Number range CHAPTER**Accuracy of the Data (2005)**

INTRODUCTION

The data contained in these Profiles are based on the American Community
Survey (ACS) sample interviewed in 2005. 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 "*Operational Overview of the 2005 American Community Survey*"
provides information on the data collection and Master Address File.

SAMPLE DESIGN

Beginning in 2005, the ACS sample expanded to include all counties and
county-equivalents in the United States, and all municipios in Puerto
Rico. The initial ACS sample is chosen in two phases, and each phase has
two stages. During the first phase, also referred to as the main phase,
the main housing unit address sample is selected for the upcoming year
and the sample is allocated to the 12 months of the sample year. During
the supplemental phase, a sample of addresses that have been added to
the Master Address File (MAF) or have become eligible for sampling after
the main sample has been chosen is selected and is allocated to the last
nine months of the year. The main sample is typically selected during
the summer of the preceding year, while the supplemental sample is
chosen in January of the sample year.

First stage sampling defines the universe for the second stage of
sampling through two steps. First, all addresses that were in a first
stage sample within the past four years are excluded. This ensures that
no address is in sample more than once in any five year period. The
second step is to select a 20% systematic sample of "new" units, i.e.
those units that have never appeared on a previous MAF extract or have
become eligible. Each new address is systematically assigned to either
the current year of to one of four backsamples. This procedure is
designed to maintain five equal partitions of the universe.

Second stage sampling uses seven distinct sampling rates. These rates
are applied to each block in the nation and Puerto Rico by calculating a
measure of size (MOS) for each of the following sampling entities:

-   Counties

-   Places (active, functioning governmental units)

-   School Districts (elementary, secondary, and unified)

-   American Indian Areas

-   Alaska Native Village Statistical Areas

-   Hawaiian Homelands

-   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)

-   Census Designated Places -- in Hawaii only

    The MOS for all areas except American Indian and Alaska Native
    Village Statistical Areas is an estimate of the number of occupied
    housing units in the area. For American Indian and Alaska Native
    Village Statistical Areas the MOS is the estimated number of
    occupied housing units (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 MOS of all
    entities it is a part of.

    The estimated number of occupied HUs for each Census Tract
    (TRACTMOS) is also calculated.

    These two measures, MOS and TRACTMOS are used to assigned the
    initial sampling rates as shown in Table 1 below.

    **Table 1**. **Initial Sampling Rate Categories for the United
    States and Puerto Rico**

+-----------------------------------------+-------------+-------------+
| **Sampling Rate Category**              | **Initial   |             |
|                                         | Sampling    |             |
|                                         | Rates**     |             |
+-----------------------------------------+-------------+-------------+
|                                         | **United    | **Puerto    |
|                                         | States**    | Rico**      |
+-----------------------------------------+-------------+-------------+
| Blocks in smallest governmental units   | 10.0%       | 10.0%       |
|                                         |             |             |
| (MOS\<200)                              |             |             |
+-----------------------------------------+-------------+-------------+
| Blocks in smaller governmental units    | 6.9%        | 8.1%        |
|                                         |             |             |
| (200≤ MOS\<800)                         |             |             |
+-----------------------------------------+-------------+-------------+
| Blocks in small governmental units      | 3.6%        | 4.1%        |
|                                         |             |             |
| (800≤ MOS≤1200)                         |             |             |
+-----------------------------------------+-------------+-------------+
| Blocks in large tracts                  | 1.6%        | 2.0%        |
|                                         |             |             |
| (MOS \>1200, TRACTMOS ≥ 2000) where     |             |             |
| Mailable addresses ≥ 75% and predicted  |             |             |
| levels of completed interviews prior to |             |             |
| sub-sampling \> 60%                     |             |             |
+-----------------------------------------+-------------+-------------+
| Other Blocks in large tracts            | 1.7%        |             |
|                                         |             |             |
| (MOS \>1200, TRACTMOS ≥ 2000)           |             |             |
+-----------------------------------------+-------------+-------------+
| All other blocks                        | 2.1%        | 2.7%        |
|                                         |             |             |
| (MOS \>1200, TRACTMOS \<2000)           |             |             |
|                                         |             |             |
| where Mailable addresses ≥ 75% and      |             |             |
| predicted levels of completed           |             |             |
| interviews prior to sub-sampling \> 60% |             |             |
+-----------------------------------------+-------------+-------------+
| All other blocks (MOS \>1200, TRACTMOS  | 2.3%        |             |
| \<2000)                                 |             |             |
+-----------------------------------------+-------------+-------------+

Once each block is assigned to a sampling stratum, a systematic sample
of addresses is selected from the second-stage universe within each
county, county equivalent, and municipio.

Sub-Sampling The Unmailable And Non-Responding Addresses

All addresses determined to be unmailable are sampled for the Computer
Assisted Personal Interview (CAPI) phase of data collection at a rate of
2-in-3. Unmailable addresses do not go to the Computer Assisted
Telephone Interview (CATI) phase of data collection. All other
non-responding addresses where a telephone number is obtained go to
CATI. Subsequent to CATI, all addresses for which no response has been
obtained prior to CAPI are sampled for based on the expected rate of
completed interviews at the tract level using the following sampling
rates.

**Table 2. CAPI Sub-Sampling Rates for the United States and Puerto
Rico**

+--------------------------------------------------------+-------------+
| #### Address and Tract Characteristics                 | **CAPI      |
|                                                        | S           |
|                                                        | ub-Sampling |
|                                                        | Rate**      |
+--------------------------------------------------------+-------------+
| **United States**                                      |             |
+--------------------------------------------------------+-------------+
| Unmailable addresses and addresses in Remote Alaska    | 2-in-3      |
+--------------------------------------------------------+-------------+
| Mailable addresses in tracts with predicted levels of  | 1-in-2      |
| completed interviews prior to CAPI sampling between 0% |             |
| and 35%                                                |             |
+--------------------------------------------------------+-------------+
| Mailable addresses in tracts with predicted levels of  | 2-in-5      |
| completed interviews prior to CAPI sampling greater    |             |
| than 35% and less than 51%                             |             |
+--------------------------------------------------------+-------------+
| Mailable addresses in other tracts                     | 1-in-3      |
+--------------------------------------------------------+-------------+
|                                                        |             |
+--------------------------------------------------------+-------------+
| **Puerto Rico**                                        |             |
+--------------------------------------------------------+-------------+
| Unmailable addresses                                   | 2-in-3      |
+--------------------------------------------------------+-------------+
| Mailable addresses -- June through December            | 1-in-2      |
+--------------------------------------------------------+-------------+
| Mailable addresses -- January through May              | 1-in-3      |
+--------------------------------------------------------+-------------+

Beginning in 2005, differential CAPI sub-sampling rates were used for
mailable addresses, instead of a flat rate of 1-in-3.

ESTIMATION PROCEDURE

Number range CHAPTERThe estimates that appear in this product were
obtained from a ratio estimation procedure that resulted in the
assignment of two sets of weights: a weight to each sample person record
and a weight to each sample housing unit record. For any given
tabulation area, a characteristic total was estimated by summing the
weights assigned to the persons, households, families or housing units
possessing the characteristic in the tabulation area. Estimates of
person characteristics were based on the person weight. Estimates of
family, household, and housing unit characteristics were based on the
housing unit weight.

Number range CHAPTEREach sample person or housing unit record was
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 had the value 40, all characteristics of that person or
housing unit would be tabulated with the weight of 40.

Estimation strata were formed by grouping counties of similar
demographic and social characteristics using Census 2000 data. The
characteristics considered in the stratification included;

-   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 stratum was also required to meet a threshold of 400 expected
person interviews in the 2005 ACS. The stratification process then
attempted to minimize the differences on the characteristics listed
above between the counties within a stratum. The process also tried to
preserve as many counties that met the threshold to form their own
estimation areas. In total, there were 2,006 estimation strata formed
from the 3,219 counties and county equivalents including Puerto Rico.

The estimation procedure used to assign the weights was then performed
independently within each of the ACS estimation strata.

1.  Initial Housing Unit Weighting Factors - This process produced the
    following factors:

    -   Base Weight (BW) - This initial weight was assigned to every
        housing unit is the inverse of its block's sampling rate.

    -   CAPI Subsampling Factor (SSF) - The weights of the CAPI cases
        were adjusted to reflect the results of CAPI subsampling. This
        factor was 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 were unmailable. A two-thirds sample of
        these were sent directly to CAPI and for these cases SSF = 1.5.

-   Variation in Monthly Response by Mode (VMS) - This factor made 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 was computed and assigned
    based on the following groups.

    Strata x Month

-   Noninterview Factor (NIF) - This factor adjusted the weight of all
    responding occupied housing units to account for both responding and
    nonresponding housing units. The factor was computed in two stages.
    The first factor, NIF1, is a ratio adjustment that was computed and
    assigned to occupied housings units based on the following groups.

    Strata x Building Type x Tract

    A second factor, NIF2, is a ratio adjustment that was computed and
    assigned to occupied housing units based on the following groups.

    Strata x Building Type x Month

    NIF was then computed by applying NIF1 and NIF2 for each occupied
    housing unit. Vacant housing units were assigned a value of NIF =
    1.0. Nonresponding housing units were now assigned a weight of 0.0.

-   Noninterview Factor - Mode (NIFM) - This factor adjusted the weight
    of just the responding CAPI occupied housing units to account for
    both CAPI respondents and all nonrespondents. This factor was
    computed as if NIF had not already been assigned to every occupied
    housing unit record. This factor was not used directly but rather as
    part of computing the next factor: MBF. NIFM was computed and
    assigned to occupied CAPI housing units based on the following
    groups.

    Strata x Building Type x Month

    Mail and CATI cases received a value of NIFM = 1.0. Vacant housing
    units received a value of NIFM = 1.0.

-   Mode Bias Factor (MBF) - This factor made the total weight of the
    housing units in the groups below the same as if NIFM had been used
    instead of NIF. MBF was computed and assigned to occupied housing
    units based on the following groups.

    Strata x Tenure (Owner or renter) x Month x Marital Status
    (married/widowed or single)

    Vacant housing units received a value of MBF = 1.0. MBF is applied
    to the weights computed through NIF.

    -   Number range CHAPTERHousing unit Post-stratification Factor
        (HPF1) - This factor made the total weight of all housing units
        agree with the 2005 independent housing unit estimates at the
        stratum level.

2.  Person Weighting Factors - Initially the person weight of each
    person in an occupied housing unit was the product of the weighting
    factors of their associated housing unit (BW x . . . x HPF1). At
    this point everyone in the household would have the same weight.
    These person weights were then individually adjusted based on each
    person\'s age, race, sex, and Hispanic origin as described below.

-   Person Post-Stratification Factor (PPSF) - This factor was applied
    to individuals based on their age, race, sex and Hispanic origin. It
    adjusted the person weights so that the weighted sample counts
    matched independent population estimates by age, race, sex, and
    Hispanic origin at the stratum level. Because of collapsing of
    groups in applying this factor, only total population is assured of
    agreeing with the official 2005 intercensal population estimates at
    the stratum level.

    This used the following groups:

> Number range CHAPTERStrata x Race (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)) x Sex x Age Groups.

-   Rounding - The final product of all person weights (BW x . . . x
    HPF1 x PPSF) was rounded to an integer. Rounding was performed so
    that the sum of the rounded weights was within one person of the sum
    of the unrounded weights for any of the groups listed below:

    County

    County x Race

    County x Race x Hispanic Origin

    County x Race x Hispanic Origin x Sex

    County x Race x Hispanic Origin x Sex x Age

    County x Race x Hispanic Origin x Sex x Age x Tract

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

    For example, the number of White, Hispanic, Males, Age 30 estimated
    for a county using the rounded weights was within one of the number
    produced using the unrounded weights.

3.  Number range CHAPTERFinal Housing Unit Weighting Factors - This
    process produced the following factors:

-   Principal Person Factor (PPF) - This factor adjusted for
    differential response depending on the race, Hispanic origin, sex,
    and age of the principal person in the household. The principal
    person was defined as the female spouse of the responding
    householder. If there was no such person, then the responding
    householder was the principal person. The value of PPF for a housing
    unit was the PPSF of the principal person.

-   Final Housing Unit Controls (HPF2) - The final product of the
    principal person weights (BW x . . . x HPF1 x PPF) was then assigned
    to the housing unit. The total number of weighted housing unit
    counts are then made to agree to the 2005 independent housing unit
    estimates at the stratum level.

-   Rounding - The final product of all housing unit weights (BW x . . .
    x PPF x HPF2) was rounded to an integer. Rounding was performed so
    that total rounded weight was within one housing unit of the total
    unrounded weight for any of the groups listed below:

    County

    County x Tract

    County x Tract x Block

    CONFIDENTIALITY OF THE DATA

Number range CHAPTERThe 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.

-   Number range CHAPTERTitle 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.

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-   Disclosure Limitation: Disclosure limitation is the process for
    protecting the confidentiality of data. A disclosure of data occurs
    when someone can use published statistical information to identify
    either an individual that has provided information under a pledge of
    confidentiality. For data tabulations the Census Bureau uses
    disclosure limitation 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.

-   Number range CHAPTERData Swapping: Data swapping is a method of
    disclosure limitation 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 for Census 2000.

    ERRORS IN THE DATA

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-   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 procedures provides the basis
    for the statistical analysis of sample 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 data. For example, operations such as editing, reviewing, or
    keying data from questionnaires 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
    housing units. One way of accomplishing this is by following up on
    mail nonrespondents during the CATI and CAPI phases.

    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.

Number range CHAPTERConfidence 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 1.65 times the
> estimated standard error below the estimate to 1.65 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 / 1.65

Lower Confidence Bound = Estimate - Margin of Error

Upper Confidence Bound = Estimate + Margin of Error

> 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 population estimate 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 must be kept in
> mind when creating these bounds. Another of these natural limits would
> be 100% 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 has
no sampling error. When using any of the formulas in the following
section, use a standard error of zero for these controlled estimates.

Number range CHAPTER

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

-   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. 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
    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.

    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. Exceptions
    include:

1.  The estimate of the number or proportion of people, households,
    housing units or families in a geographic area with a specific
    characteristic is zero. A special procedure is used to estimate the
    standard error.

2.  There are no sample observations available to compute an estimate of
    a median, a proportion, or some other ratio, or an estimate of its
    standard error. The estimate is represented in the tables by "-" and
    the margin of error by "\*\*" (two asterisks).

3.  Only a small number of identical values are reported and used to
    calculate a median, aggregate, mean, or per capita amount. In this
    case, there are too few sample observations to compute a stable
    estimate of the standard error. The margin of error is represented
    in the tables by "\*" (one asterisk).

4.  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
and differences between two sample estimates. The estimate of the
standard error of a sum or difference is approximately the square root
of the sum of the two individual standard errors squared; that is, for
standard errors $\text{SE}(\hat{X})$and $\text{SE}(\hat{Y})$ of
estimates $\hat{X}$ and $\hat{Y}$:

> $$\text{SE}(\hat{X} + \hat{Y}) = \text{SE}(\hat{X} - \hat{Y}) = \sqrt{\lbrack\text{SE}(\hat{X})\rbrack^{2} + \lbrack\text{SE}(\hat{Y})\rbrack^{2}}$$

Number range CHAPTERThis method, however, will underestimate
(overestimate) the standard error if the two items in a sum are highly
positively (negatively) correlated or if the two items in a difference
are highly negatively (positively) correlated.

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:

> $$
> $\text{SE}\left( \frac{\hat{X}}{\hat{Y}} \right) = \frac{1}{\hat{Y}}\sqrt{\lbrack\text{SE}(\hat{X})\rbrack^{2} + \frac{{\hat{X}}^{2}}{{\hat{Y}}^{2}}\lbrack\text{SE}(\hat{Y})\rbrack^{2}}$

Number range CHAPTERProportions/percents -- For a proportion (or
percent), a ratio where the numerator *is* a subset of the denominator,
a slightly different estimator is used. Note the difference between the
formulas for the standard error for proportions (below) and ratios
(above) - the plus sign in the previous formula has been replaced with a
minus sign. If the value under the square root sign is negative, use the
ratio standard error formula above, instead. If
$\hat{P} = \frac{\hat{X}}{\hat{Y}}$, then

> $$\text{SE}(\hat{P}) = \frac{1}{\hat{Y}}\sqrt{\lbrack\text{SE}(\hat{X})\rbrack^{2} - \frac{{\hat{X}}^{2}}{{\hat{Y}}^{2}}\lbrack\text{SE}(\hat{Y})\rbrack^{2}}$$

If $\hat{Q} = \text{100}\text{\%} \times \hat{P}$ (P is the proportion
and Q is its corresponding percent), then
$\text{SE}(\hat{Q}) = \text{100}\text{\%} \times \text{SE}(\hat{P})$.

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

> $$\text{SE}(\hat{X} \times \hat{Y}) = \sqrt{{\hat{X}}^{2} \times \lbrack\text{SE}(\hat{Y})\rbrack^{2} + {\hat{Y}}^{2} \times \lbrack\text{SE}(\hat{X})\rbrack^{2}}$$

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 (Est~1~ and Est~2~) and their
respective standard errors (SE~1~ and SE~2~), calculate

$$Z = \frac{\text{Est}_{1} - \text{Est}_{2}}{\sqrt{\left( \text{SE}_{1} \right)^{2} + \left( \text{SE}_{2} \right)^{2}}}$$

If Z \> 1.65 or Z \< -1.65, then the difference can be said to be
statistically significant at the 90% confidence level. 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, Census 2000
100% counts and long form estimates, estimates from other Census Bureau
surveys, and estimates from other sources. Not all estimates have
sampling error -- Census 2000 100% counts do not, for example, although
Census 2000 long form estimates do -- but they should be used if they
are available 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 to determine statistical
significance, because there are circumstances where that method will not
give the correct test result. The Z calculation above is recommended in
all cases.

All statistical testing in ACS data products is based on the 90%
confidence level. Users should understand that all testing is 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.

Number range CHAPTEREXAMPLES OF STANDARD ERROR CALCULATIONS

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

Example 1 - Calculating the Standard Error from the Confidence Interval

> The estimated number of males, never married is 34,171,130 from
> summary table B12001 for the United States for 2004. The margin of
> error is 81,645.

Standard Error = Margin of Error / 1.65

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

SE(34,171,130) = 81,645 / 1.65 = 49,482.

Example 2 - Calculating the Standard Error of a Sum

> 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
> 34,171,130. From summary table B12001 we have the number of females,
> never married is 29,943,646 with a margin of error of 74,944. So, the
> estimated number of people who have never been married is 34,171,130 +
> 29,943,646 = 64,114,776. To calculate the 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 49,482. The standard error for the number of females never
> married is calculated using the margin of error:

SE(29,943,646) = 74,944 / 1.65 = 45,421.

So using the formula for the standard error of a sum or difference we
have:

SE(64,114,776) =
$\sqrt{\text{49},\text{482}^{2} + \text{45},\text{421}^{2}}$= 67,168

> Caution: This method, however, will underestimate (overestimate) the
> standard error if the two items in a sum are highly positively
> (negatively) correlated or if the two items in a difference are highly
> negatively (positively) correlated.
>
> To calculate the lower and upper bounds of the 90 percent confidence
> interval around 64,114,776 using the standard error, simply multiply
> 67,168 by 1.65, then add and subtract the product from 64,114,776.
> Thus the 90 percent confidence interval for this estimate is
> \[64,114,776 - 1.65(67,168)\] to \[64,114,776 + 1.65(67,168)\] or
> 64,003,949 to 64,225,603.

Example 3 - Calculating the Standard Error of a 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 29,943,646 and the number of
> people who have never been married is 64,114,776 To calculate the
> 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
> females never married from example 2 as 49,482 and the standard error
> for the number of people never married calculated from example 2 as
> 67,168.

The estimate is (29,943,646 / 64,114,776) \* 100% = 46.7%

So, using the formula for the standard error of a proportion or percent,
we have:

SE(46.7%) = 100% \*
$\left( \frac{1}{\text{64},\text{114},\text{776}}\sqrt{\text{49},\text{482}^{2} - 0\text{.}\text{467}^{2} \times \text{67},\text{168}^{2}} \right)$
= 0.06%

> To calculate the lower and upper bounds of the 90 percent confidence
> interval around 46.7 using the standard error, simply multiply 0.06 by
> 1.65, then add and subtract the product from 46.7. Thus the 90 percent
> confidence interval for this estimate is

\[46.7 - 1.65(0.06)\] to \[46.7 + 1.65(0.06)\], or 46.6% to 46.8%.

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.

-   Undercoverage \-- It is possible for some sample housing units or
    persons to be missed entirely by the survey. The undercoverage of
    persons and housing units can introduce biases into the data. A
    major way to avoid undercoverage 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 was the Master Address File (MAF). The MAF
    is created by combining the Delivery Sequence File of the United
    States Postal Service, and the address list for Census 2000. An
    attempt is made to assign all appropriate geographic codes to each
    MAF address via an automated procedure using the Census Bureau TIGER
    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 personal visit CAPI assignments.

    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. In some
    multi-unit structures the interviewer could not locate the exact
    sample unit or found a different number of units than expected. In
    these cases the interviewers were instructed to list the units in
    the building and follow a specific procedure to select a replacement
    sample unit.

-   Respondent and Interviewer Error \-- The person answering the
    questionnaire or responding to the questions posed by an interviewer
    could serve as a source of error, although the questions were
    phrased as clearly as possible based on testing, and detailed
    instructions for completing the questionnaire were provided to each
    household. In addition, respondents\' answers were edited for
    completeness, and problems were followed up as necessary.

    -   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.

    -   Item Nonresponse \-- Nonresponse to particular questions on the
        survey questionnaire and instrument allows for the introduction
        of 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 biases is
        afforded by minimizing nonresponse. In the ACS, nonresponse for
        the CATI and CAPI operations was reduced substantially by the
        requirement that the automated instrument receive a response to
        each question before the next one could be asked. For mail
        responses, the automated clerical review and follow-up
        operations were aimed at obtaining a response for every question
        on selected questionnaires. Values for any items that remain
        unanswered were imputed by computer using reported data for a
        person or housing unit with similar characteristics.

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-   Automated Clerical Review \-- 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, as well as questionnaires with too many omissions to be
    accepted as returned.

-   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, and follow-up by telephone; the manual coding of
    write-in responses; and the electronic data processing. The various
    field, coding and computer operations undergo a number of quality
    control checks to insure their accurate application.

-   Automated Editing \-- After data collection was completed, any
    remaining incomplete or inconsistent information was imputed during
    the final automated 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 lacking 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 assign an entry for a person or housing unit that was
    consistent with entries for persons or housing units with similar
    characteristics. Assigning acceptable values in place of blanks or
    unacceptable entries enhances the usefulness of the data.