 13, UNITED STATES CODE ................................................................................................. 3
DISCLOSURE AVOIDANCE ............................................................................................................ 3
DATA SWAPPING .......................................................................................................................... 3
SYNTHETIC DATA ........................................................................................................................ 4
PUMAS ....................................................................................................................................... 4
ADDITIONAL MEASURES .............................................................................................................. 4
SAMPLE DESIGN........................................................................................................................ 4
SAMPLE DESIGN FOR HOUSING UNITS ......................................................................................... 5
SAMPLE DESIGN FOR GROUP QUARTERS ..................................................................................... 6
WEIGHTING ................................................................................................................................ 6
GROUP QUARTERS PERSON WEIGHTING ...................................................................................... 6
HOUSING UNIT AND HOUSEHOLD PERSON WEIGHTING ............................................................... 7
ESTIMATION............................................................................................................................... 9
ERRORS IN THE DATA........................................................................................................... 11
SAMPLING ERROR ...................................................................................................................... 11
NONSAMPLING ERROR ............................................................................................................... 11
MEASURING SAMPLING ERROR ........................................................................................ 12
STANDARD ERROR ..................................................................................................................... 12
CONFIDENCE INTERVALS ........................................................................................................... 12
LIMITATIONS .............................................................................................................................. 13
APPROXIMATING STANDARD ERRORS WITH REPLICATE WEIGHTS ............................................ 14
APPROXIMATING GENERALIZED STANDARD ERRORS WITH DESIGN FACTORS ........................... 16
EXAMPLES OF STANDARD ERROR CALCULATIONS USING GENERALIZED STANDARD ERROR
FORMULAS ................................................................................................................................. 21
WORKING WITH DOLLAR AMOUNTS .............................................................................. 24
ADJUSTMENT FACTORS ON THE PUMS FILE .............................................................................. 24
COMPARING PUMS FILES FROM DIFFERENT PERIODS ............................................................... 25
REFERENCES ............................................................................................................................ 25
PUMAS AFFECTED BY TEL SUPPRESSION...................................................................... 26

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CONFIDENTIALITY OF THE DATA
The Census Bureau has implemented a series of steps to protect the confidentiality of the data.
Title 13 of the 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. 1 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.

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

1

The Census Bureau’s Disclosure Review Board approved the 2013-2017 PUMS 5-year data for release with DRB
Clearance number CDDRB-FY19-071.

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individuals. Data swapping procedures were first used in the 1990 Census, and were used
again in the 2000 Census 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.

PUMAs
The Census Bureau takes further steps to prevent the identification of specific individuals,
households, or housing units. The main disclosure avoidance method used is to limit the
geographic detail shown in the files. The smallest geographic unit that is identified is the
Public Use Microdata Area (PUMA), which is based on a population size of initially around
100,000 or more. No geography smaller than the PUMA can be identified on a PUMS file.

Additional Measures
Other disclosure avoidance measures used in the PUMS files includes top-coding, age
perturbation, weight perturbation and collapsing of detail for categorical variables. The
answers to open-ended questions, where an extreme value might identify an individual are topcoded (or bottom-coded). Top-coding (and bottom-coding) substitutes the value of extreme
cases with the mean of the highest (or lowest) cases. Top-coded questions include age, income
and housing unit value. Age perturbation disguises original data by randomly adjusting the
reported ages for a subset of individuals. Weight perturbation disguises the probability of
selection for some records. Users should exercise caution when forming estimates near topcoded or bottom-coded values. More information on the variables that receive top or bottom
coding in the 2017 PUMS can be found at: https://www.census.gov/programssurveys/acs/technical-documentation/pums/documentation.html.

SAMPLE DESIGN
The 2013-2017 ACS 5-year PUMS sample is the same sample found in each of the 1-year PUMS
files for the years 2013, 2014, 2015, 2016, and 2017. It contains five percent of the housing
units and five percent of the GQ persons plus some imputed GQ persons in the United States,
District of Columbia, and Puerto Rico weighted to represent the average population during five
years.
The PUMS GQ sample for data years 2013, 2014, 2015, 2016, and 2017 contained additional
imputed records to represent the not-in-sample GQs, which effectively double the total number
of records from those years.

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See the Accuracy of the Data for the 2017 PUMS 1-year for a further explanation of the PUMS
sampling of these imputed GQ records. By including these imputed records in the 2013-2017
ACS 5-year PUMS, the PUMS will agree better with the 2013-2017 5-year full sample ACS for
population totals by state and PUMA. More details about the methodology of the large-scale
whole person imputation into not-in-sample GQ facilities can be found in the 2017 ACS 1-year
Accuracy of the Data at: https://www.census.gov/programs-surveys/acs/technicaldocumentation/code-lists.html.

Sample Design for Housing Units
The sampling for HUs (and persons from HUs) was performed independently on the ACS
samples of HUs for each of the years 2013, 2014, 2015, 2016 and 2017 as follows:
1. Records of HUs were sorted within each state by: PUMA, ACS weighting area, interview
mode, type of vacant, tenure, building type, household type, householder demographics
(race, Hispanic origin, sex and age), county, tract, and housing unit weight.
2. Systematic sampling was applied to ACS HUs as described below:
a. Within each state, a random number was chosen between zero and the sampling
interval. A counter was initialized with the random number.
b. At each record, the value of the counter was incremented by one and compared to
the sampling interval.
i. If the counter’s new value was greater than the sampling interval, the HU
record was selected for the PUMS and a flag was set to 1. The counter
was decreased by the sampling interval with the new value passed to the
next record.
ii. If the counter was less than the sampling interval, the HU record was not
selected for the PUMS and the value of the counter was passed to the next
record without altering its value.
3. All HUs selected for PUMS were placed in the PUMS HU sample file.
The PUMS HU sample file was matched to the ACS sample of persons. All persons in selected
HUs were placed in the PUMS person sample.
The 2013-2017 5-year ACS Housing Unit estimates for all states may be found on American
FactFinder:
https://factfinder.census.gov/bkmk/table/1.0/en/ACS/17_5YR/B25001/0100000US.04000.

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The PUMS HU sample size may be found in the 2013-2017 ACS 5-year PUMS Record Counts
file located on the PUMS Technical Documentation page (https://www.census.gov/programssurveys/acs/technical-documentation/pums/documentation.html).

Sample Design for Group Quarters
The sampling for PUMS GQ persons was originally performed on the ACS sample of GQ
persons for each of the years 2013, 2014, 2015, 2016 and 2017 as follows:
1. GQ persons were sorted within each state by the size of their GQ facility (large vs
small), the type of GQ facility, PUMA, demographics (race, Hispanic origin, sex and
age), county, tract, and GQ person weight.
2. Systematic sampling was applied as described above under HUs.
3. All selected GQ persons were added to the PUMS person sample. All imputed records
derived from the selected record were also placed in the PUMS person sample.
4. A placeholder record was also placed in the PUMS HU file for each PUMS GQ person
record.
The 2013-2017 ACS 5-year estimates for Group Quarters may be found at:
https://factfinder.census.gov/bkmk/table/1.0/en/ACS/17_5YR/B26001/0100000US.04000.
The PUMS estimates for Group Quarters may be found in the PUMS Estimates for User
Verification on the PUMS Technical Documenation page: https://www.census.gov/programssurveys/acs/technical-documentation/pums/documentation.html.

WEIGHTING
Group Quarters Person Weighting
The procedure used to assign the weights to the GQ persons is performed independently within
state. The steps are as follows:
Initial Weight for GQ Persons
The 5-year PUMS initial weight is the product of the 1-year ACS unrounded weights for the
record divided by five and the PUMS subsampling factor. For 2012 and later records, each
imputed record received the same subsampling factor as its donor interview. Note that for
these data, the ACS weights for sample and imputed records added together represent the GQ
universe.

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GQ Person Weighting Factors
GQ Person Post-stratification Factor
This factor adjusts the GQ person weights so that the weighted sample counts equal the
published ACS estimates at the state level. The GQ imputed records are not distinguished
from the GQ sample records when forming the cells used for the GQ Person Poststratification Factor Adjustment. Since this adjustment is done at the state level and since
noise is added for disclosure avoidance reasons, only state level PUMS GQ person
estimates will agree with published ACS estimates. This adjustment uses the following
groups:
State × Institutional/noninstitutional × Sex × Age Category
Rounding for GQ Person Weights
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 ACS total GQ person estimate
for the state.

Housing Unit and Household Person Weighting
The estimation procedure used to assign the HU and person weights is performed independently
within each PUMA.
Initial Weight for Persons and HUs
The 5-year PUMS initial weight is equal to the product of the ACS 1-year final weight for
the record and the PUMS subsampling factor divided by five.
Person Weighting Factors
The person weights are adjusted to agree better with ACS published estimates for
householders, spouses, race, Hispanic origin, sex and age by a series of two steps that are
repeated until a stopping criterion is met. This is an iterative proportional fitting or raking
process. The person weights are individually adjusted at each step as described below.
The two steps are as follows:
Spouse Equalization/Householder Equalization Raking Factor
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 householder. All
persons are assigned to one of four groups:

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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 ACS estimate of married-couple or unmarried-partner households using the ACS
housing unit weight. The weights of persons in the third group are adjusted so that the sum
is equal to the ACS estimate of occupied housing units not having a partner using the
housing unit weight. The weights of persons in the fourth group are adjusted to agree with
the ACS total population minus the first three groups. 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
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 ACS population
estimates by age, race, sex, and Hispanic origin at the PUMA level. Because of collapsing
of groups in applying this factor, only total population is assured of agreeing precisely with
the published ACS 2013-2017 population estimates at the PUMA level.
This uses the following groups within each PUMA (note that there are 13 Age groupings):
Race / Ethnicity (non-Hispanic White, non-Hispanic Black, non-Hispanic American Indian
or Alaskan Native, non-Hispanic Asian, non-Hispanic Native Hawaiian or Pacific Islander,
and Hispanic (any race)) × Sex × Age Groups
These two steps are repeated several times until the estimates at the PUMA level achieve
their optimal consistency with regard to the spouse and householder equalization. The final
Person Weighting Factor is then equal to the product of the factors from all of the iterations
of these two adjustments. The unrounded person weight is then equal to the product of
Person Weighting Factor times the initial person weight.
Rounding of Person Weights
The person weight after the Person Weighting Factor has been applied 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 ACS total person from HU’s estimates within state and PUMA.

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Householder Adjustment Factor (HHRF)
This factor applied to occupied housing units is the same as the Person Weighting factor
from the person weighting. After this stage, the weight of the housing unit is identical to the
unrounded person weight of the householder after the Person Weighting Factor is applied.
Housing Unit Control factor
This factor adjusts PUMS housing unit estimates to agree with the published ACS housing
unit estimates for housing units with married couples (or partners), occupied housing units
without partners and vacant housing units.
Rounding of Housing Unit weights
The Housing Unit weight after the Housing Unit Control Factor is applied is rounded to an
integer. Rounding is performed so that the sum of the rounded weights is within one housing
unit of the sum of the ACS total HU’s estimates within state and PUMA.
For a detailed description of how the original ACS weights are computed, see the 2013-2017
ACS Multiyear Accuracy of the Data at: https://www.census.gov/programssurveys/acs/technical-documentation/code-lists.html

ESTIMATION
To produce estimates or tabulations of characteristics from the PUMS, add the weights of all
persons or HUs that possess the characteristic of interest. 2 For instance, if the characteristic of
interest is “total number of black teachers”, determine the race and occupation of all persons and
cumulate the weights of those who match the characteristics of interest. To get estimates of
proportions, divide the weighted estimate of persons or HUs with a given characteristic by the
weighted estimate of the base. For example, the proportion of “black teachers” is obtained by
dividing the weighted estimate of black teachers by the estimate of teachers.
PUMS estimates are expected to be different from published ACS estimates that are based on the
full set of data because of the additional sampling. The exception will be characteristics
controlled by the ratio-estimate factors.

2

Users should exercise caution when forming estimates near top-coded or bottom-coded values. More information
on the variables that receive top or bottom coding in the PUMS can be found at: https://www.census.gov/programssurveys/acs/technical-documentation/pums/documentation.html.

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Note that the housing unit file contains some records with blank weights. These are the GQ
placeholder records. 3 The housing unit weights were set to zero for these records since they are
not housing units, but persons. For confidentiality reasons, the GQ data are not provided at the
level of an address but only at the person-level. All of the GQ person data are included in the
PUMS person file except variable for the “Yearly food stamp/Supplemental Nutrition Assistance
Program (SNAP) recipiency” (FS), which are the only data, included on the GQ placeholder
records in the housing unit file. For food stamp recipiency estimates of persons in GQs, you will
need to match the placeholder records to the person file to get the person weights.
A note to GQ data users. There are limitations to the usefulness of GQ estimates at the PUMA
level. The PUMS weighting controls the GQ estimates to agree with the ACS state level
estimates. Depending on the application or analysis, GQ data users should consider working
with state level estimates rather than PUMAs.
In a limited number of geographies, the ACS PUMS file has suppressed variables affecting
PUMS data. The suppression was due to nonsampling error or issues with interpreting the
recode. Three variables were affected. In 2015, the telephone variable (TEL) was suppressed in
five PUMAS. In 2016, the telephone variable (TEL) was suppressed in fourteen PUMAS. In
these cases, the nonsampling errors could not be edited or corrected before the publication of the
PUMS file. Specific PUMAs with suppressed data for TEL are listed in at the end of this
document. In addition, beginning in 2012, the complete plumbing facilities recode (PLM) was
suppressed in Puerto Rico.
Within the PUMAs listed in the user note, the variable TEL were assigned the value ‘8’ to
identify it as suppressed due to data problems. In order to estimate the telephone rate for the
states impacted by the suppressed values, use the estimate having the value of ‘1’ divided by the
sum of the estimate having the value of ‘1’ and the estimate having the value of ‘2’. In the
examples shown, the fertility rate is the number out of 1000 and the telephone rate is a percent:

The variable PLM was assigned the value of ‘9’ in all PUMAs in Puerto Rico to mean “not
applicable”. See the table at the end of this document for a list of PUMA codes affected by the
TEL variables.

3

To identify HU and GQ placeholder records on the PUMS housing file, see the TYPE variable in the PUMS data
dictionary: https://www.census.gov/programs-surveys/acs/technical-documentation/pums/documentation.html.

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In 2016, the questions pertaining to business on property (BUS) and presence of a flush toilet
(TOIL) were removed from the ACS questionnaire. As such, there is not data for data year 2016
for variables BUS and TOIL on the 2013-2017 ACS 5-year PUMS file. Rather, they are each
assigned a value of ‘9’ for data year 2016 cases. Likewise, the corresponding allocation flags,
FBUSP and FTOILP, were assigned a value of zero on the 2013-2017 file.
Data users should use caution when dealing with the variables BUS, TOIL, FBUSP and FTOILP.
Five year estimates for these variables cannot be derived from the 2013-2017 ACS 5-year PUMS
file, as only four years’ worth of data are provided. Any five year estimates derived for BUS,
TOIL, FBUSP and FTOILP from the 2013-2017 file will be inaccurate.
These variables were retained on the file as they were previously included as components of the
variables SVAL and PLM, respectively. The removal of the variables caused a change in the
components of these variables between data years 2015 and 2016. For 2015, BUS was included
as a component in SVAL; for 2016 and later, SVAL does not consider the presence of a business
on the property. Likewise, PLM required a flush toilet in 2015; in 2016 and later, complete
plumbing is not defined by TOIL. Keeping the 2013, 2014, and 2015 data values on the 20132017 ACS 5-year PUMS files allows users to create recodes of PLM and SVAL that will be
comparable across all five data years. For more information, please reference the BUS and TOIL
User Note at: https://www.census.gov/programs-surveys/acs/technical-documentation/usernotes.html.

ERRORS IN THE DATA
Every sample survey is subject to two types of error: sampling error and nonsampling error.

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 HUs and persons within those HUs. 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.
Estimates made with PUMS data are subject to additional sampling error because the PUMS
data consists of a subset of the full ACS sample. Thus standard errors of PUMS estimates can
be larger than standard errors that would be obtained using all of the ACS data.

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

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data. For example, operations such as data entry from questionnaires and editing may
introduce error into the 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 HUs. One way of accomplishing this is by following up on mail
nonrespondents during the CATI and CAPI phases.
More information about the control of nonsampling error can be found in ACS Multiyear
Accuracy of the Data (2013-2017) at: https://www.census.gov/programs-surveys/acs/technicaldocumentation/code-lists.html.

MEASURING SAMPLING ERROR
Standard Error
Standard Error is a measure of the deviation of a sample estimate from the average of all
possible samples. Sampling error and some types of nonsampling error are estimated by the
standard error. The sample estimate and its estimated standard error permit the construction of
interval estimates with a prescribed confidence that the interval includes the average result of
all possible samples.
Two methods are provided for calculating the standard errors of PUMS estimates: a Successive
Difference Replicate (SDR) method (using replicate weights) and a Generalized Variance
Function method (using design factors). Replicate weights have been provided with the ACS
PUMS files since the 2005 PUMS. Design factors (a type of generalized variance function) is
a method used by the Census 2000 PUMS and also in use by the ACS PUMS since 2000. It is
important to keep in mind that there will be differences between the standard error
approximations computed by these two methods. Using the replicate weights will produce a
more accurate estimate of a standard 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.

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For example, if all possible samples that could result under the PUMS 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:
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;
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.
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.
These intervals are referred to as 68 percent, 90 percent, and 95 percent confidence intervals,
respectively. An example of how to construct a 90 percent confidence interval follows:
Add and subtract 1.645 times the standard error to the estimate to yield the lower and upper
bounds of a 90% confidence interval around the estimate (EST).
LB=Lower bound = EST - 1.645*SE(EST)
UB=Upper bound = EST + 1.645*SE(EST)
The 90% confidence interval is the interval (LB, UB).

Limitations
The user should be careful when computing and interpreting standard errors and confidence
intervals.
Nonsampling Error
The estimated standard errors included in this data product do not include all 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.

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Very Small (Zero) or Very Large Estimates
The value of almost all PUMS 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 that, 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 proportions 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.

Approximating Standard Errors with Replicate Weights
Replicate weights can be used to calculate what are referred to as successive difference
replicate (SDR) or direct standard errors. Standard errors for the published ACS tabulations
are calculated using the SDR method. Direct standard errors will often be more accurate than
generalized standard errors, although they may be more inconvenient for some users to
calculate. The advantage of using SDR method is that a single formula is used to calculate the
standard error of many types of estimates.
Each PUMS housing unit and person record contains 80 replicate weights. These replicate
weights were formed from the ACS replicate weights adjusted for PUMS subsampling and
ratio adjustments. For any estimate X, 80 replicate estimates are also computed using the
replicate weights. For this discussion, we refer to X as the ‘full sample estimate.’ The first
replicate estimate, X1, is computed using the first replicate weight, the second replicate
estimate, X2, is computed using the second replicate weight, and so on. Each replicate estimate
is computed using the replicate weights in the same way that the full sample estimate X is
computed.
NOTE: When programming the replicate weight standard errors, users will find the eighty
replicate weights can be positive, zero or negative. The negative replicate weights are partly
due to the addition of the Group Quarters (GQ) population to the full ACS weighting process.
Within a weighting cell, GQ estimates were subtracted from population totals, sometimes
resulting in negative values for the cell. The cells were collapsed in such a way as to prevent a
final cell from being zero or negative for the full sample weights. The full sample weights are
never negative. This restriction was not placed on the replicate weights since their only
purpose is to represent the variability of the sample. PUMS replicate weights are based on
ACS replicate weights so negative values may occur. Keep in mind that the replicate weights
are only to be used to estimate the variance with the formula provided in the PUMS accuracy
document.

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The standard error of X can be computed after the replicate estimates X1 through X80 are
computed. The standard error is estimated using the sum of squared differences between each
replicate estimate Xr and the full sample estimate X. The standard error formula is:

If X is zero, then use the generalized variance method for zero estimates given the Standard
Errors for Totals and Percentages section of this document, to approximate the standard error.
Data users who wish to see worked examples may consult the documentation for the ACS
Variance Replicate Tables, located here: https://www.census.gov/programssurveys/acs/technical-documentation/variance-tables.html.
The standard error can be used to form a 90% confidence interval around the estimate (X) as
follows:
LB=Lower bound = X - 1.645*SE(X)
UB=Upper bound = X + 1.645*SE(X)
The 90% confidence interval is the interval (LB, UB).
As previously mentioned, we consider the replicate weight SEs to be more accurate than the
design factor SEs. For exceptions, please note the following:
After using replicate weight SEs, some users may notice that occasionally the SE is zero for an
estimate. The user may want to know if this is accurate. Except for controlled estimates, all
PUMS estimates are based on a sample of the population and should not have a SE of zero.
However, if the estimate is a controlled count (or total) such as total population or total GQ
population in a state, there is no sampling variability in the estimate. It is expected that the
replicate weight SE and MOE will be zero for some controlled estimates.
If your estimate is a median, the replicate weight method may yield a SE of zero. This occurs
when several records in the middle of the distribution were rounded to the same value, or when
the characteristic contains few records, such as a median based on less than five records.
Rounding by respondents, as well as rounding by PUMS edits may mask the variability in the
median. In order to yield a more adequate standard error for that case, use the design factor
method to estimate the SE of a median.

 P a g e   16
Examples of PUMS estimates with replicate weight standard errors are found in the document
PUMS Estimates for User Verification at: https://www.census.gov/programssurveys/acs/technical-documentation/pums/documentation.html.

Approximating Generalized Standard Errors with Design Factors
Note on the Design Factors
Note that beginning in 2017, the design factors are no longer included in this document.
They are published in a comma separated value (CSV) file located at:
https://www.census.gov/programs-surveys/acs/technicaldocumentation/pums/documentation.html/.
Totals and Percentages
The design factors provided in comma-separated value (CSV) file entitled “2013-2017 ACS
5-year PUMS Design Factors (Attachment A)” can be used to approximate the standard
errors of most sample estimates of totals and proportions. Design factors are given by
subject for the United States, all 50 states, the District of Columbia, and Puerto Rico. The
term "subject" refers to a characteristic, such as age for persons and tenure for HUs. The
design factors reflect the effects of the actual sample design and estimation procedures used
for the ACS. To approximate the standard error for most estimates, use the following
formulas:
Total Formula

Where:
DF = Design Factor
N = Size of Population in the Geographic Area
Y = Estimate of Characteristic Total
Percent Formula

Where:
DF = Design Factor
B = Denominator of Estimated Percentage
p = Estimated Percentage
The values of N and the design factor can be determined as follows:

 P a g e   17
1. For the value of N, obtain the number of persons, number of households or number of
HUs, respectively for the geographies you are interested in. If the estimate is of HUs
then use the number of HUs; if the estimate is of families or households then use the
number of households; otherwise use the number of persons.
2. Select the appropriate table from the comma separated value (CSV) PUMS Design
Factors (Attachment A) file, located at: https://www.census.gov/programssurveys/acs/technical-documentation/pums/documentation.html. Use the design factor
for the United States when estimating characteristics for the United States or geographic
areas that cover more than one state. Use the table for a specific state when estimating
characteristics for that state or geographic areas that are contained entirely within that
state.
3. Then use the selected characteristic to obtain the appropriate design factor for the
characteristic; for example, educational attainment or ancestry. If the estimate is a
combination of two or more characteristics, we suggest the following guideline: Use the
largest design factor for this combination of characteristics. The only exception to this
is for items crossed with race or Hispanic Origin. For an item(s) crossed with race
or Hispanic Origin, use the largest design factor not including the race or Hispanic
Origin design factor.
An inspection of the formulas used to calculate the simple random sampling standard errors
suggests that when dealing either with zero estimates or with very small estimates of totals
and percentages, the standard error estimates approach zero. This is also the case for very
large estimates of totals and percentages. Zero or small estimates, like any other sample
estimates, are still subject to sampling variability and therefore an estimated standard error of
zero or close to zero is not adequate. Use the recommended procedures below for estimates
that fit the following descriptions:
1. An estimated total is less than 425 or within 425 of the total size of the tabulation
area. Use a basic standard error of 110 multiplied by the design factor for the type of
estimate.
2. An estimated percentage is less than 2 or greater than 98. Use a value of 2 for the
estimated percentage in the percent formula.
3. The denominator of a percentage is zero. There are no sample observations
available to compute an estimate of a proportion or an estimate of its standard error.
Sums and Differences
For the sum or difference between two estimates, the standard error is approximately the
square root of the sum of the two individual standard errors squared:

 P a g e   18

This method is, however, an approximation as the two estimates of interest in a sum or a
difference are likely to be correlated. If the two quantities X and Y are positively correlated,
this method underestimates the standard error of the sum of X and Y and overestimates the
standard error of the difference between the two estimates. If the two estimates are
negatively correlated, this method overestimates the standard error of the sum and
underestimates the standard error of the difference.
Ratios
Frequently, the statistic of interest is the ratio of two variables, where the numerator is not a
subset of the denominator. An example is the ratio of students to teachers in public
elementary schools. The standard error of the ratio between two sample estimates is
estimated as follows:

If the ratio is a proportion, that is, the numerator is a subset of the denominator, then follow
the procedure outlined in the Standard Errors for Totals and Percentages section of this
document.
Medians
The sampling variability of an estimated median of a variable depends on the form of the
distribution and the size of its base. The standard error of an estimated median is
approximated by constructing a 68 percent confidence interval. Estimate the 68 percent
confidence limits of a median based on sample data using the following procedure. 4
1. Obtain the weighted frequency distribution for the selected variable using user
defined categorical values. Cumulate these frequencies to yield the base. In general,
variables pertaining to income could use a category as small as $2,500, for example
4

The design factor method shown here for medians is preferred over the replicate weight method whenever the
replicate weight method gives a standard error of zero. This may happen due to having several records in the middle
of the range that have exactly the same value. Be aware that PUMS dollar values are rounded to the nearest 100 for
values between 1,000 and 50,000 and rounded to the nearest 1,000 above 50,000. This increases the number of
respondents with exactly the same value. The amount of rounding done by respondents is unknown, but could be
substantial. Since rounding may cause the number of records with exactly the same value to increase, and might
cause all 80 replicates to yield the same median, the replicate weight formula can give a standard error of zero. To
avoid this, it is possible to calculate the medians using a categorical method with linear interpolation for all 80
replicates, OR simply use the design factor method to estimate the standard errors.

 P a g e   19
$0-$2,499, $2,500-$4,999, etc. In Example 3, only sixteen rows are used (for
simplicity), which causes the income category widths to be larger than ideal. Other
variables such as gross rent should use smaller category widths than the income
variables.
2. Determine the standard error of a 50 percent proportion using the formula in the
Standard Errors for Totals and Percentages section of this document.

3. Subtract from and add to 50 percent the standard error determined in step 2.
p_lower = 50 – SE(50 percent)
p_upper = 50 + SE(50 percent)
4. Determine the categories in the distribution that contain p_lower and p_upper.
If p_lower and p_upper fall in the same category, follow step 5. If p_lower
and p_upper fall in different categories, go to step 6.
5. If p_lower and p_upper fall in the same category, do the following:
•
•
•
•

Define A1 as the smallest value in that category.
Define A2 as the smallest value in the next (higher) category.
Define C1 as the cumulative percent of units strictly less than A1.
Define C2 as the cumulative percent of units strictly less than A2.
Use the following formulas to determine the lower and upper bounds for a
confidence interval about the median:

6. If p_lower and p_upper fall in different categories, do the following:
For the category containing p_lower: Define A1, A2, C1, and C2 as described
in step 5. Use these values and the formula in step 5 to obtain the lower
bound.

 P a g e   20
For the category containing p_upper: Define new values for A1, A2, C1, and
C2 as described in step 5. Use these values and the formula in step 5 to obtain
the upper bound.
7. Use the lower and upper bounds determined in steps 5 or 6 to calculate the standard
error of the median.
SE(median) = 1/2 X (Upper Bound – Lower Bound)
Means
A mean is defined here as the average quantity of some characteristic (other than the number
of people, HUs, households, or families) per person, housing unit, household, or family. For
example, a mean could be the average annual income of females age 25 to 34. The standard
error of a mean can be approximated by the formula below. Because of the approximation
used in developing this formula, the estimated standard error of the mean obtained from this
formula will generally underestimate the true standard error.

Where:
B is the base (denominator) of the mean
s2 is the sample variance of the characteristic based on weighted
data.
The value of s2 can be computed using the formula:

Where:
wi is the weight of the ith sample record
yi is the value of the characteristic for the ith sample record
n is the number of sample records
Note that

is the weighted estimate of persons/HUs in the sample (ex. the number of

females age 25 to 34, and
is the weighted aggregate estimate for the characteristic
of interest (ex. the aggregate income of females age 25 to 34).

 P a g e   21

Examples of Standard Error Calculations using Generalized Standard Error
Formulas
We will present some examples based on the 2009-2013 PUMS 5-year data to demonstrate the
use of the generalized standard error formulas.
Example 1 – Using Design Factors to Estimate the Standard Error of a Total
The estimated number of people 15 years or over who were never married is 2,136,436 from the
PUMS data for the state of Virginia. To calculate the standard error, we use the total formula
given in the section Standard Errors for Totals and Percentages. In this formula, Y is our
estimate of 2,136,436 and N is the total PUMS population for the state of Virginia, which is
8,256,630. The design factor for “Marital Status” is 1.4.

To calculate the margin of error, simply multiply 7,679.46 by 1.645 to get 12,632.72. To obtain
the lower and upper bounds of the 90 percent confidence interval around 2,136,436 using the
margin of error, simply add and subtract 12,632.72 from 2,136,436. Thus the 90 percent
confidence interval for this estimate is [2,136,436 - 12,632.72] to [2,136,436 + 12,632.72] or
2,123,803.28 to 2,149,068.72.
Example 2 – Using Design Factors to Estimate the Standard Error of a Proportion or
Percentage
The estimated percent of people 25 years or over with a bachelor’s degree or higher in Louisiana
is 22.4190 =100*(681,488/3,039,780) from the PUMS data. To calculate the standard error, we
use the percent formula given in the section Standard Errors for Totals and Percentages. Use the
denominator of the percentage, 3,039,780 in the formula. The design factor for “Educational
Attainment” is 1.5.

To calculate the margin of error, multiply 0.1564 by 1.645 to get 0.2573. To obtain the lower
and upper bounds of the 90 percent confidence interval around 22.4190 percent using the margin
of error, simply add and subtract 0.2573 from 22.4190. Thus the 90 percent confidence interval
for this estimated percentage is [22.4190 - 0.2573] to [22.4190 + 0.2573] or 22.16 to 22.68.

 P a g e   22
Example 3 – Calculating the Standard Error of a Median
Users need to form a weighted frequency distribution for the variable of interest. Table 3 below
shows one possible weighted frequency distribution for adjusted household income in
Massachusetts.
Table 1: A Possible Distribution Frequency for Adjusted Household Income in MA
Adjusted Household
Frequency
Cumulative
Cumulative
Income
Frequency
Percent
Less than $10,000
153,739
153,739
6.03
$10,000 to $14,999
130,852
284,591
11.16
$15,000 to $19,999
113,550
398,141
15.62
$20,000 to $24,999
105,230
503,371
19.74
$25,000 to $29,999
95,824
599,195
23.50
$30,000 to $34,999
102,957
702,152
27.54
$35,000 to $39,999
90,972
793,124
31.11
$40,000 to $44,999
87,818
880,942
34.55
$45,000 to $49,999
83,793
964,735
37.84
$50,000 to $59,999
174,388
1,139,123
44.68
$60,000 to $74,999
231,284
1,370,407
53.75
$75,000 to $99,999
318,700
1,689,107
66.25
$100,000 to $124,999
244,795
1,933,902
75.85
$125,000 to $149,999
178,104
2,112,006
82.83
$150,000 to $199,999
202,797
2,314,803
90.79
$200,000 or more
234,913
2,549,716
100.00
The base is the cumulative sum of the weighted frequencies, which is 2,549,716.
Determine the standard error of a 50 percent proportion, using as the denominator the
cumulative sum of the weighted frequencies, 2,549,716. For this example, the design
factor for household income is 1.5.

Calculate p_lower and p_upper.
p_lower = 50 – SE(50 percent) = 49.80
p_upper = 50 + SE(50 percent) = 50.20
Determine the categories that contain p_lower and p_upper. The first category with a cumulative
percentage that is greater than 49.80 is $60,000 to $74,999. The first category with a cumulative

 P a g e   23
percentage that is greater than 50.20 is $60,000 to $74,999. Since p_lower and p_upper fall in
the same category, follow the instructions given in step 5 of the section Standard Errors for
Medians.
Define A1, A2, C1, and C2: A1 = 60,000, A2 = 75,000, C1 = 44.68 and C2 = 53.75. Calculate
the lower bound and upper bound using these values.

Finally, calculate the standard error of the median:
SE(median) = 1/2 x (69,129.00 – 68,467.48) = 330.76
Example 4 – Calculating the Standard Error of a Mean
Suppose we wish to estimate mean adjusted person income of females age 25 to 34 in Alabama.
Table 4 below summarizes the computation of the terms in the formula for s2. The PUMS data
for Alabama has 12,893 records for females age 25 to 34 that have a non-missing value for
person income.
Table 2: Computations for Mean Adjusted Person Income of Females Age 25 to 34 in
Alabama
Sample
yi
wi
wi yi
wi yi2
Record
1
21,462
5
107,309
2,303,061,466
2
21,462
13
279,004
5,987,959,811
3
9,658
32
309,051
2,984,767,660
4
65,459
21
1,374,633
89,981,762,995
.
.
12,893
55,169.65
17
937,884
51,742,728,026
274,162,709
317,090
6,575,359,529
302,151,315,109,878
Total
Note: The inflation-adjusted income shown in column 2 was the person income multiplied by the ADJINC
variable. Unrounded adjusted income was used to compute the products in the third and fourth columns.

The mean adjusted income is:

 P a g e   24

and s2 is computed as follows:

The design factor for person income in Alabama, is 1.6. The standard error of the mean can now
be calculated:

WORKING WITH DOLLAR AMOUNTS
Dollar variables must be adjusted into a common year before using them to form estimates.
Generally, the older years are adjusted to the most recent year covered by the analysis. Data in
the current ACS 5-year PUMS were collected over five years, so the adjustment will inflate
2013, 2014, 2015, and 2016 dollars into 2017 values.

Adjustment Factors on the PUMS File
The PUMS data dictionary for 2013-2017 describes two adjustment factors on the 5-year file
that put dollar values into 2017 dollars:
ADJINC – inflation adjustment factor for income variables, such as household income,
self-employment income, retirement income and wages.
ADJHSG – inflation adjustment factor for most housing dollar variables, such as utility
costs, rent, food stamps, and condominium fees.
For more details, see the PUMS data dictionary at https://www.census.gov/programssurveys/acs/technical-documentation/pums/documentation.html/.
For example, multiply the household income variable by ADJINC to adjust household income
into 2017 dollars. One reason this adjustment is needed is because interviews in the ACS were
conducted throughout the year for a reference period that included twelve previous months.
Application of the adjustment factor will convert amounts to 2017 dollars.

 P a g e   25
For example, multiply ADJHSG times the monthly rent to adjust rent into 2017 dollars. All
records get this adjustment, although the records interviewed in 2017 have a factor of 1.
Note that the values of ADJINC and ADJHSG are the same for all sample cases from the same
year. This is for disclosure avoidance reasons, that is, so that the month of interview cannot be
identified by the adjustment factor.

Comparing PUMS files from Different Periods
When comparing dollar estimates from the 2013-2017 ACS 5-year PUMS file to estimates
from other years, an additional adjustment is necessary to convert the amounts into dollars
from a common year (after applying the adjustment factor described in the previous
paragraphs). We use the CPI-U-RS adjustment factors from the Bureau of Labor Statistics.
These factors can be found in the first table in the PDF file at:
https://www.bls.gov/cpi/research-series/ [For example, to express year 2000 dollars in terms of
2017 dollars, multiply the 2000 dollars by 361.0/252.9 = 1.427].

REFERENCES
[1]. ACS Multiyear Accuracy of the Data (2013-2017)
https://www.census.gov/programs-surveys/acs/technical-documentation/code-lists.html
[2]. Design and Methodology of the American Community Survey:
https://census.gov/programs-surveys/acs/methodology/design-and-methodology.html
[3]. PUMS Accuracy of the Data for 1-Year PUMS files:
https://www.census.gov/programs-surveys/acs/technicaldocumentation/pums/documentation.html/
[4]. PUMS Data Dictionary:
https://www.census.gov/programs-surveys/acs/technicaldocumentation/pums/documentation.html/
[5]. Updated CPI-U-RS, All Items, 1977-2017
https://www.bls.gov/cpi/research-series/

 P a g e   26

PUMAs Affected by TEL Suppression
Year
2015
2015
2015
2015
2015
2016
2016
2016
2016
2016
2016
2016
2016
2016
2016
2016
2016
2016
2016

State FIPS Code
05
12
12
21
55
17
37
37
37
45
45
48
48
48
48
48
48
48
48

PUMA
01100
10700
10900
02600
01601
02100
04600
04700
05100
00603
00604
01901
01902
01903
06801
06802
06803
06804
06807