Number 6—Revised
March 1995
Revised to clarify computations in tables III and V

From the CENTERS FOR DISEASE CONTROL AND PREVENTION/National Center for Health Statistics

Direct Standardization
(Age-Adjusted Death Rates)
Lester R. Curtin, Ph.D. and Richard J. Klein, M.P.H.

Introduction
Most population-based mortality objectives and
subobjectives in Healthy People 2000 are tracked using
age-adjusted rates from the National Vital Statistics System
(appendix table I). The exceptions are deaths from
alcohol-related motor vehicle crashes, all motor vehicle
crashes, and work-related injuries (objectives 4.1, 9.3, and
10.1), which are monitored with crude death rates from other
data systems. In addition, objectives that refer to specific age
groups are tracked with age-specific rather than age-adjusted
rates.
Although the age-adjusted death rate (ADR) is one of
the most frequently used indexes of mortality, there is often
confusion concerning the basic concepts of its construction,
use, and interpretation. Some of the persistent issues include
the appropriateness of the ADR as a summary measure, the
validity of comparisons between ADRs, the method of
calculation, and the appropriateness of alternate summary
measures.

Why use age-adjusted death rates?
The total number of health events (for example, the
number of deaths) occurring in a population is useful for
determining the magnitude of a public health problem.
However, the absolute number of deaths is seldom useful for
comparisons between population groups (for example,
comparing males and females) or for comparing trends.
Assuming equal risk, a larger population group will tend to

generate more events (deaths) than a smaller group simply
because of its size. Therefore, to compare relative differences
in mortality among population groups, or for a given
population group over time, the number of deaths must be
related to the ‘‘population at risk’’ of dying to produce death
rates. The population of interest may be the entire population
of an area or a population subgroup (for example, people in
a certain age group).
The simplest death rate is the crude death rate (CDR),
defined as the total number of deaths divided by the midyear
population. CDRs are usually expressed as a rate per 1,000
or 100,000 population. CDRs for individual age cohorts,
called age-specific death rates (ASDRs), are the ratio of the
number of deaths in a given age group to the population of
that age group, again usually expressed per 1,000 or 100,000
population.
To compare the relative health of population groups or
to assess change in mortality over time, two criteria must be
considered. First, rates should relate the number of events to
the population at risk. Second, because many health
outcomes vary by age, the effect of the population’s age
distribution must be taken into account.
Although it does relate the number of events to the
population, the crude rate does not take into account the age
distribution of the population. As such, it is not an
appropriate measure for comparing differences between
population groups or for assessing change in mortality over
time. Because death rates for most diseases generally
increase with age, a population group with a relatively

U.S. DEPARTMENT OF HEALTH AND HUMAN SERVICES
Public Health Service
Centers for Disease Control and Prevention
National Center for Health Statistics

CENTERS FOR DISEASE CONTROL
AND PREVENTION

Table A. Crude death rate comparison
Community A

Community B

Age

Deaths

Population

Rate1

Deaths

Population

Rate1

0–34 years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35–64 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
120
360

1,000
3,000
6,000

20
40
60

180
150
70

6,000
3,000
1,000

30
50
70

Total. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

500

10,000

50

400

10,000

40

1Per 1,000 population.

Table B. Age-adjusted death rate calculation
Community A

Community B

Age

Standard
population

Rate1

Rate ×
population

Rate

Rate ×
population

0–34 years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35–64 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3,000
3,000
4,000

20
40
60

60
120
240

30
50
70

90
150
280

Total. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10,000

42

420

52

520

1Per 1,000 population.

younger age distribution will tend to have fewer total deaths
from a given disease than a comparably sized population
group with an older age distribution. Similarly, even if the
age-specific risks of dying for a group remain unchanged
between two time points, the number of deaths will increase
as the population ages.
As an alternative to crude rates, ASDRs can be used.
The most comprehensive and reliable method of comparing
death rates over time or between different population groups
is to compare individual ASDRs for all age groups of
interest. However, this method often requires an extremely
large number of comparisons and tends to overwhelm both
the investigator and the intended audience.
Because the crude death rate is not appropriate and
ASDRs provide too much detailed information, a summary
measure that controls for a population’s age distribution is
needed. A commonly used measure is the ADR (1,2).
Age-adjusted rates were developed in 1841 for the
analysis of mortality data (3). In the 19th century, mortality
data provided the most useful, and often the only, measure
of the health of a population. About that time, it was
observed that a community could have ASDRs that were
lower than the national average at each age interval, but,
because the community’s population was older, the overall
CDR was higher for the community than for the Nation.
Table A presents a hypothetical comparison to illustrate
this situation.
In each of the two comparably-sized communities in
table A, the ASDRs increase with age; at each age the
age-specific rates are higher for community B than for
community A. Yet, the total (or crude) death rate is lower for
community B. This occurs because community A is an older
population; 60 percent of its citizens are 65 years and over.

In contrast, only 10 percent of community B’s population is
in the oldest age group. Because the death rate is highest in
the oldest age group, there are fewer total deaths in
community B.

Direct standardization
There are two basic methods of standardization, or
age-adjustment; both were introduced in the 19th century.
These two methods have become known as the direct and
indirect methods. (Indirect standardization is discussed in a
later section.) When the direct standardization method is
applied to ASDRs, the resultant summary index is called the
ADR. Two assumptions are made when this index is
computed for a population: The population’s observed
age-specific rates are assumed to be valid, and the age
distribution of the population is assumed to be that of a
standard, or reference, or population.
Table B illustrates the calculation of the ADR using the
hypothetical data from table A. Specific computational
formulae for the ADR are given in the technical appendix.
To calculate the ADR, the standard population and the
age-specific death rate for each age interval are multiplied
and these products are summed. In this example, the total
for community A is 420. This sum is divided by the total
standard population (10,000 in this case) to obtain the ADR.
As with crude rates, the ADR is usually expressed in terms
of a rate per 1,000 or per 100,000 population. Thus, the
ADR for community A is 42 deaths per 1,000 population
and the ADR for community B is 52 per 1,000. Note that,
although the crude rate for community A was larger than that
for community B, the ADR for community A is smaller than
the ADR for community B. This is consistent with each of
2

‘‘When not to adjust.’’) Although the magnitude of the
ADRs may be greatly affected by the choice of a standard
population, relative mortality, as measured by trends, race
ratios, and sex ratios, is generally unaffected (8).
Despite this, controversy continues over which standard
population to use when age adjusting death rates to measure
temporal changes in cause-specific mortality. Examination of
the issue shows that standard populations generally yield
only a small effect on trend comparisons by cause of death
(9). Thus, any standard population is adequate so long as
comparison populations are not very ‘‘unusual’’ or
‘‘abnormal’’ with respect to the population under study (10).
This means that the age distribution of the standard
population should be somewhat similar to the population of
interest.

the age-specific rates for community A being smaller than
those of community B.
Because of the method of computation, the age-adjusted
rate is often interpreted as the hypothetical death rate that
would have occurred if the observed age-specific rates were
present in a population whose age distribution is that of the
standard population. It is very important to realize that the
ADR is an artificial measure whose absolute value has no
intrinsic meaning. The ADR is useful for comparison
purposes only, not to measure absolute magnitude. (To
compare absolute magnitude, crude rates are used.) It is also
important to note that in order to compare two age-adjusted
rates, the same standard population must have been used.

Selection of a standard population
After the decision to use an ADR is made, the standard
population must be selected. There are two basic types of
standard populations, internal and external. Internal standards
are created from the data to be used in the analysis; for
example, the average age distribution of all populations to be
compared. The use of an internal standard has certain
statistical advantages for the ADR (4). However, if an
internal standard is used, the results cannot be directly
compared to other studies that use adjusted rates computed
using a different standard population.
External standards are standard populations drawn from
sources outside the analysis. For example, the National
Center for Health Statistics (NCHS) typically uses a standard
based on the 1940 United States population (5). This U.S.
standard population is usually given in terms of a ‘‘standard
million’’ in 10-year age groups. The U.S. standard million
population is presented in appendix table I. A specific
example of age-adjustment for one of the Healthy People
2000 mortality objectives using the standard million and
10-year ASDR is given in appendix table II. The calculation
of the variances of the age-adjusted rates in table II are
shown in appendix table III.
NCHS publishes a large number of ADRs based on the
U.S. standard population. This standard is also used to track
those mortality objectives in Healthy People 2000 and those
Health Status Indicators which are monitored with
age-adjusted rates. Several States use the same standard
population in their publications. As long as the identical
standard is used, ADRs from various national and State
publications can be compared. But, if different standard
populations are used to compute the ADR, then these ADRs
are not comparable. Thus, there are considerable advantages
to using the U.S. standard when computing State and local
ADRs.
In recent years there have been discussions about
whether the 1940 standard should be supplanted by a more
contemporary ‘‘standard’’ that more closely reflects the U.S.
population’s current (or future) age distribution (6,7). In
considering this issue, it is important to remember that the
actual magnitude of the ADR is beside the point. The ADR
is an index number used for relative comparisons that should
not be affected by the choice of a standard. (See section on

Small number issues
One problem with ADR is that rates based on small
numbers of deaths will exhibit a large amount of random
variation. (See the technical appendix for more detail.)
Therefore, if the number of deaths is small, mortality data
should be aggregated over a number of years, or several
small geographic areas must be combined into larger areas
before computing the ADR (11). A very rough guideline is
that there should be at least 25 total deaths over all age
groups.

When not to adjust
The general consensus of the scientific literature is that,
if it is appropriate to standardize, then the selection of the
standard population should not affect relative comparisons.
However, standardization is not appropriate when agespecific death rates in the populations being compared do
not have a consistent relationship (12).
For example, evaluating trends in age-adjusted cancer
death rates over time can be difficult because the ASDRs for
younger ages have been decreasing while death rates at older
ages are increasing. If a relatively young standard population
is used, the trend in ADR may show a small increase or
even a decrease; if a relatively older standard population is
used, cancer mortality shows a much larger increase. Thus,
using a more current (i.e., older) population than the 1940
standard, such as the 1990 U.S. population, as a standard
population yields a much greater increase in the cancer
mortality trend curve than does an analysis using the 1940
standard. Under these circumstances, a single summary
measure is likely to be inappropriate for describing trends
over time. Here, one should not use ADR, but should look at
trends among ASDRs.
This does not mean it is inappropriate to publish ADR
for cancer mortality. Within a defined time interval, e.g.,
1990, geographic or race-sex comparisons may still be
appropriate. It can be noted that when age-adjusted rates are
computed using two distinct standards and the comparisons
are different, then it is not appropriate to standardize in the
first place. Again, only age-specific comparisons may be
3

Summary

valid. Kitagawa illustrates this situation with an example of
the mortality of white males living in metropolitan counties
compared with those residing in nonmetropolitan counties in
1960 (13). In this case, ASDRs for white males under age 40
were lower in metropolitan counties than in nonmetropolitan
counties. After age 40, the reverse was true. A summary
index, such as the ADR, does not adequately describe the
mortality differentials in the two groups. In cases such as
these, the ADR is an imprecise indicator of mortality; the
age-specific comparisons would be a better choice.

This paper describes some of the issues related to the
computation and use of age-adjusted rates. The following
points were made:
+ The age-adjusted rate is an index measure, the magnitude
of which has no intrinsic value. It should be used for
comparison purposes only.
+ If it is appropriate to use age-adjustment, then the
comparison should not be affected by the selection of a
standard population. Conversely, if the comparison can be
affected by the choice of a standard population, then it is
not appropriate to age-adjust for that comparison.
+ The standard population should not be ‘‘abnormal’’ or
‘‘unnatural’’ when compared to populations under study.
Considering the amount of published material, there are
advantages to using the U.S. standard population.
+ Standardization is not a substitute for the examination of
age-specific rates.

Indirect standardization
Because of concerns with the use of ADR, some
mortality analysts prefer indirect standardized rates. Indirect
standardization is generally thought of as an approximation
to direct standardization. That is, when data needed to
compute a direct measure (e.g., ASDRs) are not available,
there may still be enough information to compute an
indirectly standardized measure. However, indirect
standardization has intrinsic value and should be considered
on its own merits, not solely as an approximation to direct
standardization (14,15).
For indirect standardization, a standard set of
age-specific death rates are assumed to apply to the observed
population. For example, the age-specific U.S. death rates
could be applied to the age-specific local area population.
This technique yields an ‘‘expected’’ number of deaths in a
population, assuming the standard set of ASDRs was
operating in the population.
An indirect adjusted death rate (IADR) can be computed
from the expected number of deaths, but the index most
often used is the ratio of the expected to the actual observed
number of deaths. This ratio is called the standardized
mortality ratio (SMR). The mathematics of indirect
standardization and an example of the calculation of an
SMR are given in the appendix.

While standardization is most often applied to a series
of age-specific death rates, direct or indirect standardization
can also be applied to variables other than age. For example,
infant mortality rates can be adjusted for birthweight
distribution (16). Age-adjustment can also be used to
monitor other measures of health at the local level, such as
incidence or prevalence of disease.
Throughout the history of the ADR, the utility of the
measure has often come into question. Any summary index,
including direct or indirect standardization, will mask
age-specific differences. Therefore, some authors have
stressed the importance of comparing individual age-specific
rates rather than attempting to summarize differences among
the age-specific rates (17,18). A summary index, however, is
more easily compared than an entire table of age-specific
rates. Thus, the age-adjusted rate continues to be an integral
part of the analysis of mortality trends and differentials.
Accepting this, the need for a summary index must be
balanced with recognition of the limitations of summary
measures.

4

References

11. Kleinman JC. Age-adjusted mortality indices for small areas:
Applications to health planning. American Journal of Public
Health, 67:834–40. 1977.
12. Fleiss JL. Statistical methods for rates and proportions. John
Wiley and Sons, New York. 1973.
13. Kitagawa EM. Theoretical considerations in the selection of a
mortality index and some empirical comparison. Human Biology,
38:293–308. 1966.
14. Inskip H, Beral V, Fraser P. Methods for age-adjustment of rates.
Statistics in Medicine 2:455–66. 1983.
15. Tukey JW. Statistical mapping: What should and should not be
plotted. Proceedings of the 1976 workshop on automated cartography and epidemiology. National Center for Health Statistics.
DHEW (PHS) 79–1254. 1979.
16. Foster JE, Kleinman JC. Adjusting neonatal mortality rates for
birthweight. National Center for Health Statistics. Vital Health
Stat 2(94). 1982.
17. Woosley TD. Adjusted death rates and other indices of mortality.
Chapter 4 in Vital Statistics Rates in the United States,
1900–1940.Washington: U.S. Government Printing Office. 1959.
18. Elveback LR. Discussion of indexes of mortality and tests of
their statistical significance. Human Biology 38:322–24. 1966.
19. Chiang CL. Standard error of the age-adjusted death rate. U.S.
Department of Health, Education, and Welfare: Vital Statistics
Special Reports 47:271–85. 1961.
20. Keyfitz N. Sampling variance of standardized mortality rates.
Human Biology 38:309–17. 1966.

1. Shyrock HS, Siegel JS. The methods and materials of demography, vol 2. U.S. Bureau of the Census. Washington: U.S.
Government Printing Office. 1971.
2. Spiegelman M. Introduction to demography. Rev. ed. Cambridge, MA. Harvard University Press. 1968.
3. Neison FGP. On a method recently proposed for conducting
inquiries into the comparative sanatory condition of various
districts. Journal of the Royal Statistical Society of London (now
the Royal Statistical Society), vol 7, pp 40–68. 1844.
4. Kalton G. Standardization: A technique to control for extraneous
variables. Applied Statistics, 17:118–36. 1968.
5. National Center for Health Statistics. Vital Statistics of the
United States, 1989, vol II, mortality, part A. Washington: Public
Health Service. 1992.
6. Johnson R. Proposed new standard population. Proceedings of
the social statistics section, American Statistical Association, pp
176–81. 1990.
7. Feinleib MF, Zarate AO, eds. Reconsidering age adjustment
procedures: Workshop proceedings. National Center for Health
Statistics. Vital Health Stat 4(29). 1992.
8. Spiegelman M, Marks HH. Empirical testing of standards for the
age adjustment of death rates by the direct method. Human
Biology, 38:280–92. 1966.
9. Curtin LR, Maurer J, Rosenberg HM. On the selection of
alternative standards for the age-adjusted death rate: Proceedings of the social statistics section, American Statistical Association, pp 218–23. 1980.
10. Wolfenden HH. On the theoretical and practical considerations
underlying the direct and indirect standardization of death rates.
Population Studies, 16:188–90. 1962.

5

Appendix
Table I. Healthy People 2000 mortality objectives
Objective
number

Cause of death

1.1

Coronary heart disease

1.1a

[Blacks]

2.1
2.1a

See 1.1
See 1.1a

2.2

Cancer (all sites)

ICD–9
identifying codes

Objective
number

Cause of death

410–414, 402,
429.2

9.3
9.3a

Motor vehicle crashes
[Ages 14 and younger]

9.3b
9.3c
9.3d

[Ages 15–24]
[Ages 70 and older]
[American Indians/Alaska Natives]

9.3e
9.3f

[Motorcyclists]
[Pedestrians]

9.4
9.4a
9.4b

Falls and fall-related injuries
[Ages 65–84]
[Ages 85+]

9.4c

[Black males 30–69]

9.5

Drowning

9.5a
9.5b
9.5c

[Ages 0–4]
[Males 15–34]
[Black males]

9.6
9.6a
9.6b
9.6c
9.6d

Residential fires
[Ages 0–4]
[Ages 65 and older]
[Black males]
[Black females]

E890–E899

10.1
10.1a
10.1b
10.1c
10.1d

Work-related injuries1
[Mine workers]
[Construction workers]
[Transportation workers]
[Farm workers]

E800–E999

13.7

Cancer of the oral cavity and pharynx

140–149

14.3
14.3a

Maternal mortality
[Blacks]

630–676

15.1
15.1a
15.2

See 1.1
See 1.1a
Stroke

15.2a

[Blacks]

16.1

See 2.2

16.2
16.3
16.4

See 3.2
Breast cancer in women
Cancer of the uterine cervix

174
180

16.5

Colorectal cancer

153.0–154.3,
154.8,159.0

17.9
17.9a

Diabetes-related deaths1
[Blacks]

17.9b

[American Indians/Alaska Natives]
Epidemic-related pneumonia and
influenza deaths for ages 65+

140–208

3.1

See 1.1

3.1a
3.2
3.3

See 1.1a
Lung cancer
Chronic obstructive pulmonary disease

162.2–162.9
490–496

4.1

Alcohol-related motor vehicle crashes

E810–E819

4.1a
4.1b
4.2
4.2a
4.2b
4.3

[American Indians/Alaska Natives]
[Ages 15–24]
Cirrhosis
[Black males]
[American Indians/Alaska Natives]
Drug-related deaths

6.1
6.1a
6.1b
6.1c
6.1d

Suicides
[Ages 15–19]
[Males 20–34]
[White males 65 and older]
[American Indian/Alaska Native males]

E950–E959

7.1
7.1a
7.1b
7.1c
7.1d
7.1e
7.1f

Homicides
[Children 0–3]
[Spouses 15–34]
[Black males 15–34]
[Hispanic males 15–34]
[Black females 15–34]
[American Indians/Alaska Natives]

E960–E969

7.2

See 6.1

7.2a
7.2b
7.2c
7.2d

See
See
See
See

7.3

571

292, 304,
305.2–305.9,
E850–E858,
E950.0–E950.5,
E962.0,
E980.0–E980.5

6.1a
6.1b
6.1c
6.1d

Firearm injuries

E922.0–E922.3,
E922.8–E922.9,
E955.0–E955.4,
E965.0–E965.4,
E970,
E985.0–E985.4

Knife injuries

E920.3, E956,
E966, E986, E974

20.2

9.1
9.1a
9.1b

Unintentional injuries
[American Indians/Alaska Natives]
[Black males]

E800–E949

1Healthy People 2000 uses multiple cause-of-death data.

9.1c

[White males]

6

ICD–9
identifying codes
E810–E825

E880–E888

E830, E832, E910

430–438

250

480–487

are based on the 1940 U.S. population and are called the
‘‘standard million.’’ As the name implies, the standard
million weights sum to one million. The standard million is
shown in table II.
The age-adjusted rates shown in most NCHS
publications and those used to track the Healthy People 2000
objectives are computed using the standard million and
ASDRs in 10-year age groups. A specific calculation for
stroke mortality (Healthy People 2000 objective 15.2) for
males and females is shown in table III. For illustrative
purposes, the deaths and populations are those of a
hypothetical medium-sized State.

This appendix presents examples of the computation of
the age-adjusted death rate (ADR), indirect adjusted death
rate (IADR), and the standard mortality ratio (SMR). These
examples demonstrate that each standardized index is a
weighted average of the age-specific rates. For the ADR, the
weights are determined by the standard population. A
discussion of the variability of the ADR is also included.
Suppose the data are aggregated into i = 1, 2, ..., I age
groups. Let:
di = the number of deaths in the i-th age interval, and
pi = the population size in the i-th age interval.
The total number of deaths is
d =

∑i

di

p =

∑i

pi

In this example, di = deaths in 10-year age-groups,
m i = 10-year ASDR per 100,000, and wsi = psi / Σi psi
(weights on a unit basis).

the total population is

Age specific death rates (ASDRs) are defined as
ASDR =

Indirect standardization

number of deaths for age interval i
midyear population for age interval i

For direct standardization, the observed ASDRs and a
standard population are used. For indirect standardization,
the observed population and a standard set of ASDRs are
used. Indirect standardized rates are sometimes calculated
and presented, but more often an SMR is presented. The
indirect standardized rate and SMR are defined as follows:

thus,
ASDR = mi = the death rate in the i-th age interval.

The age-specific death rate is given by
m i = di / p i

SMR =

In this form, the death rate (m) on a unit basis (i.e., per
person) will be between 0 and 1. ASDRs are usually
expressed as a rate per 1,000 or per 100,000 population. For
example, if there are 10 deaths in an age group that has a
total population of 1,000 persons, the ASDR on a unit basis
is 0.01; per 1,000 it is 10, per 100,000 it is 1,000.
The annual crude death rate is defined as the total
number of deaths over all ages divided by the midyear
population. The crude death rate is then

or
SMR =

IADR = SMR * (crude rate for the standard population)

or

Again, it is usually expressed per 1,000 or per 100,000
population.
Algebraically, the direct standardized (or age-adjusted)
rate is a weighted average of the age-specific death rates. To
compute the ADR, the standard population is used to determine a set of weights. For convenience, let
psi = population in age group i in the standard population
and let the standard weights be given by

IADR =

SMR =

400
= 1.33
300

IADR = 1.33 * 50 = 67

Each index has advantages and disadvantages. Indirect rates
can be used when age-specific numbers of deaths are not
available or when the number of deaths is small. Also, the
indirect standardized rates have smaller variability. However,
indirect rates may not be comparable across areas; they can be
used for comparisons of areas only if age and area effects are
independent.

Then the ADR is given by

∑i

∑i msi * pi

Then

psi

[NOTE: That in this form 0<wsi<1 and the wsi sum to 1. The
weights are often expressed as a standard million so that wsi
sum to 1,000,000.]

ADR =

Ms * ∑i di

For the data in table A, the age-specific rates for community A
can be used as the standard rates. Then the crude rate and the
indirect standardized rates are the same for community A (50
per 1,000); the SMR for community A is 1. The calculation for
the IADR and SMR for community B is shown in table IV.

psi

∑i

∑i di
∑i msi * pi

where msi are the standard ASDRs on a unit basis. The indirect
adjusted death rate is then

m = total deaths / total population

wsi =

number of observed deaths
number of expected deaths

wsi * mi

The ASDRs used by NCHS to compute the ADR are
rounded to one decimal place. The weights used by NCHS
7

Variability

(27.1, 28.7)

Because the 95-percent confidence intervals do not overlap,
the difference between the ADRs for males and females is
statistically significant at the 0.05 level.
Some care has to be exercised when both the rates are low
and the number of deaths is small. In this case, the above
formula can result in the lower bound being less than zero, and
death rates cannot be negative. One way to avoid this is to use
log transformations of the rates; another way is to use a
discrete distribution function. These methods are beyond the
scope of this report. The NCHS Office of Research and
Methodology can provide assistance on computing variances
for ADRs based on small frequencies.
When ASDRs are based on sufficiently small numbers, a
simple Poisson approximation may be used to compute the
variance of the ASDRs, as follows:

The numbers of deaths reported for a community
represent complete counts. As such, numbers of deaths and
death rates are not subject to sampling error, although they
are subject to errors in the registration process. However,
when used for analytic purposes, such as comparison of rates
over time or for different areas, the number of events that
actually occurred may be considered as one of a large series
of possible results that could have arisen under the same
circumstances. The probable range of values may be
estimated from the actual figures according to certain
statistical assumptions. From these assumptions the standard
errors of ASDRs and ADRs can be calculated (19,20).
The variance of an ASDR is assumed to be determined
by a binomial distribution. This assumes that the chance of
dying in an age interval is constant within the age interval
and that everyone has the same chance of dying; this is an
assumption of homogeneity. Under the homogeneity
assumption, the variance of an age-specific rate on a unit
basis is given by
Variance (mi ) =

m2i
di
where mi is the ASDR on a unit basis for the i-th age group,
and di is the corresponding number of deaths

mi * (1 – mi )

NOTE: If the rates are per 100,000, then the (1–mi) term
becomes (100,000–mi).

In these cases, the resultant Poisson age-specific variances can be
used in the formulae described in this section to compute the
variance of the ADR. More information on random variation
can be found in the annual vital statistics volumes (5).

The variance of an ADR can be defined as weighted average
of the variances of the ASDRs. Under the assumption that
ASDR are independent, or that the covariance (mi , mj) = 0 for
i not equal to j, then the standardized rates are simply
weighted averages of the age-specific rates, and the variance is
given by

Table II. Standard million age distribution used
to adjust death rates to the U.S. population
in 1940

Variance (ADR) =

pi

∑i w2i * Variance (mi )

An example of the calculation of the variances of the ageadjusted rates for males and females computed in table III is
shown in table V. In order to obtain meaningful variances, the
number of deaths and the populations for males and females
used in table V are those of the same hypothetical mediumsized State used in table III. The variance of an age-adjusted
death rate for the entire U.S. population is extremely small.
Confidence intervals can be formed using the variances. If
the number of deaths is large enough (again, a rough principle
is 25 or more) then a 95-percent confidence interval for the
age-adjusted rate is formed as:
(ADR – 1.96 * √var (mi) , ADR + 1.96 * √var (mi) )

For the example in table III, the 95-percent confidence interval
for the ADR for strokes for males for the hypothetical State is
[33.0 – (1.96 * 1.05) , 33.0 + (1.96 * 1.05)]

or
(30.9, 35.1)

For females, the 95-percent confidence interval is
[27.9 – (1.96 * 0.80) , 27.9 + (1.96 * 0.80)]

or
8

Age

Standard
million
(psi )

Unit basis
(psi / 1,000,000)

All ages . . . . . . . . . . . . .

1,000,000

1.00000

Under 1 year
1–4 years . .
5–14 years. .
15–24 years .

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.
.

15,343
64,718
170,355
181,677

0.015343
0.064718
0.170355
0.181677

25–34
35–44
45–54
55–64

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162,066
139,237
117,811
80,294

0.162066
0.139237
0.117811
0.080294

65–74 years . . . . . . . . . . .
75–84 years . . . . . . . . . . .

48,426
17,303

0.048426
0.017303

85 years and over . . . . . . .

2,770

0.002770

years .
years .
years .
years .

Revised Table III
Table III. Age-adjusted death rate calculation for stroke (ICD-9 430–438) for males and females: Hypothetical
medium-sized State
Age

di

pi
(thousands)

mi
(per 100,000)

wsi
(unit basis)

mi * wsi

Males
Under 1 year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

38

2.6

0.015343

0.0398918

1–4 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5–14 years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15–24 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

–
1
2

150
322
344

–
0.3
0.6

0.064718
0.170355
0.181677

0.0000000
0.0511065
0.1090062

25–34 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35–44 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45–54 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8
21
46

443
379
256

1.8
5.5
18.0

0.162066
0.139237
0.117811

0.2917188
0.7658035
2.1205980

55–64 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65–74 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75–84 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103
254
371

189
136
57

54.5
186.8
650.9

0.080294
0.048426
0.017303

4.3760230
9.0459768
11.2625227

85 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212

12

1,766.7

0.002770

4.8937590

Age-adjusted rate = Sum (mi * wsi ) = 33.0
(per 100,000)
Females
Under 1 year . . .
1–4 years . . . . .
5–14 years. . . . .
15–24 years . . . .
25–34 years . . . .
35–44 years . . . .
45–54 years . . . .
55–64 years . . . .
65–74 years . . . .
75–84 years . . . .
85 years and over

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1
–
1
2
7
21
41
83
245
553
661

36
143
309
337
458
401
267
208
178
100
34

2.8
–
0.3
0.6
1.5
5.2
15.4
39.9
137.6
553.0
1,944.1

0.015343
0.064718
0.170355
0.181677
0.162066
0.139237
0.117811
0.080294
0.048426
0.017303
0.002770

0.0429604
0.0000000
0.0511065
0.1090062
0.2430990
0.7240324
1.8142894
3.2037306
6.6634176
9.5685590
5.3851570

Age-adjusted rate = Sum (mi * wsi ) = 27.8
(per 100,000)

Table IV. Calculation of SMR and indirect adjusted death rate
Age

msi
(per 1,000)

msi
(unit basis)

di

pi

msi * pi

0–34 years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35–64 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
40
60

0.02
0.04
0.06

180
150
70

6,000
3,000
1,000

120
120
60

Total. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120

0.05

400

10,000

300

9

Revised Table V
Table V. Variance calculation for the age-adjusted death rate calculation for stroke (ICD-9 430–438) for males
and females: Hypothetical medium-sized State
Age

di

mi
(per 100,000)

pi
(thousands)

wsi
(unit basis)

w2i* Var(mi )

Var (mi )

Males
Under 1 year . . . . . . . . . . . . . . . . . . . . . .

1

38

2.6

0.015343

6.84193

0.00161

1–4 years . . . . . . . . . . . . . . . . . . . . . . . .
5–14 years. . . . . . . . . . . . . . . . . . . . . . . .
15–24 years . . . . . . . . . . . . . . . . . . . . . . .

–
1
2

150
322
344

–
0.3
0.6

0.064718
0.170355
0.181677

0.00000
0.09317
0.17442

0.00000
0.00270
0.00576

25–34 years . . . . . . . . . . . . . . . . . . . . . . .
35–44 years . . . . . . . . . . . . . . . . . . . . . . .
45–54 years . . . . . . . . . . . . . . . . . . . . . . .

8
21
46

443
379
256

1.8
5.5
18.0

0.162066
0.139237
0.117811

0.40631
1.45111
7.02998

0.01067
0.02813
0.09757

55–64 years . . . . . . . . . . . . . . . . . . . . . . .
65–74 years . . . . . . . . . . . . . . . . . . . . . . .
75–84 years . . . . . . . . . . . . . . . . . . . . . . .

103
254
371

189
136
57

54.5
186.8
650.9

0.080294
0.048426
0.017303

28.82026
137.09637
1134.49700

0.18581
0.32150
0.33966

85 years and over . . . . . . . . . . . . . . . . . . .

212

12

1,766.7

0.002770

14462.39759

0.11097

w21

* Var (mi) = 1.10
Variance of age-adjusted death rate = Sum
Standard error of ADR = Square root of variance = 1.05
Females
Under 1 year . . .
1–4 years . . . . .
5–14 years. . . . .
15–24 years . . . .
25–34 years . . . .
35–44 years . . . .
45–54 years . . . .
55–64 years . . . .
65–74 years . . . .
75–84 years . . . .
85 years and over

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1
–
1
2
7
21
41
83
245
553
661

36
143
309
337
458
401
267
208
178
100
34

2.8
–
0.3
0.6
1.5
5.2
15.4
39.9
137.6
553.0
1,944.1

0.015343
0.064718
0.170355
0.181677
0.162066
0.139237
0.117811
0.080294
0.048426
0.017303
0.002770

7.77756
0.00000
0.09709
0.17804
0.32751
1.29669
5.76690
19.17504
77.19700
549.94191
5606.77868

0.00183
0.00000
0.00282
0.00588
0.00860
0.02514
0.08004
0.12362
0.18103
0.16465
0.04302

Variance of age-adjusted death rate = Sum w21 * Var (mi ) = 0.64
Standard error of ADR = Square root of variance = 0.80

10

DEPARTMENT OF
HEALTH & HUMAN SERVICES
Public Health Service
Centers for Disease Control and Prevention
National Center for Health Statistics
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