PB92-963373

The overarching mandate of the Comprehensive Environmental Response, Compensation, and Liability Act

(CERCLA) is to protect human health and the environment from current and potential threats posed by

uncontrolled releases of hazardous substances. To help meet this mandate, the U.S. Environmental Protection

Agency’s (EPA’s) Office of Emergency and Remedial Response has developed a human health risk assessment

process as part of its remedial response program. This process is described in the Risk Assessment Guidance for

Superfund: Volume I — Human Health Evaluation Manual (RAGS/HHEM). Part A of RAGS/HHEM

addresses the baseline risk assessment, and describes a general approach for estimating exposure to individuals

from hazardous substance releases at Superfund sites.

This bulletin explains the concentration term in the exposure/intake equation to remedial project managers

(RPMs), risk assessors, statisticians, and other personnel. This bulletin presents the general intake equation as

presented in RAGS/HHEM Part A, discusses basic concepts concerning the concentration term, describes

generally how to calculate the concentration term, presents examples to illustrate several important points, and

lastly, identifies where to get additional help.

THE CONCENTRATION TERM

How is the concentration term used?

RAGS/HHEM Part A presents the

Superfund risk assessment in four "steps":

(1) data collection and evaluation; (2) exposure

assessment; (3) toxicity assessment; and, (4) risk

characterization. The concentration term is

calculated for use in the exposure assessment step.

Highlight 1 presents the general equation

Superfund uses for calculating exposure, and

illustrates that the concentration term (C) is one of

several parameters needed to estimate contaminant

intake for an individual.

For Superfund assessments, the concentration

term (C) in the intake equation is an estimate of the

arithmetic average concentration for a contaminant

based on a set of site sampling results. Because of

the uncertainty associated with estimating the true

average concentration at a site, the 95 percent upper

confidence limit (UCL) of the arithmetic mean

should be used for this variable. The 95 percent

UCL provides reasonable confidence that the true

site average will not be underestimated.

Why use an average value for the concentration

term?

An estimate of average concentration is used

because:

United States Office of Solid Waste and Publication 9285.7-08I

Environmental Protection Emergency Response May 1992

Agency Washington, D.C. 20460

 Supplemental Guidance to

RAGS: Calculating the

Concentration Term

 Office of Emergency and Remedial Response Intermittent Bulletin

 Hazardous Site Evaluation Division, OS-230 Volume 1 Number 1

Supplemental Guidance to RAGS is a bulletin series on risk assessment of Superfund sites. These bulletins serve as supplements to

Risk Assessment Guidance for Superfund: Volume IHuman Health Evaluation Manual. The information presented is intended as

guidance to EPA and other government employees. It does not constitute rulemaking by the Agency, and may not be relied on to create

a substantive or procedural right enforceable by any other person. The Government may take action that is at variance with these

bulletins.
2

(1) carcinogenic and chronic noncarcinogenic

toxicity criteria1

 are based on lifetime

average exposures; and,

(2) Average concentration is most

representative of the concentration that

would be contacted at a site, over time.

For example, if you assume that an exposed

individual moves randomly across an exposure

area, then the spatially-averaged soil concentration

can be used to estimate the true average

concentration contacted over time. In this example,

the average concentration contacted over time

would equal the spatially averaged concentration

over the exposure area. While an individual may

not actually exhibit a truly random pattern of

movement across an exposure area, the assumption

of equal time spent in different parts of the area is a

simple but reasonable approach.

When should an average concentration be used?

The two types of exposure estimates now being

required for Superfund risk assessments, a

reasonable maximum exposure (RME) and an

average, should both use an average concentration.

To be protective, the overall estimate of intake

(see Highlight 1) used as a basis for action at

 

1

 When acute toxicity is of most concern, a longterm average concentration generally should not be

used for risk assessment purposes, as the focus

should be to estimate short-term, peak

concentrations.

Superfund sites should be an estimate in the highend of the intake/dose distribution. One high-end

option is the RME used in the superfund program.

The RME, which is defined as the highest exposure

that could reasonably be expected to occur for a

given exposure pathway at a site, is intended to

account for both uncertainty in the contaminant

concentration and variability in exposure

parameters (e.g., exposure frequency, averaging

time). For comparative purposes, agency guidance

(U.S. EPA, Guidance on Risk Characterization for

Risk Managers and Risk Assessors, February 26,

1992) states that an average estimate of exposure

also should be presented in risk assessments. For

decision-making purposes in the Superfund

program, however, RME is used to estimate risk.2

Why use an estimate of the arithmetic mean

rather than the geometric mean?

The choice of the arithmetic mean

concentration as the appropriate measure for

estimating exposure derives from the need to

estimate an individual’s long-term average

exposure. Most Agency health criteria are based on

the long-term average daily dose, which is simply

the sum of all daily doses divided by the total

number of days in the averaging period. This is

the definition of an arithmetic mean. The

 

2

 For additional information on RME, see

RAGS/HHEM Part A and the National Oil and

Hazardous Substances Pollution contingency plan

(NCP), 55 Federal Register 8710, March 8, 1990.

Highlight 1

GENERAL EQUATION FOR ESTIMATING EXPOSURE

TO A SITE CONTAMINANT

BW AT

CR EFD I C 1

×

× = ×

where:

I = Intake (i.e., the quantitative measure of exposure in RAGS/HHEM)

C = Contaminant Concentration

CR = Contact (Intake) Rate

EFD = Exposure Frequency and Duration

BW = Body Weight

AT = Averaging Time
3

arithmetic mean is appropriate regardless of the

pattern of daily exposures over time, or the type of

statistical distribution that might best describe the

sampling data. The geometric mean of a set of

sampling results, however, bears no logical

connection to the cumulative intake that would

result from long-term contact with the site

contaminants, and it may differ appreciably from—

and be much lower than—the arithmetic mean.

Although the geometric mean is a convenient

parameter for describing central tendencies of

lognormal distributions, it is not an appropriate

basis for estimating the concentration term used in

Superfund exposure assessments. The following

simple example may help clarify the difference

between the arithmetic and geometric mean, when

used for an exposure assessment:

Assume the daily exposure for a trespasser

subject to random exposure at a site is 1.0,

0.01, 1.0, 0.01, 1.0, 0.01, 1.0, and 0.01

units/day, over an 8-day period. Given these

values, the cumulative exposure is simply

their summation, or 4.04 units. Dividing this

by 8 days of exposure results in an arithmetic

mean of 0.505 units per day. This is the value

we would want to use in a risk assessment for

this individual, not the geometric mean of 0.1

units per day. Viewed another way,

multiplication of the geometric mean by the

number of days equals 0.8 units, considerably

lower than the known cumulative exposure of

4.04 units.

UCL AS AN ESTIMATE OF THE

AVERAGE CONCENTRATION

What is a 95 percent UCL?

The 95 percent UCL of a mean is defined as a

value that, when calculated repeatedly for randomly

drawn subsets of site data, equals or exceeds the

true mean 95 percent of the time. Although the 95

percent UCL of the mean provides a conservative

estimate of the average (or mean) concentration, it

should not be confused with a 95th percentile of site

concentration data (as shown in Highlight 2).

Why use the UCL as the average concentration?

Statistical confidence limits are the classical

tool for addressing uncertainties of a distribution

average. The 95 percent UCL of the arithmetic

mean concentration is used as the average

concentration, because it is not possible to know the

true mean. The 95 percent UCL, therefore,

accounts for uncertainties due to limited sampling

data at Superfund sites. As sampling data become

less limited at a site, uncertainties decrease, the

UCL moves closer to the true mean, and exposure

evaluations using either the mean or the UCL

produce similar results. This concept is illustrated

in Highlight 2.

Should a value other than the 95 percent UCL

be used for the concentration?

A value other than the 95 percent UCL can be

used, provided the risk assessor can document that

high coverage of the true population mean occurs

(i.e., the value equals or exceeds the true population

mean with high probability). For exposure areas

with limited amounts of data or extreme variability

in measured or modeled data, the UCL can be

greater than the highest measured or modeled

concentration. In these cases, if additional data

cannot practicably be obtained, the highest

measured or modeled value could be used as the

concentration term. Note, however, that the true

mean still may be higher than this maximum value

(i.e., the 95 percent UCL indicates a higher mean is

possible), especially if the most contaminated

portion of the site has not been sampled.

CALCULATING THE UCL

How many samples are necessary to calculate

the 95 percent UCL?

Sampling data from Superfund sites have

shown that data sets with fewer than 10 samples per

exposure area provide poor estimates of the mean

concentration (i.e., there is a large difference

between the sample mean and the 95 percent UCL),

while data sets with 10 to 20 samples per exposure

are provide somewhat better estimates of the mean,

and data sets with 20 to 30 samples provide fairly

consistent estimates of the mean (i.e., the 95

percent UCL is close to the sample mean).

Remember that, in general, the UCL approaches the

true mean as more samples are included in the

calculation.

Should the data be transformed?

EPA’s experience shows that most large or

“complete” environmental contaminant data sets
4

from soil sampling are lognormally distributed,

rather than normally distributed (see Highlights 3

and 4, for illustrations of lognormal and normal

distributions). In most cases, it is reasonable to

assume that Superfund soil sampling data are

lognormally distributed. Because transformation is

a necessary step in calculating the UCL of the

arithmetic mean for a lognormal distribution, the

data should be transformed by using the natural

logarithm function (i.e., calculate ln(x), where x is

the value from the data set). However, in cases

where there is a question about the distribution of

the data set, a statistical test should be used to

identify the best distributional assumption for the

data set. The W-test (Gilbert, 1987) is one

statistical method that can be used to determine if a

data set is consistent with a normal or lognormal

distribution. In all cases, it is valuable to plot the

data to better understand the contaminant

distribution at the site.

How do you calculate the UCL for a lognormal

distribution?

To calculate the 95 percent UCL of the

arithmetic mean for a lognormally-distributed data

set, first transform the data using the natural

logarithm function as discussed previously (i.e.,

calculate ln(x)). After transforming the data,

determine the 95 percent UCL for the data set by

completing the following four steps:

(1) Calculate the arithmetic mean of the

transformed data (which is also the log of the

geometric mean);

(2) Calculate the standard deviation of the

transformed data;

(3) Determine the H-statistic (e.g., see Gilbert,

1987); and,

(4) Calculate the UCL using the equation shown in

Highlight 5.

How do you calculate the UCL for a normal

distribution?

If a statistical test supports the assumption that the

data set is normally distributed, calculate the 95

percent UCL by completing the following four

steps:

Highlight 2

COMPARISON OF UCL AND 95th PERCENTILE

0

10

20

30

40

50

0 5 10 15 20 25 30

Conce ntration

Obs ervations

Mean

Upp e r C o nfid e n c e

Lim it (UC L)

o f the M e an

95th

P e rc e n tile

As sample size increases, the UCL of the mean moves closer to the true mean, while the 95th

percentile of the distribution remains at the upper end of the distribution.
5

Highlight 3

EXAMPLE OF A LOGNORMAL DISTRIBUTION

0

10

20

30

40

50

0 5 10 15 20 25 30

Concentration

Obs ervations

Mean

Highlight 4

EXAMPLE OF A NORMAL DISTRIBUTION

0

10

20

30

40

50

0 5 10 15 20 25 30

Conce ntration

Obs ervations

Mean
6

(1) Calculate the arithmetic mean of the

untransformed data;

(2) Calculate the standard deviation of the

untransformed data;

(3) Determine the one-tailed t-statistic (e.g., see

Gilbert, 1987); and,

(4) Calculate the UCL using the equation shown in

Highlight 6.

Use caution when applying normal distribution

calculations, if there is a possibility that heavily

contaminated portions of the site have not been

adequately sampled. In such cases, a UCL from

normal distribution calculations could fall below

the true mean, even if a limited data set at a site

appears normally distributed.

EXAMPLES

The examples show in Highlights 7 and 8

address the exposure scenario where an individual

at a Superfund site has equal opportunity to contact

soil in any sector of the contaminated area over

time. Even though the examples address only soil

exposures, the UCL approach is applicable to all

exposure pathways. Guidance and examples for

other exposure pathways will be presented in

forthcoming bulletins.

Highlight 7 presents a simple data set and

provides a stepwise demonstration of transforming

the data—assuming a lognormal distribution—and

calculating the UCL. Highlight 8 uses the same

data set to show the difference between the UCLs

that would result from assuming normal and

lognormal distribution of the data. These

Highlight 5

CALCULATING THE UCL OF THE ARITHMETIC MEAN

FOR A LOGNORMAL DISTRIBUTION

(x 0.5s sH / n 1 ) UCL e

2 + + − =

where:

UCL = upper confidence limit

e = constant (base of the natural log, equal to 2.718)

x = mean of the transformed data

s = standard deviation of the transformed data

H = H-Statistic (e.g., from table published in Gilbert, 1987)

n = number of samples

Highlight 6

CALCULATING THE UCL OF THE ARITHMETIC MEAN FOR A NORMAL DISTRIBUTION

UCL = x + t(s / n )

where:

UCL = upper confidence limit

x = mean of the untransformed data

s = standard deviation of the untransformed data

t = Student-t statistic (e.g., from table published in Gilbert, 1987)

n = number of samples
7

Highlight 7

EXAMPLE OF DATA TRANSFORMATION AND CALCULATION OF UCL

This example shows the calculation of a 95 percent UCL of the arithmetic mean concentration for

chromium in soil at a Superfund site. This example is applicable only to a scenario in which a spatially

random exposure pattern is assumed. The concentrations of chromium obtained from random sampling

in soil at this (in mg/kg) are 10, 13, 20, 36, 41, 59, 67, 110, 110, 136, 140, 160, 200, 230, and 1300.

Using these data, the following steps are taken to calculate a concentration term for the intake equation:

(1) Plot the data and inspect the graph. (You may need the help of a statistician for this part, as

well as other parts, of the calculation of the UCL.) The plot (not shown, but similar to

Highlight 3) shows a skew to the right, consistent with a lognormal distribution.

(2) Transform the data by taking the natural log of the values (i.e., determine ln(x)). For this data

set, the transformed values are: 2.30, 2.56, 3.00, 3.58, 3.71, 4.08, 4.20, 4.70, 4.70, 4.91, 4.94,

5.08, 5.30, 5.44, and 7.17.

(3) Apply the UCL equation in Highlight 5, where:

x = 4.38

s = 1.25

H = 3.163 (based on 95 percent)

n = 15

The resulting 95 percent UCL of the arithmetic mean is thus found to equal e (6.218), or 502 mg/kg.

Highlight 8

COMPARING UCLs OF THE ARITHMETIC MEAN ASSUMING DIFFERENT

DISTRIBUTIONS

In this example, the data presented in Highlight 7 are used to demonstrate the difference in the UCL

that is seen if the normal distribution approach were inappropriately applied to this data set (i.e., if, in this

example, a normal distribution is assumed).

ASSUMED DISTRIBUTION: Normal Lognormal

TEST STATISTIC: Student-t H- statistic

95 PERCENT UCL (mg/kg): 325 502
8

examples demonstrate the importance of using the

correct assumptions.

WHERE CAN I GET MORE HELP?

Additional information on Superfund’s policy

and approach to calculating the concentration term

and estimating exposures at waste sites can be

obtained in:

• U.S. EPA, Risk Assessment Guidance for

Superfund: Volume I—Human Health

Evaluation Manual (Part A), EPA/540/1-

89/002, December 1989.

• U.S. EPA, Guidance for Data Usability in

Risk Assessment, EPA/540/G-90/008,

(OSWER Directive 9285.7-05), October

1990.

• U.S. EPA, Risk Assessment Guidance for

Superfund (Part A—Baseline Risk

Assessment) Supplemental Guidance/

Standard Exposure Factors, OSWER

Directive 9285.6-03, May 1991.

Useful statistical guidance can be found in many

standard textbooks, including:

• Gilbert, R.O., Statistical Methods for

Environmental Pollution Monitoring, Van

Nostrand Reinhold, New York, New York,

1987.

Questions or comments concerning the

concentration term can be directed to:

• Toxics Integration Branch

Office of Emergency and Remedial

 Response

401 M Street, SW.

Washington, DC 20460

Phone: 202-260-9486

EPA staff can obtain additional copies of this

bulletin by calling EPA’s Superfund Document

Center at 202-260-9760. Others can obtain copies

by contacting NTIS at 703-487-4650.

NOTE

This reproduction of EPA Publication 9285.7-08I was prepared by:

Oregon Department of Environmental Quality

Waste Prevention and Management Division

811 SW Sixth Avenue

Portland, OR 97204

Additional information about the Oregon DEQ can be found on the internet at:

www.deq.state.or.us