Activities ° Specify the parameter of interest (mean, median, percentile). ° Specify the Action Level for the study. ° Develop a decision rule. decision finds that cleanup is needed, then the same decision unit or EA should be used when evaluating whether the cleanup standard has been attained. Furthermore, the size, shape, and orientation (the "sample support") of the samples used to determine that cleanup was necessary should be the same for samples used to determine whether the cleanup standard is met (though this last condition is not strictly necessary when the parameter of interest is the mean).

° Operational Considerations – The scale of the decision unit could be defined based on operational considerations, such as the need to characterize each "batch" of waste after it has been treated or the need to characterize each drum as it is being filled at the point of waste generation. As a practical matter, the scale for the decision making often is defined by the spatial boundaries – for example as defined by a container such as a drum, roll off box, truck load, etc. or the time required to fill the container.

° Other – The possibility of "hot spots" (areas of high concentration of a contaminant) may be apparent to the planning team from the history of the facility. In cases where previous knowledge (or planning team judgment) includes identification of areas that have a higher potential for contamination, a scale may be developed to specifically represent these areas.

Additional information and considerations on defining the scale of the decision making can be found in Guidance for the Data Quality Objectives Process for Hazardous Waste Site Operations EPA QA/ G 4HW (USEPA 2000a) and Guidance for the Data Quality Objectives Process EPA QA/ G 4 (USEPA 2000b).

4.5 Step 5: Develop a Decision Rule A statement must be developed that combines the parameter of interest and the Action Levels with the DQO outputs already developed. The combination of these three elements forms the decision rule and summarizes what attributes the decision maker wants to study and how the information will assist in solving the central problem. To develop the decision rule, perform the activities described in the following three sections:

4.5.1 Specify the Parameter of Interest A statistical "parameter" is a descriptive measure of a population such as the population mean, median, or a percentile (see also Section 3.2). See Table 2.

Some of the RCRA regulations specify the parameter of interest. For example, the comparable fuels sampling and analysis requirements at 40 CFR 261.38( c)( 8)( iii)( A) specify the mean as the parameter of interest, and the ground water monitoring requirements at 40 CFR 264.97 specify the parameter of interest for each statistical 4 EPA uses a narrative criteria to define most reactive wastes, and waste handlers should use their knowledge to determine if a waste is sufficiently reactive to be regulated.

40 test. Other RCRA regulations do not specify the parameter of interest, however, you can select a parameter based on what the Action Level is intended to represent. In general, if an Action Level is based on long term average health effects, the parameter of interest could be the population mean (USEPA 1992a). If the Action Level represents a value that should never (or rarely) be exceeded, then the parameter of interest could be an upper population percentile, which can serve as a reasonable approximation of the maximum value.

If the objective of the study does not involve estimation of a parameter or testing a hypothesis, then specification of a parameter is not necessary.

Table 2. Population Parameters and Their Applicability to a Decision Rule Parameter Definition Appropriate Conditions for Use Mean Average Estimate central tendency: Comparison of middle part of population to an Action Level.

Median Middle observation of the distribution; 50 th percentile; half of data are above and below May be preferred to estimate central tendency if the population contains many values that are less than the limit of quantitation. The median is not a good choice if more than 50% of the population is less than the limit of quantitation because a true median does not exist in this case. The median is not influenced by the extremes of the contaminant distribution.

Percentile Specified percent of sample that is equal to or below the given value For cases where it is necessary to demonstrate that, at most, only a small portion of a population could exceed the Action Level. Sometimes selected if the decision rule is being developed for a chemical that can cause acute health effects. Also useful when a large part of the population contains values less than the detection limit.

4.5.2 Specify the Action Level for the Study You should specify an Action Level or concentration limit that would cause the decision maker to choose between alternative actions. Examples of Action Levels follow:

° Comparable/ syngas fuel constituent specification levels specified at § 261.38 ° Land disposal restrictions concentration level treatment standards at § 268.40 and § 268.48 ° Risk based cleanup levels specified in a permit as part of a corrective action ° "Pass" or "fail" thresholds for tests for ignitability, corrosivity, reactivity 4 , and toxicity.

Also, be sure the detection or quantitation limits for the analytical methods identified in DQO Step 3 (Section 4.3) are below the Action Level, if possible. 41 Step 6: Specify Limits on Decision Errors Purpose To specify the decision maker's tolerable limits on decision error.

Activities ° Identify potential sources of variability and bias in the sampling and measurement processes (see Section 6) ° Determine the possible range on the parameter of interest. ° Choose the null hypothesis. ° Consider the consequences of making an incorrect decision. ° Specify a range of values where the consequences are minor (the "gray region") ° Specify an acceptable probability of making a decision error. If your objective is to compare "onsite" to "background" to determine if there is a statistically significant increase above background (as would be the case for monitoring releases from a land treatment unit under § 264.278), you will not need to specify an Action Level; rather, the Action Level is implicitly defined by the background concentration levels and the variability in the data. A summary of methods for determining background concentrations in soil can be found in USEPA 1995a. Methods for determining background concentrations in ground water can be found in USEPA 1989b and 1992b.

Finally, note that some studies will not require specification of a regulatory or risk based Action Level. For example, if the objective may be to identify the existence of a release, samples could be obtained to verify the presence or absence of a spill, leak, or other discharge to the environment. Identifying a potential release also could include observation of abandoned or discarded barrels, containers, and other closed receptacles containing hazardous wastes or constituents (see 61 FR No. 85, page 19442).

4.5.3 Develop a Decision Rule After you have completed the above activities, you can construct a decision rule by combining the selected population parameter and the Action Level with the scale of the decision making (from DQO Process Step 4) and the alternative action (from DQO Step 2). Decision rules are expressed as "if (criterion)..., then (action)...." A hypothetical example follows:

"If the true 95 th percentile of all possible 100 gram samples of the waste being placed in the 20 cubic yard container is less than 5.0 mg/ L TCLP lead, then the solid waste will be classified as nonhazardous waste. Otherwise, the solid waste will be classified as a RCRA hazardous waste."

Note that this is a functional decision rule based on an ideal condition (i. e., knowledge of the true concentration that equals the 95 th percentile of all possible sample analysis results). It also identifies the boundary of the study by specifying the sample unit (100 gram samples in accordance with the TCLP) and the size of the decision unit. It does not, however, specify the amount of uncertainty the decision maker is willing to accept in the estimate. You specify that in the next step.

4.6 Step 6: Specify Limits on Decision Errors Because samples represent only a portion of the population, the information available to make decisions will be incomplete; hence, decision errors sometimes will be made. Decision errors occur because decisions are made using estimates of the parameter of interest, rather than the true (and unknown) value. In fact, if you repeatedly sampled and analyzed a waste over and over in an identical manner the results would be a little different each time (see Figure 8 in Section 3). This variability 5 Statisticians sometimes refer to a Type I error as a "false positive," and a Type II error as a "false negative." The terms refer to decision errors made relative to a null hypothesis, and the terms may not necessarily have the same meaning as those used by chemists to describe analytical detection of a constituent when it is not really present (" false positive") or failure to detect a constituent when it really is present (" false negative").

6 An exception to this assumption is found in "detection monitoring" and "compliance monitoring" in which underlying media (such as soil, pore water, or ground water) at a new waste management unit are presumed "clean" until a statistically significant increase above background is demonstrated (in the case of detection monitoring) or a statistically significant increase over a fixed standard is demonstrated (in the case of compliance or assessment monitoring).

42 in the results is caused by the non homogeneity of the waste or media, slight differences in how the samples of the waste were collected and handled, variability in the analysis process, and the fact that only a small portion of the waste is usually ever sampled and tested. (See Section 6.1 for a more detailed discussion of sources of variability and bias in sampling). For example, if you conduct sampling and analysis of a solid waste and classify it as "nonhazardous" based on the results, when in fact it is a hazardous waste, you will have made a wrong decision or decision error. Alternatively, if you classify a solid waste as hazardous, when in fact it is nonhazardous, you also will have made a decision error.

There are two types of decision error. A "Type I" or "false rejection" decision error occurs if you reject the null hypothesis when it is true. (The "null hypothesis" is simply the situation presumed to be true or the "working assumption".) A "Type II" or "false acceptance" decision error occurs if you accept the null hypothesis when it is false. 5 Table 3 summarizes the four possible situations that might arise when a hypothesis is tested. The two possible true conditions correspond to the two columns of the table: the null hypothesis or "baseline assumption" is either true or the alternative is true. The two kinds of decisions are shown in the body of the table. Either you decide the baseline is true, or you decide the alternative is true. Associated with these two decisions are the two types of risk – the risk of making a Type I (false rejection) error (denoted by ) and the risk of making a Type   II (false acceptance) error (denoted by ). You can improve your chances of making correct   decisions by reducing and (which often requires more samples or a different sampling     design) and by using field sampling techniques that minimize errors related to sampling collection and handling (see also Sections 6 and 7).

Table 3. Conclusions and Consequences for a Test of Hypotheses True Condition Baseline is True Alternative is True Decision Based on Sample Data Baseline is True Correct Decision Type II (false acceptance) error (probability )    Alternative is True Type I (false rejection) error (probability )   Correct Decision For many sampling situations under RCRA, the most conservative (i. e., protective of the environment) approach is to presume that the constituent concentration in the waste or media exceeds the standard in the absence of strong evidence to the contrary. 6 For example, in 43 testing a solid waste to determine if it exhibits the TC, the null hypothesis can be stated as follows: "the concentration is equal to or greater than the TC regulatory level." The alternative hypothesis is "the concentration is less than the TC regulatory level." After completion of the sampling and analysis phase, you conduct an assessment of the data. If your estimate of the parameter of interest is less than the threshold when the true value of the parameter exceeds the threshold, you will make a decision error (a Type I error). If the estimate of the parameter of interest is greater than the threshold when the true value is less than the threshold, you also will make an error (a Type II error)  but one that has little potential adverse impacts to human health and the environment.

Note that during the planning phase and during sampling you will not know which kind of error you might make. Later, after a decision has been made, if you rejected the null hypothesis then you either made a Type I (false rejection) decision error or not; you could not have made a Type II (false acceptance) decision error. On the other hand, if you did not reject the null hypothesis, then you either made a Type II (false acceptance) error or not; you could not have made a Type I (false rejection) error. In either case, you will know which type of error you might have made and you will know the probability that the error was made.

In the RCRA program, EPA is concerned primarily with controlling errors having the most adverse consequences for human health and the environment. In the interest of protecting the environment and maintaining compliance with the regulations, there is an incentive on the part of the regulated entity to minimize the chance of a Type I decision error. The statistical methods recommended in this document emphasize controlling the Type I (false rejection) error rate and do not necessarily require specification of a Type II (false acceptance) error rate.

The question for the decision maker then becomes, what is the acceptable probability (or chance) of making a decision error? To answer this question, four activities are suggested. These activities are based on guidance found in Guidance for the Data Quality Objectives Process QA/ G 4 (USEPA 2000b) but have been tailored for more direct application to RCRA waste related studies. The Guidance for the Data Quality Objectives Process EPA QA/ G 4 also provides detailed guidance on the use of a graphical construct called a Decision Performance Curve to represent the quality of a decision process.

4.6.1 Determine the Possible Range on the Parameter of Interest Establish the possible range (maximum and minimum values) of the parameter of interest using data from a pilot study, existing data for a similar waste stream, or process knowledge (e. g., using a materials balance approach). It is desirable, but not required, to have an estimate of the standard deviation as well.

4.6.2 Identify the Decision Errors and Choose the Null Hypothesis Table 4 presents four examples of decision errors that could be made in a RCRA waste study. In the first three examples, the consequences of making a Type I error could include increased risk to human health and the environment or a potential enforcement action by a regulatory authority. The consequences of making a Type II error could include unnecessary financial and administrative resources required to manage the waste as hazardous (when, in fact, it is not) or continuing site cleanup activities when, in fact, the site is "clean." 44 Table 4. Examples of Possible Decision Errors in RCRA Waste Studies Regulatory Requirement "Null Hypothesis" (baseline condition) Possible Decision Errors Type I Error ( )   "False Rejection" Type II Error ( )   "False Acceptance"

Example 1: Under 40 CFR 261.11, conduct sampling to determine if a solid waste is a hazardous waste by the TC. The solid waste contains TC constituents at concentrations equal to or greater than their applicable regulatory levels (i. e., the solid waste is a hazardous waste). Concluding the waste is not hazardous when, in fact, it is. Deciding the waste is hazardous when, in fact, it is not.

Example 2: Under 40 CFR 268.7, conduct sampling and testing to certify that a hazardous waste has been treated so that concentrations of hazardous constituents meet the applicable LDR treatment standards. The concentration of the hazardous constituents exceeds the treatment standard (i. e., the treatment standard has not been attained). Concluding the treatment standard has been met when, in fact, it has not. Concluding the treatment standard has not been met when, in fact, it has.

Example 3: Under 40 CFR 264.101 (and proposed Subpart S  Corrective Action at SWMUs), a permittee conducts testing to determine if a remediation at a SWMU has attained the risk based cleanup standard specified in the permit.* The mean concentration in the SWMU is greater than the risk based cleanup standard (i. e., the site is contaminated).* Concluding the site is "clean" when, in fact, it is contaminated. Concluding the site is still contaminated when, in fact, it is "clean."

Example 4: Under 40 CFR 264.98( f), detection monitoring, monitor ground water at a regulated unit to determine if there is a statistically significant increase of contamination above background. The level of contamination in each point of compliance well does not exceed background. Concluding the contaminant concentration in a compliance well exceeds background when, in fact, it does not. Concluding the contaminant concentration in a compliance well is similar to background when, in fact, it is higher.

* If the cleanup standard is based on "background" rather than a risk based cleanup standard, then the hypotheses would be framed in reverse where the mean background and on site concentrations are presumed equal unless there is strong evidence that the site concentrations are greater than background. * A parameter other than the mean may be used to evaluate attainment of a cleanup standard (e. g., see USEPA 1989a).

In Example 4, however, the null hypothesis is framed in reverse of Examples 1 through 3. When conducting subsurface monitoring to detect contamination at a new unit (such as in detection monitoring in the RCRA ground water monitoring program), the natural subsurface environment is presumed uncontaminated until statistically significant increases over the background concentrations are detected. Accordingly, the null hypothesis is framed such that the downgradient conditions are consistent with the background. In this case, EPA's emphasis on the protection of human health and the environment calls for minimizing the Type II error

the mistake of judging downgradient concentrations the same as the background when, in fact, 45 they are higher. Detailed guidance on detection and compliance monitoring can be found in RCRA Ground Water Monitoring: Draft Technical Guidance (USEPA 1992c) and EPA's guidance on the statistical analysis of ground water monitoring data at RCRA facilities (USEPA 1989b and 1992b).

4.6.3 Specify a Range of Possible Parameter Values Where the Consequences of a False Acceptance Decision Error are Relatively Minor (Gray Region)

The "gray region" is one component of the quantitative decision performance criteria the planning team establishes during the DQO Process to limit impractical and infeasible sample sizes. The gray region is a range of possible parameter values near the action level where it is "too close to call." This gray area is where the sample data tend toward rejecting the baseline condition, but the evidence (data statistics) is not sufficient to be overwhelming. In essence, the gray region is an area where it will not be feasible to control the false acceptance decision error limits to low levels because the high costs of sampling and analysis outweigh the potential consequences of choosing the wrong course of action.

In statistical language, the gray region is called the "minimum detectable difference" and is often expressed as the Greek letter delta ( ). This value is an essential part of the calculations for   determining the number of samples that need to be collected so that the decision maker may have confidence in the decision made based on the data collected.

The first boundary of the gray region is the Action Level. The other boundary of the gray region is established by evaluating the consequences of a false acceptance decision error over the range of possible parameter values in which this error may occur. This boundary corresponds to the parameter value at which the consequences of a false acceptance decision error are significant enough to have to set a limit on the probability of this error occurring. The gray region (or "area of uncertainty") establishes the minimum distance from the Action Level where the decision maker would like to begin to control false acceptance decision errors.

In general, the narrower the gray region, the greater the number of samples needed to meet the criteria because the area of uncertainty has been reduced.

The quality of the decision process, including the boundaries of the gray region, can be depicted graphically using a Decision Performance Goal Diagram (DPGD). Detailed guidance on the construction and use of DPGDs is given in EPA DQO guidance documents (e. g., USEPA 2000a and 2000b) and in Data Quality Objectives Decision Error Feasibility Trials Software (DEFT) User's Guide (USEPA 2001a). Figure 12( a) and Figure 12( b) show how some of the key outputs of Step 6 of the DQO Process are depicted in a DPGD when the parameter of interest is the mean (Figure 12( a)) and a percentile (Figure 12( b) .

The DPGD given in Figure 12( a) shows how the boundaries of the gray region are set when the null hypothesis is established as "the true mean concentration exceeds the standard." Notice that the planning team has set the action level at 5 ppm and the other boundary of the gray region at 4 ppm. This implies that when the mean calculated from the sample data is less than 4 ppm (and the planning assumptions regarding variability hold true), then the data will be considered to provide "overwhelming evidence" that the true mean (unknown, of course) is below the action level. 46 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Gray Region (Relatively large decision error rates are considered tolerable.)

Action Level True value of the parameter (mean concentration, ppm) Low High 0 1 2 3 4 5 6 7 Tolerable false rejection decision error rate Tolerable false acceptance decision error rate Baseline Alternative Probability of Deciding that the Parameter Exceeds the Action Level Figure 12( a). Decision Performance Goal Diagram where the mean is the parameter of interest. Null hypothesis (baseline condition): the true mean exceeds the action level.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Gray Region (Relatively large decision error rates are considered tolerable.)

Action Level (P0)

True value of the parameter (true proportion of all possible samples of a given support that have concentrations less than the applicable standard) Low High 0. 80 0.875 0. 95 0. 975 Tolerable false acceptance decision error rate Tolerable false rejection decision error rate Alternative Baseline Probability of Deciding that the Parameter Exceeds the Action Level 0.90 0.825 0.775 1. 00 0.925 0.85 Figure 12( b). Decision Performance Goal Diagram where a percentile is the parameter of interest. Null hypothesis (baseline condition): true proportion  of all possible samples of a given support that are less than the applicable standard  is less than 0.90. 47 Now consider the DPGD given in Figure 12( b). The figure shows how the gray region is set when the null hypothesis is established as "the true proportion of samples below the concentration standard is less than 0.90." Notice in this example the planning team has set the action level at 0.90 and the other boundary of the gray region at 0.95. This implies that when the proportion of samples that comply with the standard is greater than 0.95, then the data will be considered to provide "overwhelming evidence" that the true proportion (unknown, of course) is greater than the action level of 0.90.

The term "samples" refers to all possible samples of a specified size, shape, and orientation (or sample support) drawn from the DQO decision unit. Sampling procedures and sample support can affect the measurement value obtained on individual samples and have a profound effect on the shape of the sampling distribution. Thus, the outcome of statistical procedures that examine characteristics of the upper tail of the distribution can be influenced by the sample support – more so than when the mean is the parameter of interest. Accordingly, when testing for a proportion, a complete statement of the null hypothesis should include specification of the sample support. See Sections 6.3.1 and 6.3.2 for guidance on establishing the appropriate sample support as part of the DQO Process.

4.6.4 Specify an Acceptable Probability of Making a Decision Error You can never completely eliminate decision errors or even know when they have occurred, but you can quantify the probability of making such errors. In this activity, you establish the acceptable probability of making a decision error.

The Type I error rate ( ) is a measure of the amount of "mistrust" you have in the conclusion   (Myers 1997) and is also known as the significance level for a test. The flip side of this is the amount of faith or confidence you have in the conclusion. The confidence level is denoted mathematically as . As stated previously, the Type I error (the error of falsely rejecting 1      the null hypothesis) is of greatest concern from the standpoint of environmental protection and regulatory compliance.

The probability of making a Type II error (the error of falsely accepting the null hypothesis) also can be specified. For example, if the sample data lead you to conclude that a waste does not qualify for the comparable fuels exclusion (40 CFR 261.38), when the true mean concentration in the waste is in fact below the applicable standard, then a Type II (false acceptance error) has been made. (Note that some of the statistical methods given in this document do not require specification of a Type II error rate).

As a general rule, the lower you set the probability of making a decision error, the greater the cost in terms of the number of samples required, time and personnel required for sampling and analysis, and financial resources required.

An acceptable probability level for making a decision error should be established by the planning team after consideration of the RCRA regulatory requirements, guidance from EPA or the implementing agency, the size (volume or weight) of the decision unit, and the consequences of making a decision error. In some cases, the RCRA regulations specify the Type I or Type II (or both) error rates that should be used. For example, when testing a waste to determine whether it qualifies for the comparable/ syngas fuel exclusion under 40 CFR 261.38, the regulations require that the determination be made with a Type I error rate set at 5 7 Under §261.38( c)( 8)( iii)( A), a generator must demonstrate that "each constituent of concern is not present in the waste above the specification level at the 95% upper confidence limit around the mean."

48 percent (i. e., ). 7  = 0 05 .

In other cases, the regulations do not specify any decision error limits. The planning team must specify the decision error limits based on their knowledge of the waste; impacts on costs, human health, and ecological conditions; and the potential consequences of making a decision error. For example, if the quantity of waste (that comprises a decision unit) is large and/ or heterogeneous, then a waste handler may require high confidence (e. g., 95 or 99 percent) that a high proportion of the waste or media complies with the applicable standard. On the other hand, if the waste quantity is a relatively small (e. g., a drum) and sampling and measurement error can be minimized, then the waste handler may be willing to relax the confidence level required or simply use a nonstatistical (e. g., judgmental) sampling design and reduce the number of samples to be taken.

For additional guidance on controlling errors Section 6 and EPA's DQO guidance (USEPA 2000a and 2000b).

4.7 Outputs of the First Six Steps of the DQO Process Table 5 provides a summary of the outputs of the first six steps of the DQO Process. Typically, this information will be incorporated into a QAPP, WAP, or other similar planning document (as described in Section 5.7). The DQOs can be simple and straight forward for simple projects and can be documented in just a few pages with little or no supporting data. For more complex projects, the DQOs can be more lengthy, and the supporting data may take up volumes. The team that will be optimizing the sample design( s) will need the information to support their plan development. The project manager and the individuals who assess the overall outcome of the project also will need the information to determine if the DQOs were achieved.

Keep in mind that the DQO Process is an iterative one; it might be necessary to return to earlier steps to modify inputs when new data become available or to change assumptions if achieving the original DQOs is not realistic or practicable.

The last step (Step 7) in the DQO Process is described in detail in the next section of this document. Example applications of the full DQO Process are presented in Appendix "I." 49 Table 5. Summary of Outputs of the First Six Steps of the DQO Process DQO Step Expected Outputs 1. State the Problem ° List of members of the planning/ scoping team and their role/ expertise in the project. Identify individuals or organizations participating in the project (e. g. facility name) and discuss their roles, responsibilities, and organization. ° A concise description of the problem. ° Summary of available resources and relevant deadlines.

2. Identify the Decision ° A decision statement that links the principal study question to possible actions that will solve the problem or answer the question.

3. Identify Inputs to the Decision ° A list of informational inputs needed to resolve the decision statement, how the information will be used, sources of that information, and an indication of whether the information is available for will need to be obtained. ° A list of environmental variables or characteristics that will be measured.

4. Define the Boundaries ° A detailed description of the spatial and temporal boundaries of the problem (i. e., define the population, each decision unit, and the sample support). ° Options for stratifying the population under study. ° Any practical constraints that may interfere with the study.

5. Develop a Decision Rule ° The parameter of interest that characterizes the population. ° The Action Level or other method for testing the decision rule. ° An "if ... then..." statement that defines the conditions that would cause the decision maker to choose among alternative actions.

6. Specify Limits on Decision Errors ° Potential variability and bias in the candidate sampling and measurement methods ° The baseline condition (null hypothesis) ° The boundaries of the gray region ° The decision maker's tolerable decision error rates based on a consideration of consequences of making an incorrect decision. 50 5 OPTIMIZING THE DESIGN FOR OBTAINING THE DATA This section describes DQO Process Step 7, the last step in the DQO Process. The purpose of this step is to identify an optimal design for obtaining the data. An optimal sampling design is one that obtains the requisite information from the samples for the lowest cost and still satisfies the DQOs.

You can optimize the sampling design by performing five activities that are described in detail in this section. These activities are based on those described in Guidance for the Data Quality Objectives Process EPA QA/ G 4 (USEPA 2000b), but they have been modified to more specifically address RCRA waste related studies.

In this final planning step, combine the data collection design information with the other outputs of the DQO Process and document the approach in a planning document such as a QAPP, WAP, or similar planning document. As part of this step, it may be necessary to work through Step 7 more than once after revisiting the first six steps of the DQO Process.

5.1 Review the Outputs of the First Six Steps of the DQO Process Each of the steps in the DQO Process has a series of outputs that include qualitative and quantitative information about the study. The outputs of the first six steps of the DQO Process, as described in Section 4, serve as inputs to DQO Step 7.

Review the existing information and DQO outputs (see Table 5). Determine if any data gaps exist and determine whether filling those gaps is critical to completion of the project. Data gaps can be filled by means of a "preliminary study" or "pilot study." A preliminary study or pilot can include collection of samples to obtain preliminary estimates of the mean and standard deviation. In addition, a preliminary study can help you verify waste or site conditions, identify unexpected conditions or materials present, gain familiarization with the waste and facility operations, identify how the waste can be accessed, check and document the physical state of the material to be sampled, and identify potential health and safety hazards that may be present.

Review the potential sources of variability and bias (" error") that might be introduced in the sampling design and measurement processes. See Section 6 for a discussion of sources of error in sampling and analysis. Step 7: Optimize the Design for Collecting the Data Purpose To identify a resource effective data collection design for generating data that are expected to satisfy the DQOs.

Activities ° Review the outputs of the first six steps of the DQO Process (see Section 5.1). ° Consider various data collection design options, including sampling and analytical design alternatives (see Section 5.2), and composite sampling options (see Section 5.3). ° For each data collection design alternative, determine the appropriate number of samples (see Section 5.4 or 5.5). ° Select the most resource effective design that satisfies all of the data needs for the least costs (see Section 5.6). ° Prepare a QAPP, WAP, or similar planning document as needed to satisfy the project and regulatory requirement (see Section 5.7). 51 5.2 Consider Data Collection Design Options Data collection design incorporates two interdependent activities  the sample collection design and analytical design.

Sampling Design: In developing a sampling design, you consider various strategies for selecting the locations, times, and components for sampling, and you define appropriate sample support. Examples of sampling designs include simple random, stratified random, systematic, and judgmental sampling. In addition to sampling designs, make sure your organization has documented standard operation procedures (SOPs) that describe the steps to be followed when implementing a sampling activity (e. g., equipment preparation, sample collection, decontamination). For guidance on suggested content and format for SOPs, refer to Guidance for the Preparing Standard Operating Procedures (SOPs) EPA QA/ G 6 (USEPA 2001c). Sampling QA/ QC activities also should be part of sampling design. Activities used to document, measure, and control data quality include project specific quality controls (e. g., duplicate samples, equipment blanks, field blanks, and trip blanks) and the associated quality assessments (e. g., audits, reviews) and assurances (e. g., corrective actions, reports to management). These activities typically are documented in the QAPP (see Section 5.7 and USEPA 1998a).

Analytical Design: In DQO Steps 3 and 5, an Action Level and candidate analytical methods were identified. The information should be used to develop analytical options in terms of cost, method performance, available turnaround times, and QA/ QC requirements. The analytical options can be used as the basis for designing a performance based cost effective analytical plan (e. g., deciding between lower cost field analytical methods and/ or higher cost laboratory methods). Candidate laboratories should have adequate SOPs that describe the steps to be followed when implementing an analytical activity (e. g., sample receipt procedures, subsampling, sample preparation, cleanup, instrumental analysis, data generation and handling). If field analytical techniques are used, hard copies of the analytical methods or SOPs should be available in the field. Refer to Chapter Two of SW 846 for guidance on the selection of analytical methods.

The goal of this step is to find cost effective design alternatives that balance the number of samples and the measurement performance, given the feasible choices for sample designs and measurement methods.

Sampling design is the "where, when, and how" component of the planning process. In the context of waste sampling under RCRA, there are two categories of sampling designs: (1) probability sampling and (2) authoritative (nonprobability) sampling. The choice of a sampling design should be made after consideration of the DQOs and the regulatory requirements.

Probability sampling refers to sampling designs in which all parts of the waste or media under study have a known probability of being included in the sample. In cases in which all parts of the waste or media are not accessible for sampling, the situation should be documented so its potential impacts can be addressed in the assessment phase. Probability samples can be of various types, but in some way, they all make use of randomization, which allows probability statements to be made about the quality of estimates derived from the resultant data. 52 Probability sampling designs provide the ability to reliably estimate variability, the reproducibility of the study (within limits), and the ability to make valid statistical inferences. Five types of probability sampling designs are described in Sections 5.2.1 through 5.2.5:

° Simple random sampling ° Stratified random sampling ° Systematic sampling ° Ranked set sampling ° Sequential sampling.

A strategy that can be used to improve the precision (reproducibility) of most sampling designs is composite sampling. Composite sampling is not a sampling design in and of itself, rather composite sampling is a strategy used as part of a probability sampling design or an authoritative sampling design. Composite sampling is discussed in Section 5.3.

One common misconception of probability sampling procedures is that these procedures preclude the use of important prior information. Indeed, just the opposite is true. An efficient sampling design is one that uses all available prior information to help design the study. Information obtained during DQO Step 3 (" Identify Inputs to the Decision") and DQO Step 4 (" Define the Study Boundaries") should prove useful at this stage. One of the activities suggested in DQO Step 4 is to segregate the waste stream or media into less heterogeneous subpopulations as a means of segregating variability. To determine if this activity is appropriate, it is critical to have an understanding of the various kinds of heterogeneity the constituent of concern exhibits within the waste or media (Pitard 1993). Making assumptions that a waste stream is homogeneous can result in serious sampling errors. In fact, some authors suggest the word "homogeneous" be removed from our sampling vocabulary (Pitard 1993, Myers 1997).

Table 6 provides a summary of sampling designs discussed in this guidance along with conditions for their use, their advantages, and their disadvantages. Figure 13 provides a graphical representation of the probability sampling designs described in this guidance. A number of other sampling designs are available that might perform better for your particular situation. Examples include cluster sampling and double sampling. If an alternative sampling design is required, review other publications such as Cochran (1977), Gilbert (1987), USEPA (2000c) and consult a professional statistician. Sampling Over Time or Space?

An important feature of probability sampling designs is that they can be applied along a line of time or in space (see Figure 13) or both (Gilbert 1987):

Time Sampling designs applied over time can be described by a one dimensional model that corresponds to flowing streams such as the following:

° Solid materials on a conveyor belt ° A liquid stream, pulp, or slurry moving in a pipe or from a discharge point (e. g., from the point of waste generation) ° Continuous elongated piles (Pitard 1993).

Space For practical reasons, sampling of material over a threedimensional space is best addressed as though the material consists of a series of overlapping twodimensional planes of more or less uniform thickness (Pitard 1993, Gy 1998). This is the case for obtaining samples from units such as the following:

° Drums, tanks, or impoundments containing single or multi phasic liquid wastes ° Roll off bins, relatively flat piles, or other storage units ° Landfills, soil at a land treatment unit, or a SWMU. 53 Table 6. Guidance for Selection of Sampling Designs Sampling Design Appropriate Conditions for Use Advantages Limitations Probability Sampling Simple Random Sampling (Section 5.2.1) Useful when the population of interest is relatively homogeneous (i. e., there are no major patterns or "hot spots" expected). ° Provides statistically unbiased estimates of the mean,

proportions, and the variability.

° Easy to understand and implement. ° Least preferred if patterns or trends are known to exist and are identifiable.

° Localized clustering of sample points can occur by random chance.

Stratified Random Sampling (Section 5.2.2) Most useful for estimating a parameter (e. g., the mean) of wastes exhibiting high heterogeneity (e. g.,

there are distinct portions or components of the waste with high and low constituent concentrations or characteristics). ° Ensures more uniform coverage of the entire target population.

° Potential for achieving greater precision in estimates of the mean and variance.

° May reduce costs over simple random and systematic sampling designs because fewer samples may be required.

° Enables computation of reliable estimates for population subgroups of special interest. ° Requires some prior knowledge of the waste or media to define strata and to obtain a more precise estimate of the mean.

° Statistical procedures for calculating the number of samples, the mean, and the variance are more complicated than for simple random sampling.

Systematic Sampling (Section 5.2.3) Useful for estimating spatial patterns or trends over time. ° Preferred over simple random when sample locations are random within each systematic block or interval.

° Practical and easy method for designating sample locations.

° Ensures uniform coverage of site,

unit, or process.

° May be lower cost than simple random sampling because it is easier to implement. ° May be misleading if the sampling interval is aligned with the pattern of contamination, which could happen inadvertently if there is inadequate prior knowledge of the pattern of contamination.

° Not truly random, but can be modified through use of the "random within blocks" design. 54 Table 6. Guidance for Selection of Sampling Designs (Continued)

Sampling Design Appropriate Conditions for Use Advantages Limitations Probability Sampling (continued)

Ranked Set Sampling (Section 5.2.4) ° Useful for reducing the number of samples required.

° Useful when the cost of analysis is much greater than the cost of collecting samples.

° Inexpensive auxiliary variable (based on expert knowledge or measurement) is needed and can be used to rank randomly selected population units with respect to the variable of interest.

° Useful if the ranking method has a strong relationship with accurate measurements. ° Can reduce analytical costs. ° Requires expert knowledge of waste or process or use of auxiliary quantitative measurements to rank population units.

Sequential Sampling (Section 5.2.5) ° Applicable when sampling and/ or analysis are quite expensive,

when information concerning sampling and/ or measurement variability is lacking, when the waste and site characteristics of interest are stable over the time frame of the sampling effort, or when the objective of the sampling effort is to test a specific hypothesis.

° May not be especially useful if multiple waste characteristics are of interest or if rapid decision making is necessary. ° Can reduce the number of samples required to make a decision.

° Allows a decision to be made with less sampling if there is a large difference between the two populations or between the true value of the parameter of interest and the standard. ° If the concentration of the constituent of concern is only marginally different from the action level, sequential procedures will require an increasing number of samples approaching that required for other designs such as simple random or systematic sampling. 55 Table 6. Guidance for Selection of Sampling Designs (Continued)

Sampling Design Appropriate Conditions for Use Advantages Limitations Authoritative Sampling Judgmental (Section <IP_ADDRESS>) ° Useful for generating rough estimates of the average concentration or typical property.

° To obtain preliminary information about a waste stream or site to facilitate planning or to gain familiarity with the waste matrix for analytical purposes.

° To assess the usefulness of samples drawn from a small portion of the waste or site.

° To screen samples in the field to identify "hot" samples for subsequent analysis in a laboratory. ° Can be very efficient with sufficient knowledge of the site or waste generation process.

° Easy to do and explain. ° The utility of the sampling design is highly dependent on expert knowledge of waste.

° Nonprobability based so inference to the general population is difficult.

° Cannot determine reliable estimates of variability.

Biased (Section <IP_ADDRESS>) ° Useful to estimate "worst case" or "best case" conditions (e. g., to identify the composition of a leak,

spill, or waste of unknown composition). 56 Sampling Over Space (two dimensional plan view) Sampling Over Time or Along a Transect onedimensional

Simple Random Sampling (a) Simple Random Sampling (b)

Stratified Random Sampling Strata high medium low (c) Strata Stratified Random Sampling high medium low (d)

Systematic Grid Sampling (e) Systematic Sampling (f)

Random Sampling Within Blocks (g) Random Sampling Within Segments (h)

Figure 13. Probability sampling designs over space or along an interval (modified after Cochran 1977 and Gilbert 1987) 57 Box 3. Simple Random Sampling: Procedure 1. Divide the area of the study into N equal size grids, intervals (if sampling over time), or other units. The spacing between adjacent sampling locations should be established in the DQOs, but the length should be measurable in the field with reasonable accuracy. The total number of possible sampling locations (N) should be much larger than n (the number of samples to be collected).* 2. Assign a series of consecutive numbers to each location between 1 and N. 3. Draw n integers between 1 and N from a random number table or use the random number function on a hand held calculator (i. e., generate a random number between 0 and 1 and multiply the number by N). 4. Collect samples at each of the n locations or intervals.

* For additional guidance on calculating spacing between sampling locations, see Methods for Evaluating the Attainment of Cleanup Standards, Volume I: Soil and Solid Media (USEPA 1989a). 5.2.1 Simple Random Sampling The simplest type of probability sampling is simple random sampling (without replacement), in which every possible sampling unit in the target population has an equal chance of being selected. Simple random samples, like the other samples, can be either samples in space (Figure 13( a)) or in time (Figure 13( b)) and are often appropriate at an early stage of an investigation in which little is known about nonrandom variation within the waste generation process or the site. All of the sampling units should have equal volume or mass, and ideally be of the same shape and orientation if applicable (i. e., they should have the same "sample support").

With a simple random sample, the term "random" should not be interpreted to mean haphazard; rather, it has the explicit meaning of equiprobable selection. Simple random samples are generally developed through use of a random number table (found in many statistical text books), a random number function on a hand held calculator, or by a computer.

One possible disadvantage of pure random sampling is that localized clustering of sample points can occur. If this occurs, one option is to select a new random time or location for the sample. Spatial or temporal biases could result if unknown trends, patterns, or correlations are present. In such situations, stratified random sampling or systematic sampling are better options.

5.2.2 Stratified Random Sampling In stratified random sampling, a heterogeneous unit, site, or process is divided into nonoverlapping groups called strata. Each stratum should be defined so that internally it is relatively homogeneous (that is, the variability within each stratum is less than the variability observed over the entire population) (Gilbert 1987). After each stratum is defined, then simple random sampling is used within each stratum (see Figure 13( c) and 15( d)). For very heterogeneous wastes, stratified random sampling can be used to obtain a more efficient estimate of the parameter of interest (such as the mean) than can be obtained from simple random sampling.

It is important to note that stratified random sampling, as described in this guidance, can be used when the objective is to make a decision about the whole population or decision unit. If the objective is to determine of a solid waste is a hazardous waste or to measure attainment of a treatment standard for a hazardous waste, then any obvious "hot spots" or high concentration wastes should be characterized separately from low concentration wastes to minimize mixing of 58 Box 4. Stratified Random Sampling: Procedure 1. Use prior knowledge of the waste stream or site to divide the target population into L nonoverlapping strata such that the variability within stratum is less than the variability of the entire population (for example, see Figure 13c and Figure 13d). The strata can represent area, volume, mass, or time intervals. 2. Assign a weight to each stratum. The value Wh hth of each should be determined based on its relative Wh importance to the data user, or it can be the proportion of the volume, mass, or area of the waste that is in stratum . h 3. Conduct random sampling within each stratum. hazardous waste with nonhazardous wastes and to prevent impermissible dilution (see also Appendix C). If the objective of the sampling effort is to identify nonrandom spatial patterns (for example, to create a map of contamination in shallow soils), then consider the use of a geostatistical technique to evaluate the site.

In stratified random sampling it is usually necessary to incorporate prior knowledge and professional judgment into a probabilistic sampling design. Generally, wastes or units that are "alike" or anticipated to be "alike" are placed together in the same stratum. Units that are contiguous in space (e. g., similar depths) or time are often grouped together into the same stratum, but characteristics other than spatial or temporal proximity can be employed. For example, you could stratify a waste based on particle size (such that relatively large pieces of contaminated debris are assigned to one stratum and unconsolidated fines assigned to a separate stratum). This is called stratification by component. See Appendix C of this guidance for additional information on stratification, especially as a strategy for sampling heterogeneous wastes, such as debris.

In stratified random sampling a decision must be made regarding the allocation of samples among strata. When chemical variation within each stratum is known, samples can be allocated among strata using optimum allocation in which more samples are allocated to strata that are large, more variable internally, or cheaper to sample (Cochran 1977, Gilbert 1987). An alternative is to use proportional allocation. In proportional allocation, the sampling effort in each stratum is directly proportional to the size (for example, the mass) of the stratum. See Section 5.4.2 for guidance on determining optimum and proportional allocation of samples to strata.

There are several advantages to stratified random sampling. Stratified random sampling:

° Ensures more uniform coverage of the entire target population ° Ensures that subareas that contribute to overall variability are included in the sample ° Achieves greater precision in certain estimation problems ° Generally will be more cost effective than simple random sampling even when imperfect information is used to form the strata.

There are also some disadvantages to stratified random sampling. Stratified random sampling is slightly more difficult to implement in the field and statistical calculations for stratified sampling are more complex than for simple random sampling (e. g., due to the use of weighting factors and more complex equations for the appropriate number of samples). 59 Box 5: Systematic Sampling: Procedure Sampling Over Space 1. Determine the size of the area to be sampled. 2. Denote the surface area of the sample area by . A 3. Assuming a square grid is used, calculate the length of spacing between grid nodes (L)

L A n =

where n is the number of samples. The distance L should be rounded to the nearest unit that can be easily measured in the field. 4. To determine the sampling locations, randomly select an initial sampling point within the area to be sampled. Using this location as one intersection of two gridlines, construct gridlines parallel to the original grid and separated by distance L. 5. Collect samples at each grid node (line intersection) (see Figure 13e). Alternatively, randomly select a sampling point within each grid block (see Figure 13g).

Sampling Along a Line (e. g., Over Time) 1. Determine the start time and point and the total length of time (N) over which the samples will be collected. 2. Decide how many samples (n) will be collected over the sampling period.

3. Calculate a sampling interval where . k N n =

4. Randomly select a start time and collect a sample every kth interval until n samples have been obtained (see Figure 13f). Alternatively, randomly select a sampling point within each interval (Figure 13h). 5.2.3 Systematic Sampling Systematic sampling entails taking samples at a preset interval of time or in space and using a randomly selected time or location as the first sampling point (Gilbert 1987).

Systematic sampling over space involves establishing a two dimensional grid of the unit or waste under investigation (Figure 13( e)). The orientation of the grid is sometimes chosen randomly and various types of systematic samples are possible. For example, points may be arranged in a pattern of squares (rectangular grid sampling) or a pattern of equilateral triangles (triangular grid sampling). The result of either approach is a simple pattern of equally spaced points at which sampling is to be performed. As shown in Figure 13( f), systematic sampling also can be conducted along a transect (every five feet, for example), along time intervals (every hour, for example), or by flow or batches (every 10,000 gallons, for example) (King 1993).

The systematic sampling approach is attractive because it can be easily implemented in the field, but it has some limitations such as not being truly random. You can improve on this sampling design by using random sampling within each grid block (Figure 13( g)) or within each time interval (Figure 13( h)). This approach maintains the condition of equiprobability during the sampling event (Myers 1997) and can be considered a form of stratified random sampling in which each of the boundaries of the strata are arbitrarily defined (rather than using prior information) and only one random sample is taken per stratum (Gilbert 1987). This approach is advantageous because it avoids potential problems caused by cycles or trends.

Systematic sampling also is preferred when one of the objectives is to locate "hot spots" within a site or otherwise map the pattern of concentrations over an area (e. g., using geostatistical techniques). Even without using geostatistical methods, "hot spots" or other patterns could be identified by using a systematic design (see "ELIPGRID" software in Appendix H and Gilbert 1987, page 119). On the other hand, the systematic sampling design should be used with caution whenever there is a possibility of some type of cyclical pattern in the waste unit or 60 A AAA t1 t2 t3 Time Concentration 0 Period * * * ** B BB B BB * Mean Concentration Figure 14. Potential pitfall of systematic sampling over time: cyclic trend combined with a systematic sampling design (after Cochran 1977 and Gilbert 1987) process that might match the sampling frequency, especially processes being measured over time (such as discharges from a pipe or material on a conveyor).

Figure 14 illustrates the potential disadvantage of using systematic sampling when cyclic trends are present. When there is a cyclic trend in a waste generation process, using a uniform pattern of sampling points can result in samples with very unusual properties. The sets of points labeled "A" and "B" are systematic samples for which the sampling intervals are one period and onehalf period, respectively. The points labeled "A" would result in a biased estimate of the mean but a sampling variance of zero. The points labeled "B" would result in an unbiased estimate of the mean with very small variance, even a zero variance if the starting point happened to be aligned exactly with the mean.

5.2.4 Ranked Set Sampling Ranked set sampling (RSS) (McIntyre 1952) can create a set of samples that at a minimum is equivalent to a simple random sample, but can be as much as two to three times more efficient than simple random sampling. This is because RSS uses the availability of expert knowledge or an inexpensive surrogate measurement or auxiliary variable that is correlated with the more expensive measurement of interest. The auxiliary variable can be a qualitative measure, such as visual inspection for color or an inexpensive quantitative (or semi quantitative) measure that can be obtained from a field instrument such as a photoionization detector for volatile organics or an X ray fluorescence analyzer for elemental analysis. RSS exploits this correlation to obtain a sample that is more representative of the population than would be obtained by random sampling, thereby leading to more precise estimates of the population parameters than random sampling. RSS is similar to other probabilistic sampling designs such as simple random sampling in that sampling points are identified and samples are collected. In RSS, however, only a subset of the samples are selected for analysis.

RSS consists of creating m groups, each of size m (for a total of "m x m" initial samples), then ranking the surrogate from largest to smallest within each group. One sample from each group is then selected according to a specified procedure and these m samples are analyzed for the more expensive measurement of interest (see Box 6 and Figure 15).

The true mean concentration of the characteristic of interest is estimated by the arithmetic sample mean of the measured samples (e. g., by Equation 1). The population variance and standard deviation also are estimated by the traditional equations (e. g., by Equations 2 and 3). For additional information on RSS, see USEPA 1995b, USEPA 2000c, and ASTM D 6582 Standard Guide for Ranked Set Sampling: Efficient Estimation of a Mean Concentration in Environmental Sampling. 61 5.2.5 Sequential Sampling In sequential testing procedures (Wald 1973), sampling is performed by analyzing one (or more) sample( s) at a time until enough data have been collected to meet the statistical confidence level that the material does not exceed the critical level. The expected sample size, using this sequential procedure, can be approximately 30 to 60 percent lower than a corresponding fixed sample size test with the same power. The sequential procedure is especially helpful in situations in which the contamination is very high or very low relative to the action level. In these situations, the sequential procedure will quickly accumulate enough evidence to conclude that the waste or site either meets or fails to meet the standard.

Figure 16 shows how the procedure operates in a simple example for determining the mean concentration of a constituent of concern in soil. This particular example involves clean closure of a waste management unit, however, the approach could be used for other situations in which the mean is the parameter of interest. The procedure consists of analyzing groups of samples and calculating the mean and 80 percent confidence interval (or upper 90 percent confidence limit) for the mean after analysis of each group of samples. The horizontal axis represents the number of sample units evaluated. The vertical axis represents the concentration of the contaminant; plotted are the mean and 80 percent confidence interval after analysis of n samples. The , against which the sample is to be judged, is shown as a horizontal line. AL The sampled units are analyzed first in a small lot (e. g., five samples). After each evaluation the mean and confidence interval on the mean are determined (point "a"). If the 90 percent UCL on the mean value stays above the critical value, , after successive increments are analyzed, AL the soil in the unit cannot be judged to attain the action level (point "b"). If the UCL goes below Set 1 Set 2 Set 3 Set 4 Rank 1 234 Sample sent for analysis Sample ignored m = 4 For example, if 12 samples are needed, the process is repeated 2 more times using fresh samples.

Figure 15. Ranked set sampling. After the samples are ranked in order from lowest to highest, a sample is selected for analysis from Set 1 with Rank 1, from Set 2 with Rank 2, etc. Box 6. Ranked Set Sampling: Procedure 1. Identify some auxiliary characteristic by which samples can be ranked in order from lowest to highest (e. g., by use of a low cost field screening method). 2. Randomly select samples m m × from the population (e. g., by using simple random sampling). 3. Arrange these samples into sets of m size . m 4. Within each set, rank the samples by using only the auxiliary information on the samples. 5. Select the samples to be analyzed as follows (see Figure 17): ° In Set 1, select the sample with rank 1 ° In Set 2, select the sample with rank 2, etc ... ° In Set , select the unit with rank m . m 6. Repeat Steps 1 through 5 for cycles to obtain a total of samples for analysis. r n mr =   62 a b c d Mean calculated from n samples AL  Risk based action level Confidence Interval 5 10 20 40 Concentration AL Soil does not attain AL Soil attains AL Cumulative number of samples (n)

Figure 16. Example of sequential testing for determining if concentrations of a constituent of concern in soil at a closed waste management unit are below a risk based action level (AL). the critical value line, it may be concluded that the soil attains the standard. In the figure, the total number of samples is successively increased until the 90 percent UCL falls below the critical level (points "c" and "d").

A sequential sampling approach also can be used to test a percentile against a standard. A detailed description of this method is given in Chapter 8 of Methods for Evaluating the Attainment of Cleanup Standards Volume 1: Soil and Solid Media (USEPA 1989a).

In sequential sampling, the number of samples is not fixed a priori; rather, a statistical test is performed after each analysis to arrive at one of three possible decisions: reject the hypothesis, accept the hypothesis, or perform another analysis. This strategy is applicable when sampling and/ or analyses are quite expensive, when information concerning sampling and/ or measurement variability is lacking, when the waste and site characteristics of interest are stable over the time frame of the sampling effort, or when the objective of the sampling effort is to test a specific hypothesis. It may not be especially useful if multiple waste characteristics are of interest or if rapid decision making is necessary.

In planning for a sequential sampling program, the following considerations are important:

° Pre planning the effort between the field and laboratory, including developing a system of pre planned paperwork and sample containers ° Arranging for a system of rapid delivery of samples to the laboratory ° Providing rapid turnaround in the laboratory ° Rapidly returning data to the planners, supervisors, and others responsible for decision making.

If the sequential sampling program is carried out using field methods (e. g., portable detectors), much of the inconvenience involved with shipping and return of results can be avoided.

5.2.6 Authoritative Sampling Authoritative sampling is a nonstatistical sampling design because it does not assign an equal probability of being sampled to all portions of the population. This type of sampling should be considered only when the objectives of the investigation do not include the estimation of a population parameter. For example, authoritative sampling might be appropriate when the objective of a study is to identify specific locations of leaks, or when the study is focused solely 63 on the sampling locations themselves. The validity of the data gathered with authoritative sampling is dependent on the knowledge of the sampler and, although valid data sometimes can be obtained, it is not recommended for the chemical characterization of wastes when the parameter of interest (such as the mean) is near the action level.

Authoritative sampling (also known as judgmental sampling, biased sampling, nonprobability sampling, nonstatistical sampling, purposive sampling, or subjective sampling) may be appropriate under circumstances such as the following:

° You need preliminary information about a waste stream or site to facilitate planning or to gain familiarity with the waste matrix for analytical purposes.

° You are conducting sampling for a RCRA Facility Assessment (RFA) to identify a potential or actual release to the environment.

° You have encountered a spill of an unknown chemical and need to determine the chemical makeup of the spilled material.

° You have access to only small portions of the population and judgment is applied to assess the usefulness of samples drawn from the small portion.

° You are screening samples in the field, using an appropriate field method, to identify "hot" samples for subsequent analysis in a laboratory.

° You are sampling to support case development for an enforcement agency or to "prove the positive" (see also Section 2.2.4).

With authoritative sampling, it is not possible to accurately estimate the population variance. Also, due to its subjective nature, the use of authoritative sampling by the regulated community to demonstrate compliance with regulatory standards generally is not advisable except in those cases in which a small volume of waste is in question or where the concentration is either well above or well below the regulatory threshold.

The ASTM recognizes two types of authoritative sampling: judgmental sampling and biased sampling (ASTM D 6311).

<IP_ADDRESS> Judgmental Sampling Judgmental sampling is a type of authoritative sampling. The goal of judgmental sampling is to use process or site knowledge to choose one or more sampling locations to represent the "average" concentration or "typical" property.

Judgmental sampling designs can be cost effective if the people choosing the sampling locations have sufficient knowledge of the waste. If the people choosing the sampling locations intentionally distort the sampling by a prejudiced selection, or if their knowledge is wanting, judgmental sampling can lead to incorrect and sometimes very costly decisions. Accurate and useful data can be generated from judgmental sampling more easily if the population is relatively homogeneous and the existence of any strata and their boundaries is known. The disadvantages of judgmental sampling designs follow: 1 Some authors use the term "discrete sample" to refer to an individual sample that is used to form a composite sample. The RCRA regulations often use the term "grab sample." For the purpose of this guidance, the terms "discrete," "grab," and "individual" sample have the same meaning.

64 ° It can be difficult to demonstrate that prejudice was not employed in sampling location selection ° Variances calculated from judgmental samples may be poor estimates of the actual population variance ° Population statistics cannot be generated from the data due to the lack of randomness.

An example application of judgement sampling is given in Appendix C of Guidance for the Data Quality Objectives Process for Hazardous Waste Site Operations (USEPA 2000a).

<IP_ADDRESS> Biased Sampling Biased sampling is the type of authoritative sampling that intends not to estimate average concentrations or typical properties, but to estimate "worst" or "best" cases (ASTM D 6051 96). The term "biased," as used here, refers to the collection of samples with expected very high or very low concentrations. For example, a sample taken at the source of a release could serve as an estimate of the "worst case" concentration found in the affected media. This information would be useful in identifying the constituent of concern and estimating the maximum level of contamination likely to be encountered during a cleanup.

At times, it may be helpful to employ a "best case" or both a "best case" and "worst case" biased sampling approach. For example, if there is a range of wastes and process knowledge can be used to identify the wastes likely to have the lowest and highest contamination levels, then these two extremes could be sampled to help define the extent of the problem.

Biased sampling, while having the ability to cost effectively generate information, has similar disadvantages to that of judgmental sampling.

5.3 Composite Sampling Composite sampling is a strategy in which multiple individual or "grab" samples (from different locations or times) are physically combined and mixed into a single sample so that a physical, rather than a mathematical, averaging takes place. 1 Figure 17 illustrates the concept of composite samples. For a well formed composite, a single measured value should be similar to the mean of measurements of the individual components of the composite (Fabrizio, et al. 1995). Collection of multiple composite samples can provide improved sampling precision and reduce the total number of analyses required compared to noncomposite sampling. This strategy is sometimes employed to reduce analysis costs when analysis costs are large relative to sampling costs. The appropriateness of using composite sampling will be highly dependent on the DQOs (Myers 1997), the constituent of concern, and the regulatory requirements. To realize the full benefits of composite sampling, field and laboratory personnel must carefully 65 Composite Individual Field Samples Composite Figure 17. Forming composite samples from individual samples (from USEPA 1995c). follow correct procedures for sample collection, mixing, and subsampling (see Sections 6 and 7).

5.3.1 Advantages and Limitations of Composite Sampling A detailed discussion of the advantages and limitations of composite sampling is presented in the Standard Guide for Composite Sampling and Field Subsampling for Environmental Waste Management Activities (ASTM D 6051 96) and EPA's Guidance for Choosing a Sampling Design for Environmental Data Collection, EPA QA/ G 5S (USEPA 2000c). Additional information on composite sampling can be found in Edland and van Belle (1994), Gilbert (1987), Garner, et al. (1988 and 1989), Jenkins, et al. (1996 and 1997), Myers (1997), and USEPA (1995c).

Advantages Three principal advantages to using composite sampling (see ASTM D 6051 96) follow:

° It can improve the precision (i. e., reduce between sample variance) of the estimate of the mean concentration of a constituent in a waste or media (see Section 5.3.5)

° It can reduce the cost of estimating a mean concentration, especially in cases in which analytical costs greatly exceed sampling costs or in which analytical capacity is limited ° A "local" composite sample, formed from several increments obtained from a localized area, is an effective way to increase the sample support, which reduces grouping and segregation errors (see also Section <IP_ADDRESS>)

° It can be used to determine whether the concentration of a constituent in one or more individual samples used to form a composite might exceed a fixed standard (i. e., is there a "hot spot"?) (see Section 5.3.6).

Limitations Composite sampling should not be used if the integrity of the individual sample values changes because of the physical mixing of samples (USEPA 1995c). The integrity of individual sample values could be affected by chemical precipitation, exsolvation, or volatilization during the pooling and mixing of samples. For example, volatile constituents can be lost upon mixing of samples or interactions can occur among sample constituents. In the case of volatile constituents, compositing of individual sample extracts within a laboratory environment may be a reasonable alternative to mixing individual samples as they are collected. 66 Listed below are some additional conditions under which compositing usually is not advantageous:

° When regulations require the use of discrete or grab samples. For example, compliance with the LDR numeric treatment standards for non wastewaters typically is to be determined using "grab" samples rather than composite samples. Grab samples processed, analyzed, and evaluated individually normally reflect maximum process variability, and thus reasonably characterize the range of treatment system performance. Typically, grab samples are used to evaluate LDR non wastewaters and composite samples are used to evaluate LDR wastewaters, except when evaluating wastewaters for metals (D004 through D011) for which grab samples are required [40 CFR 268.40( b)].

° When data users require specific data points to generate high end estimates or to calculate upper percentiles ° When sampling costs are much greater than analytical costs ° When analytical imprecision outweighs sampling imprecision and population heterogeneity ° When individual samples are incompatible and may react when mixed ° When properties of discrete samples, such as pH or flash point, may change qualitatively upon mixing. (Compositing of individual samples from different locations to be tested for hazardous waste characteristic properties, such as corrosivity, reactivity, ignitability, and toxicity, is not recommended)

° When analytical holding times are too short to allow for analysis of individual samples, if testing of individual samples is required later (for example, to identify a "hot" sample) (see Section 5.3.6)

° When the sample matrix impedes correct homogenization and/ or subsampling ° When there is a need to evaluate whether the concentrations of different contaminants are correlated in time or space.

5.3.2 Basic Approach To Composite Sampling The basic approach to composite sampling involves the following steps:

° Identify the boundaries of the waste or unit. The boundaries may be spatial, temporal, or based on different components or strata in the waste (such as battery casings and soil)

° Conduct sampling in accordance with the selected sampling design and collect a set of n x g individual samples where g is the number of individual samples used to form each composite and n is the number of such composites 2 By the Central Limit Theorem (CLT), we expect composite samples to generate normally distributed data. The CLT states that if a population is repeatedly sampled, the means of all the sampling events will tend to form a normal distribution, regardless of the shape of the underlying distribution.

67 A B C C B A B A C n   g = 9 individual field samples na nb nc n = 3 composite samples Decision Unit Boundary Subsamples analyzed xa xb xc Figure 18. A basic approach to composite sampling. The figure shows how composite sampling can be integrated into a simple random sampling design. Random samples with the same letter are randomly grouped into composite samples to obtain an estimate of the unit wide mean. ° Group either randomly or systematically the set of n x g individual samples into n composite samples and thoroughly mix and homogenize each composite sample ° Take one or more subsamples from each composite ° Analyze each subsample for the constituent( s) of concern.

The n composite samples can then be used to estimate the mean and variance (see Section 5.3.5) or identify "hot spots" in the waste (see Section 5.3.6).

5.3.3 Composite Sampling Designs Composite sampling can be implemented as part of a statistical sampling design, such as simple random sampling and systematic sampling. The choice of a sampling design to use with compositing will depend upon the study objectives.

<IP_ADDRESS> Simple Random Composite Sampling Figure 18 shows how composite sampling can be integrated into a simple random sampling design. In this figure, the decision unit could represent any waste or media about which a decision must be made (such as a block of contaminated soil at a SWMU). Randomly positioned field samples are randomly grouped together into composite samples. The set of composite samples can then be used to estimate the mean and the variance.

Because the compositing process is a mechanical way of averaging out variabilities in concentrations from location to location over a unit, the resulting concentration data should tend to be more normally distributed than individual samples (Exner, et al. 1985). This is especially advantageous because the assumption of many statistical tests is that the underlying data exhibit an approximately normal distribution. 2 68 A B C D A B C D A B C D A B C D A B C D A B C D Decision Unit Boundary Figure 19. Systematic composite sampling across a unit or site. Samples with the same letter are pooled into composites.

A A A A B B B B C C C C D D D D E E E E F F F F Decision Unit Boundary Figure 20. Systematic sampling within grid blocks or intervals. Samples with the same letter are pooled into a composite sample. <IP_ADDRESS> Systematic Composite Sampling A systematic composite sampling design is shown in Figure 19. The design can be used to estimate the mean concentration because each composite sample is formed from field samples obtained across the entire unit. For example, each field sample collected at the "A" locations is pooled and mixed into one composite sample. The process is then repeated for the "B," "C," and "D" locations. The relative location of each individual field sample (such as "A") should be the same within each block.

This design is particularly advantageous because it is easy to implement and explain and it provides even coverage of the unit. Exner, et al. (1985) demonstrated how this design was used to make cleanup decisions for blocks of soil contaminated with tetrachlorodibenzo p dioxin.

A second type of systematic composite involves collecting and pooling samples from within grid blocks, time intervals, or batches of waste grouped together (see Figure 20).

If there is spatial correlation between the grid blocks, compositing within grids can be used to estimate block to block variability (Myers 1997) or improve the estimate of the mean within a block or interval (if multiple composite samples are collected within each block). In fact, compositing samples collected from localized areas is an effective means to control "short range" (small scale) heterogeneity (Pitard 1993). When this type of compositing is used on localized areas in lieu of "grab" sampling, it is an attractive option to improve representativeness of individual samples (Jenkins, et al. 1996).

Systematic sampling within time intervals could be used in cases in which compositing occurs as part of sample collection (such as sampling of liquid effluent with an autosampling device into a single sample container over a specified time period). 3 ASTM D 6051, Standard Guide for Composite Sampling and Field Subsampling for Environmental Waste Management Activities, also provides a procedure for estimating the precision of a single composite sample.

69 If the individual field sample locations are independent (that is, they have no temporal or spatial correlation), then compositing within blocks can be an efficient strategy for estimating the population mean. If the assumption of sample independence cannot be supported, then an alternative design should be selected if the objective is to estimate the mean.

5.3.4 Practical Considerations for Composite Sampling In creating composite samples from individual field samples, it is possible that a relatively large volume of material will need to be physically mixed at some point  either in the field or in the laboratory. Thorough mixing is especially important when the individual samples exhibit a high degree of heterogeneity.

Once the individual samples are mixed, one or more subsamples must be taken because the entire composite sample usually cannot be analyzed directly. A decision must be made as to where the individual samples will be combined into the composite samples. Because large samples (e. g., several kilograms or more) may pose increased difficulties to the field team for containerization and shipping and pose storage problems for the laboratory due to limited storage space, there may be a distinct advantage to performing mixing or homogenization in the field. There are, however, some disadvantages to forming the composite samples in the field. As pointed out by Mason (1992), the benefits of homogenization may be temporary because gravity induced segregation can occur during shipment of the samples. Unless homogenization (mixing), particle size reduction, and subsampling are carried out immediately prior to analysis, the benefits of these actions may be lost. Therefore, if practical, it may be best to leave the mixing and subsampling operations to laboratory personnel.

See Section 7.3 of this document and ASTM standards D 6051 and D 6323 for guidance on homogenization, particle size reduction, and subsampling.

5.3.5 Using Composite Sampling To Obtain a More Precise Estimate of the Mean When analytical error is minor compared to sampling error, then composite sampling can be a resource efficient mechanism for increasing the precision of estimates of the population mean. If composite sampling is to be used to estimate the mean with a specified level of confidence, then multiple composite samples can be used to estimate the mean and variance. Alternately, confidence limits can be constructed around the sample analysis result for a single composite sample if an estimate of the variance of the fundamental error is available (see Gy 1998, page 73). 3 See Section <IP_ADDRESS> for a discussion of fundamental error.

The population mean ( ) can be estimated from the analysis of composite samples (each µ n made from individual samples). The population mean ( ) is estimated by the sample mean g µ () by x x n xi i n = =   1 1 Equation 6 70 The sample variance ( ) can then be calculated by s 2 s n x x i i n 2 1 2 1 1 =     =   () Equation 7 Note that Equations 6 and 7 are the same as Equations 1 and 2, respectively, for the mean and variance. When the equations are used for composite sampling, is the measurement value xi from a subsample taken from each composite sample rather than each individual sample. n Use of these equations assumes equal numbers of individual field samples ( ) are used to g form each composite, and equal numbers of subsamples are taken from each composite sample and analyzed. If these assumptions are not correct, an alternative approach described in Gilbert (1987, page 79) can be used.

By increasing the number of individual field samples ( ) per composite sample, there will be a g corresponding decrease in the standard error ( ), thus improving the precision of the estimate sx of the mean. Edland and van Belle (1994) show that by doubling the number of individual samples per composite (or laboratory) sample, the expected size of the confidence interval around the mean decreases by a factor of , which is a 29 percent decrease in the 1 2 / expected width of the confidence interval. One of the key assumptions underlying the above discussion is that variances between the samples greatly exceed the random error variance of the analytical method (Garner, et al. 1988).

Williams, et al. (1989) demonstrated the benefits of using composite sampling to obtain a more precise estimate of the mean. One of their objectives was to study the efficiency of using composite sampling as compared to collecting individual samples for the purpose of estimating the mean concentration at a site. Five sites known to have radium contamination in shallow soils were extensively sampled. At each site, shallow soil samples were collected at approximately uniformly spaced points over the entire site. Three types of samples were taken: (1) individual 500 gram samples, (2) composite samples consisting of ten 50 gram aliquots uniformly spaced over the site, and (3) composite samples consisting of twenty 25 gram aliquots uniformly spaced over the site. The samples were measured for 226 Ra. The results indicated the individual samples yielded the least precision, even when more than twice as many individual samples were collected. Sixty six individual samples produced a standard error of 1.35, while the thirty 10 aliquot composites and the thirty 20 aliquot composite samples produced standard errors of 0.76 and 0.51 respectively. The results demonstrate that composite sampling can produce more precise estimates of the mean with fewer analytical samples.

Box 7 provides an example of how a mean and variance can be estimated using composite sampling combined with systematic sampling. 71 5.3.6 Using Composite Sampling To Locate Extreme Values or "Hot Spots"

One disadvantage of composite sampling is the possibility that one or more of the individual samples making up the composite could be "hot" (exceed a fixed standard), but remain undetected due to dilution that results from the pooling process. If the sampling objective is to determine if any one or more individual samples is "hot," composite sampling can still be used. 1 n · g = 20 …..

2 One measurement taken on each composite sample 5 g = 4 n = 5 (composites) t1 t2 t3 t4 t5 t6 t7 t8 t17 t18 t19 t20 Sampling Point Waste Preparation Process Fuel Storage Tank Figure 21. Example of systematic composite sampling Box 7. Example of How To Estimate the Mean and Variance Using Systematic Composite Sampling (Assume Samples Are Independent)

Under 40 CFR 261.38, a generator of hazardous waste derived fuel is seeking an exclusion from the definition of solid and hazardous waste. To prepare the one time notice under 40 CFR 261.38( c), the generator requires information on the mean and variance of the concentrations of constituents of concern in the waste as generated. The generator elects to use composite samples to estimate the mean and variance of the nonvolatile constituents of concern.

Using a systematic sampling design, a composite sample is prepared by taking an individual (grab) sample at regular time intervals t1 through t4. The set of four grab samples are thoroughly mixed to form a composite, and one subsample is taken from each composite for analysis. The process is repeated until five composite samples are formed (see Figure 21). (Note: If the assumption of independent samples cannot be supported, then a simple random design should be used in which the 20 grab samples are randomly grouped to form the five composites).

The analytical results for one of the constituents of concern, in ppm, are summarized as follows for the composite samples (n1 through n5): 2.75, 3.71, 3.28, 1.95, and 5.10.

Using Equations 6 and 7 for the mean and variance of composite samples, the following results are obtained:

x n x ppm s n x x i i n i i n = = =

=     = + + + + = =

=     1 1679 5 3 36 1 1 1 4 0 3721 01225 0 0064 199 303 138 1 2 1 2 . .

() . . . ...

The standard error is obtained as follows:

s s n ppm x = = = 117 5 052 . . 72 A procedure for detecting hot spots using composite sampling is given below. The approach assumes the underlying distribution is normal and the composite samples were formed from equal sized individual samples.

Let be some "action level" or regulatory threshold that cannot be exceeded in an individual AL sample. Note that must be large relative to the quantitation limit for the constituent of AL concern. For a measurement from a composite sample formed from individual samples, xi g the following rules apply, assuming analytical and sampling error are negligible:

° If , then no single individual sample can be x AL g i < > AL ° If , then at least one must, and as many as all individual samples may, x AL i > be > AL ° If , then at least one of the individual samples must be . x AL g i > g > AL As a general rule, we can say that no more than individual samples can be . g x AL i   > AL If one or more of the composites are "hot" (i. e., ), then it might be desirable to go back > AL and analyze the individual samples used to form the composite. Consider saving splits of each individual field sampling so individual samples can be analyzed later, if needed.

If compositing is used to identify a hot spot, then the number of samples that make up the composite should be limited to avoid overall dilution below the analytical limit. It is possible for a composite sample to be diluted to a concentration below the quantitation limit if many of the individual samples have concentrations near zero and a single individual sample has a concentration just above the action level. Mason (1992) and Skalski and Thomas (1984) suggest the maximum number of identically sized individual samples ( ) that can be used to g form such a composite should not exceed the action level ( ) divided by the quantitation limit AL ( ). But the relationship of indicates that the theoretical maximum number of QL g ALQL   / samples to form a composite can be quite high, especially given a very low quantitation limit. As a practical matter, the number of individual samples used to form a composite should be kept to a minimum (usually between 2 and 10).

An example of the above procedure, provided in Box 8, demonstrates how a "hot" drum can be identified through the analysis of just nine samples (five composites plus four individual analyses), resulting in considerable savings in analytical costs over analysis of individual samples from each of the 20 drums. 73 5.4 Determining the Appropriate Number of Samples Needed To Estimate the Mean This section provides guidance for determining the appropriate number of samples ( ) needed n to estimate the mean. The procedures can be used when the objective is to calculate a confidence limit on the mean. If the objective is to estimate a percentile, see Section 5.5.

To calculate the appropriate number of samples, it is necessary to assemble existing data identified in DQO Step 3 (" Identify Inputs to the Decision") and Step 6 (" Specify Limits on Decision Errors"). If the parameter of interest is the mean, you can calculate using equations n presented in the following sections or by using EPA's DEFT software (USEPA 2001a). …..

One measurement taken on each composite sample Point of Waste Generation 1 2 5 Composite Samples Grab Samples Waste Figure 22. Composite sampling strategy for locating a "hot" drum Box 8. How To Locate a "Hot Spot" Using Composite Sampling  Hypothetical Example A secondary lead smelter produces a slag that under some operating conditions exhibits the Toxicity Characteristic (TC) for lead. At the point of generation, a grab sample of the slag is taken as the slag is placed in each drum. A composite sample is formed from the four grab samples representing a set of four drums per pallet. The process is repeated until five composite samples representing five sets of four drums (20 drums total) have been prepared (see Figure 22).

The generator needs to know if the waste in any single drum in a given set of four drums contains lead at a total concentration exceeding 100 ppm. If the waste in any single drum exceeds 100 ppm, then its maximum theoretical TCLP leachate concentration could exceed the regulatory limit of 5 mg/ L. Waste in drums exceeding 100 ppm total lead will be tested using the TCLP to determine if the total leachable lead equals or exceeds the TC regulatory limit.

The sample analysis results for total lead are measured as follows (in ppm) in composite samples n1 through n5 : 6, 9, 18, 20, and 45.

Using the approach for locating a "hot spot" in a composite sample, we observe that all of the composite samples except for n5 are less than or 100 ppm/ 4 (i. e., 25 AL g / ppm). The result for n5 (45 ppm) is greater than 25 ppm, indicating a potential exceedance of the TC regulatory level. A decision about the set of drums represented by n5 can be made as follows:

No more than individual samples can be , or no more than or 1 (round g x AL i   > AL () . 4 45 100 18 ppm ppm =

down) individual sample exceeds 100 ppm total lead.

We now know that it is possible that one of the four drums on the fifth palette exceeds 100 ppm, but we do not know which one. As a practical matter, analysis of all four of the individual samples should reveal the identity of the "hot" drum (if, indeed, one exists); however, the above process of elimination could be repeated on two new composite samples formed from samples taken from just the four drums in question. 4 One exception is when sequential sampling is used in which the number of samples is not fixed a priori; rather, the statistical test is performed after each round of sampling and analysis (see Section 5.2.5).

74 Alternative equations can be found in the statistical literature and guidance, including ASTM (Standard D 6311), Cochran (1977), Gilbert (1987), and USEPA (2000a, 2000b, and 2000d).

The equations presented here should yield the approximate minimum number of samples needed to estimate the mean within the precision and confidence levels established in the DQO Process; however, it is prudent to collect a somewhat greater number of samples than indicated by the equations. 4 This is recommended to protect against poor preliminary estimates of the mean and standard deviation, which could result in an underestimate of the appropriate number of samples to collect. For analytes with long holding times (e. g., 6 months), it may be possible to process and store extra samples appropriately until analysis of the initially identified samples is completed and it can be determined if analysis of the additional samples is warranted.

It is important to note that the sample size equations do not account for the number or type of control samples (or quality assessment samples) required to support the QC program associated with your project. Control samples may include blanks (e. g., trip, equipment, and laboratory), field duplicates, spikes, and other samples used throughout the data collection process. Refer to Chapter One of SW 846 for recommendations on the type and number of control samples needed to support your project. It is best to first determine how each type of control sample is to be used, then to determine the number of that type based on their use (van Ee, et al. 1990).

A key assumption for use of the sample size equations is that you have some prior estimate of the total study error, measured as the sample standard deviation ( ) or sample variance ( ). s s 2 Since total study error includes variability associated with the sampling and measurement methods (see Section 6), it is important to understand the relative contributions that sampling and analysis activities make to the overall estimate of variability. Lack of prior information regarding population and measurement variability is one of the most frequently encountered difficulties in sampling. It quickly resembles a "chicken and the egg" question for investigators – you need an estimate of the standard deviation to calculate how many samples you need, yet you cannot derive that estimate without any samples. To resolve this seemingly paradoxical question, two options are available:

Option 1. Conduct a pilot study. A pilot study (sometimes called an exploratory or preliminary study) is the preferred method for obtaining estimates of the mean and standard deviation, as well as other relevant information. The pilot study is simply phase one of a multi phase sampling effort (Barth, et al. 1989). For some pilot studies, a relatively small number of samples (e. g., four or five or more) may provide a suitable preliminary estimate of the standard deviation.

Option 2. Use data from a study of a similar site or waste stream. In some cases, you might be able to use sampling and analysis data from another facility or similar operation that generates the same waste stream and uses the same process.

If neither of the above options can provide a suitable estimate of the standard deviation ( ), a s crude approximation of still can be obtained using the following approach adopted from s 75 USEPA 1989a (page 6 6). The approximation is based on the judgment of a person knowledgeable of the waste and his or her estimate of the range within which constituent concentrations are likely to fall. Given a range of constituent concentrations in a waste, but lacking the individual data points, an approximate value for may be computed by dividing the s range (the estimated maximum concentration minus the minimum concentration) by 6, or . This approximation method should be used only if no other alternative is s Range   /6 available. The approach is based on the assumption that more than 99 percent of all normally distributed measurements will fall within three standard deviations of the mean; therefore, the length of this interval is . 6s 5.4.1 Number of Samples to Estimate the Mean: Simple Random Sampling In Step 6 of the DQO Process (" Specify Limits on Decision Errors"), you established the width of the gray region ( ) and acceptable probabilities for making a decision error ( and ).       Using this information, along with an estimate of the standard deviation ( ), calculate the s appropriate number of samples ( ) for simple random sampling using n n z zsz = + +       () 1 1 2 2 2 1 2 2        Equation 8 where = the quantile of the standard normal distribution (from the last row of z1      pth Table G 1, Appendix G), where is the probability of making a Type I   set in DQO Step 6 (Section 4.6.4). = the quantile of the standard normal distribution (from the last row of z1      pth Table G 1, Appendix G), where is the probability of making a Type II   error set in DQO Step 6 (Section 4.6.4). = an estimate of the standard deviation. s = the width of the gray region from DQO Step 6.   An example application of Equation 8 is presented in Box 9.

Two assumptions underlie the use of Equation 8. First, it is assumed that data are drawn from an approximately normal distribution. Second, it is assumed the data are uncorrelated. In correlated data, two or more samples taken close to each other (in time or in space) will have similar concentrations (Gilbert 1987). In situations in which spatial or temporal correlation is expected, some form of systematic sampling is preferred.

If the underlying population appears to exhibit a lognormal distribution, normal theory sample size equations (such as Equation 8) still can be used though they will tend to underestimate the minimum number of samples when the geometric standard deviation ( ) is low (e. g., exp( ) sy 2). If the underlying distribution is known to be lognormal, the method given by Land (1971,   1975) and Gilbert (1987) for calculating confidence limits for a lognormal mean can be solved "in reverse" to obtain . (A software tool for performing the calculation, MTCAStat 3.0, is n published by the Washington Department of Ecology. See Appendix H). Also, techniques described by Perez and Lefante (1996 and 1997) can be used to estimate the sample sizes needed to estimate the mean of a lognormal distribution. Otherwise, consult a professional statistician for assistance. 76 Box 9. Number of Samples Required to Estimate the Mean Using Simple Random Sampling: Hypothetical Example Under 40 CFR 261.38, a generator of hazardous waste derived fuel is seeking an exclusion from the definition of solid and hazardous waste. To prepare the one time notice under 40 CFR 261.38( c), the generator plans to conduct waste sampling and analysis to support the exclusion. The output of the first six steps of the DQO Process are summarized below:

Step 1: State the Problem: The planning team reviewed the applicable regulations, historical analyses, and process chemistry information. The problem is to determine whether Appendix VIII constituents present in the waste are at concentration levels less than those specified in Table 1 of §261.38.

Step 2: Identify the Decision: If the waste attains the specification levels, then it will be judged eligible for the exclusion from the definition of hazardous and solid waste.

Step 3: Identify Inputs to the Decision: Sample analysis results are required for a large number of constituents present in the waste, however, most constituents are believed to be present at concentrations well below the specification levels. Historically, benzene concentrations have been most variable, therefore, the planning team will estimate the number of samples required to determine if the specification level for benzene is attained.

Step 4: Define the Boundaries: The DQO decision unit is defined as the batch of waste generated over a one week period. Samples will be taken as the waste exits the preparation process and prior to storage in a fuel tank (i. e., at the point of generation).

Step 5: Develop a Decision Rule: The RCRA regulations at 40 CFR 261.38( c)( 8)( iii)( A) specify the mean as the parameter of interest. The "Action Level" for benzene is specified in Table 1 of §268.38 as 4,100 ppm. If the mean concentration of benzene within the DQO decision unit is less than or equal to 4,100 ppm, then the waste will be considered eligible for the exclusion (for benzene). Otherwise, the waste will not be eligible for the exclusion for benzene. (Note that the demonstration must be made for all Appendix VIII constituents known to be present in the waste).

Step 6: Specify Limits on Decision Errors: In the interest of being protective of the environment, the null hypothesis was established as "the mean concentration of benzene within the decision unit boundary exceeds 4,100 ppm," or Ho: mean (benzene) > 4,100 ppm. The boundaries of the gray region were set at the Action Level (4,100 ppm) and at a value less than the Action Level at 3000 ppm. The regulations at §261.38( c)( 8)( iii)( A) specify a Type I (false rejection) error rate ( ) of 0.05. The regulations do not specify a Type II (false acceptance) error rate ( ),     but the planning team deemed a false acceptance as of lesser concern than a false rejection, and set the false acceptance rate at 0.25. Sample analysis results from previous sampling and analyses indicate the standard deviation ( ) of benzene concentrations is about 1,200 ppm. s What is the appropriate number of samples to collect and analyze for a simple random sampling design?

Solution: Using Equation 8 and the outputs of the first six steps of the DQO Process, the number of samples is determined as:

n z zsz = + +

= + =     ()

(. () (. .( 1 1  2 2 2 1  2 2 2 2 2 2 1645+ 0.674) (1200) 4100  3000 1645) 2 7 75 8        round up) where the values for and are obtained from the last row of Table G 1 in Appendix G. z1      z1      77 x Wx st h h h L = =   1 5.4.2 Number of Samples to Estimate the Mean: Stratified Random Sampling An important aspect of a stratified random sampling plan is deciding how many samples to collect within each of the strata (Gilbert 1987). There are many ways to design a stratified random sampling plan; the development here makes the following assumptions (refer to Section 5.2.2 for a description of terms and symbols used below):

° Weights for each stratum ( ) are known in advance. One possible way to Wh assign weights to each stratum is to calculate the ratio between the waste volume classified as the stratum and the total waste volume. hth ° The number of possible sample units (i. e., physical samples) of a certain physical size is much larger than the number of sample units that will be collected and analyzed. As a general guide, this assumption should be reasonable as long as the ratio between the stratum waste volume and the volume of the individual samples is at least 100. Otherwise, you may need to consider formulas that include the finite population correction (see Cochran 1977, page 24).

° The number of sample units to be collected and analyzed in each stratum, due to analytical costs and other considerations, generally will be fairly small.

° A preliminary estimate of variability ( ) is available for each stratum. If this is sh 2 not the case, one can use an estimate of the overall variability ( ) as a s 2 substitute for the separate stratum estimates. By ignoring possible differences in the variance characteristics of separate strata, the sample size formulas given below may tend to underestimate the necessary number of samples for each strata ( ). nh Given a set of stratum weights and sample measurements in each stratum, the overall mean ( ) and overall standard error of the mean ( ) (i. e., for the entire waste under study) are xst sxst computed as follows for a stratified random sample:

Equation 9 and s W s n x h h L h h st = =   2 1 2 Equation 10 Note that and in these formulas represent the arithmetic mean and sample variance for xh sh 2 the measurements taken within each stratum.

In general, there are two approaches for determining the number of samples to take when stratified random sampling is used: optimal allocation and proportional allocation. 78 <IP_ADDRESS> Optimal Allocation In optimal allocation, the number of samples assigned to a stratum ( ) is proportional to the nh relative variability within each stratum and the relative cost of obtaining samples from each stratum. The number of samples can be determined to minimize the variance for a fixed cost or to minimize the cost for a prespecified variance.

Optimal allocation requires considerable advance knowledge about the relative variability within each stratum and the costs associated with obtaining samples from each stratum; therefore, we recommend the use of proportional allocation (see below) as an alternative. For more complex situations in which optimal allocation is preferred, consult a statistician or see Cochran (1977, page 96), Gilbert (1987, page 50), or USEPA (1989a (page 6 13)).

<IP_ADDRESS> Proportional Allocation In proportional allocation, the number of samples assigned to a stratum ( ) is proportional to nh the stratum size, that is, . To determine the total number of samples ( ) so that a n nW h h = n true difference ( ) between the mean waste concentration and the Action Level can be   detected with Type I error rate and Type II error rate , use the following equation:     n t t W s df df h h h L = +     =   1 1 2 2 2 1    ,,

  Equation 11 To use this formula correctly, the degrees of freedom ( ) connected with each quantile df t (from Table G 1, Appendix G) in the above equation must be computed as follows:

df W s W s nW h h L h h h h L =               = =     2 1 2 2 4 1 1 Equation 12 Because the degrees of freedom also depend on n, the final number of samples must be computed iteratively. Then, once the final total number of samples is computed, the number of samples for each stratum is determined by multiplying the total number of samples by the stratum weight. An example of this approach is presented in Box 10.

If only an overall estimate of is available in the preliminary data, Equation 11 reduces to: s 2 n t ts df df = +     1 1 2 2 2    ,,

  Equation 13 and Equation 12 reduces to df W nW h h h L =   =   1 1 2 1 Equation 14 79 Box 10. Number of Samples Required to Estimate the Mean Using Stratified Random Sampling – Proportional Allocation: Hypothetical Example Under the RCRA Corrective Action program, a facility owner has conducted a cleanup of a solid waste management unit (SWMU) in which the contaminant of concern is benzene. The cleanup involved removal of all waste residues, contaminated subsoils, and structures. The facility owner needs to conduct sampling and analysis to confirm that the remaining soils comply with the cleanup standard.

Step 1: State the Problem: The planning team needs to confirm that soils remaining in place contain benzene at concentrations below the risk based levels established by the authorized state as part of the cleanup.

Step 2: Identify the Decision: If the soils attain the cleanup standard, then the land will be used for industrial purposes. Otherwise, additional soil removal will be required.

Step 3: Identify Inputs to the Decision: A sampling program will be conducted, and sample analysis results for benzene will be used to make the cleanup attainment determination.

Step 4: Define the Boundaries: The DQO decision unit is the top 6 inches of soil within the boundary of the SWMU. Based on prior sample analysis results and field observations, two strata are identified: fine grained soils in 20 percent of the unit (" Stratum 1"), and coarse grained soils comprising the other 80 percent of the unit (" Stratum 2"). Based on the relative mass of the two strata, a weighting factor is assigned to each stratum such that Wh hth and . W1 02 = . W2 08 = .

Step 5: Develop a Decision Rule: The parameter of interest is established as the mean, and the Action Level for benzene is set at 1.5 mg/ kg. If the mean concentration of benzene within the DQO decision unit is less than or equal to 1.5 mg/ kg, then the unit will be considered "clean." Otherwise, another layer of soil will be removed.

Step 6: Specify Limits on Decision Errors: In the interest of being protective of the environment, the null hypothesis is established as "the mean concentration of benzene within the decision unit boundary exceeds 1.5 mg/ kg," or Ho: mean (benzene) > 1.5 mg/ kg. The boundaries of the gray region are set at the Action Level (1.5 mg/ kg) and at a value less than the Action Level at 1.0 mg/ kg. The Type I error rate ( ) is set at 0.10 and the Type   II error rate ( ) is set at 0.25. Sample analysis results from initial non composite samples provided an   n = 8 estimate of the overall standard deviation of , and the standard deviations ( ) within each stratum of s = 183 . sh hth and (and and ). s1 25 = . s2 13 = . s1 2 625 = . s2 2 169 = .

What is the appropriate number of samples to collect and analyze for a stratified random sampling design?

Solution: Using Equation 12 for the degrees of freedom under proportional allocation:

( ) ( ) ( ) ( ) ( ) df1 2 2 2 02 625 08 169 02 625 8 02 1 08 169 8 08 1 2 3 2 = × + × ×   + ×                   =   (. . ) (. . ) .. . .. . .

Then, looking up the t quantiles (from Table G 1, Appendix G) with 2 degree of freedom and taking (i. e.,   = 05 . 1.5 ppm  1.0 ppm), the total sample size (using Equation 12) works out to [ ] ( ) ( ) n1 2 2 1886 0816 05 02 625 169 76 = + × + 08× = ..

. (. . ) (. . )

Since the equations must be solved iteratively, recompute the formulas using . The same calculations give n = 76 and . After two more iterations, the sample size stabilizes at . Using the proportional df 2 48 = n2 41 = n = 42 allocation with one should take 42( 0.2) = 8.4 (round up to 9) measurements from the first stratum and n = 42 42( 0.8) = 33.6 (round up to 34) measurements from the second stratum. Since four samples already were collected from each stratum, at least five additional random samples should be obtained from the first stratum and at least thirty additional random samples should be collected from the second stratum. 80 In the example in Box 10, stratified random sampling provides a more efficient and costeffective design compared to simple random sampling of the same unit. If simple random sampling were used, a total of 52 samples would be required. With stratified random sampling, only 42 samples are required, thereby reducing sampling and analytical costs.

5.4.3 Number of Samples to Estimate the Mean: Systematic Sampling Despite the attractiveness and ease of implementation of systematic sampling plans, whether via a fixed square, rectangular, or triangular grid, or through the use of systematic random sampling, methods for estimating the standard error of the mean are beyond the scope of this guidance (for example, see Cochran 1977) and often involve more advanced geostatistical techniques (for example, see Myers 1997). An alternate approach is to treat the set of systematic samples as though they were obtained using simple random sampling. Such an approach should provide reasonable results as long as there are no strong cyclical patterns, periodicities, or significant spatial correlations between adjacent sample locations. If such features are present or suspected to be present, consultation with a professional statistician is recommended.

By regarding the systematic sample as a simple random sample, one can simply use the algorithm and formulas for simple random sampling described in Section 5.4.1 (Equation 8) to estimate the necessary sample size. As with all the sampling designs described in this section, you should have a preliminary estimate of the sample variance before using the sample size equation.

5.4.4 Number of Samples to Estimate the Mean: Composite Sampling In comparison to noncomposite sampling, composite sampling may have the effect of minimizing between sample variation, thereby reducing somewhat the total number of composite samples that must be submitted for analysis.

The appropriate number of composite samples to be collected from a waste or media can be estimated by Equation 8 for simple random and systematic composite sampling. Equation 11 can be used when composite sampling will be implemented with a stratified random sampling design (using proportional allocation). Any preliminary or pilot study conducted to estimate the appropriate number of composite samples should be generated using the same compositing scheme planned for the confirmatory study. If the preliminary or pilot study data were generated using random "grab" samples rather than composites, then the sample variance ( ) in the s 2 sample size equations should be replaced with where is the number of individual or s g 2 g grab samples used to form each composite (Edland and Van Belle 1994, page 45).

Additional guidance on the optimal number of samples required for composite sampling and the number of subsample aliquots required to achieve maximum precision for a fixed cost can be found in Edland and van Belle (1994, page 36 and page 44), Exner, et al. (1985, page 512), and Gilbert (1987, page 78). 81 5.5 Determining the Appropriate Number of Samples to Estimate A Percentile or Proportion This section provides guidance for determining the appropriate number of samples ( ) needed n to estimate an upper percentile or proportion with a prespecified level of confidence. The approaches can be used when the objective is to determine whether the upper percentile is less than a concentration standard or whether a given proportion of the population or decision unit is less than a specified value.

Two methods for determining the appropriate number of samples are given below: (1) Section 5.5.1 provides a method based on the assumption that the population is large and the samples are drawn at random from the population, and (2) Section 5.5.2 provides a method with similar assumptions but only requires specification of the level of confidence required and the number of exceedances allowed (usually zero). For both methods, it is assumed that the measurements can be expressed as a binary variable – that is, that the sample analysis results can be interpreted as either in compliance with the applicable standard (" pass") or not in compliance with the applicable standard (" fail").

5.5.1 Number of Samples To Test a Proportion: Simple Random or Systematic Sampling This section provides a method for determining the appropriate number of samples when the objective is to test whether a proportion or percentile of a population complies with an applicable standard. A population proportion is the ratio of the number of elements of a population that has some specific characteristic to the total number of elements. A population percentile represents the percentage of elements of a population having values less than some value. The number of samples needed to test a proportion can be calculated using n z GR GR z AL AL =   +                       1 1 2 1 1    () ()   Equation 15 where = false rejection error rate   = false acceptance error rate   = the percentile of the standard normal distribution (from the last row of z p pth Table G 1 in Appendix G) = the Action Level (e. g., the proportion of all possible samples of a given AL support that must comply with the standard) = other bound of the gray region, GR = width of the gray region ( ), and   GR AL   = the number of samples. n An example calculation of using the approach described here is presented in Box 11. n 82 Box 11. Example Calculation of the Appropriate Number of Samples Needed To Test a Proportion – Simple Random or Systematic Sampling A facility is conducting a cleanup of soil contaminated with pentachlorophenol (PCP). Based on the results of a field test method, soil exceeding the risk based cleanup level of 10 mg/ kg total PCP will be excavated, classified as a solid or hazardous waste, and placed into roll off boxes for subsequent disposal, or treatment (if needed) and disposal. The outputs of the first six steps of the DQO Process are summarized below.

Step 1: State the Problem: The project team needs to decide whether the soil being placed in each roll off box is a RCRA hazardous or nonhazardous waste.

Step 2: Identify the Decision: If the excavated soil is hazardous, it will be treated to comply with the applicable LDR treatment standard and disposed as hazardous waste. If it is nonhazardous, then it will be disposed as solid waste in a permitted industrial waste landfill (as long as it is not mixed with a listed hazardous waste).

Step 3: Identify Inputs to the Decision: The team requires sample analysis results for TCLP PCP to determine compliance with the RCRA TC regulatory threshold of 100 mg/ L.

Step 4: Define the Boundaries: The DQO "decision unit" for each hazardous waste determination is defined as a roll off box of contaminated soil. The "support" of each sample is in part defined by SW 846 Method 1311 (TCLP) as a minimum mass of 100 grams with a maximum particle size of 9.5 mm. Samples will be obtained as the soil is excavated and placed in the roll off box (i. e., at the point of generation).

Step 5: Develop a Decision Rule: The project team wants to ensure with reasonable confidence that little or no portions of the soil in the roll off box are hazardous waste. The parameter of interest is then defined as the 90 th percentile. If the 90 th percentile concentration of PCP is less than 100 mg/ L TCLP, then the waste will be classified as nonhazardous. Otherwise, it will be considered hazardous.

Step 6: Specify Limits on Decision Errors: The team establishes the null hypothesis (Ho) as the "true proportion (P) of the waste that complies with the standard is less than 0.90," or Ho: P < 0.90. The false rejection error rate ( ) is   set at 0.10. The false acceptance error rate ( ) is set at 0.30. The Action Level ( ) is 0.90, and the other   AL boundary of the gray region ( ) is set at 0.99. GR How many samples are required?

Solution: Using Equation 15 and the outputs of the first six steps of the DQO Process, the number of samples ( ) n is determined as:

=   +                     =   0524 0991 099 1282 0901 090 099 090 235 24 2 . .( .) . .( .) .. .

where the values for and are obtained from the last row of Table G 1 in Appendix G. z1      z1      83 5.5.2 Number of Samples When Using a Simple Exceedance Rule If a simple exceedance rule is used (see Section <IP_ADDRESS>), then it is possible to estimate the number of samples required to achieve a prespecified level of confidence that a given fraction of the waste or site has a constituent concentration less than the standard or does not exhibit a characteristic or property of concern. The approach is based on the minimum sample size required to determine a nonparametric (distribution free) one sided confidence bound on a percentile (Hahn and Meeker 1991 and USEPA 1989a).

If the exceedance rule specifies no exceedance of the standard in any sample, then the number of samples that must achieve the standard can be obtained from Table G 3a in Appendix G. The table is based on the expression:

n = log( ) log( p)   Equation 16 where alpha ( ) is the probability of a Type I error and is the proportion of the waste or site   p that must comply with the standard. Alternatively, the equation can be rearranged so that statistical performance ) can determined for a fixed number of samples: (1      () 1   = 1    p n Equation 17 Notice that the method does not require specification of the other bound of the gray region, nor does it require specification of a Type II (false acceptance) error rate ( ).   If the decision rule allows one exceedance of the standard in a set of samples, then the number of samples required can be obtained from Table G 3b in Appendix G.

An example application of the above equations is presented in Box 12. See also Appendix F, Section F. 3.2.

Box 12. Example Calculation of Number of Samples Needed When a Simple Exceedance Rule Is Used – Simple Random or Systematic Sampling What is the minimum number of samples required (with no exceedance of the standard in any of the samples) to determine with at least 90 percent confidence ) that at least 90 percent of all possible samples from (1 090   =   . the waste (as defined by the DQO decision unit) are less than the applicable standard?

From Table G 3a, we find that for and that 22 samples are required. Alternately, using 1 090   = a . p = 090 . Equation 16, we find n = = =     =   log( ) log(p) . ) . ) .   log(010 log(090 1 0.0457 218 22 If only 11 samples were analyzed (with no exceedance of the standard in any of the samples), what level of confidence can we have that at least 90 percent of all possible samples are less than the standard? Using Equation 17, we find () . 1 11090 11 0.3138 0.6862   =   =   = 1  =   p n Rounding down, we can say with at least 68 percent confidence that at least 90 percent of all possible samples would be less than the applicable standard. 84 5.6 Selecting the Most Resource Effective Design If more than one sampling design option is under consideration, evaluate the various designs based on their cost and the ability to achieve the data quality and regulatory objectives. Choose the design that provides the best balance between the expected cost and the ability to meet the objectives. To improve the balance between meeting your cost objectives and achieving the DQOs, it might be necessary to modify either the budget or the DQOs. As can be seen from the sample size equations in Section 5.4 and 5.5, there is an interrelationship between the appropriate number of samples and the desired level of confidence, expected variability (both population and measurement variability), and the width of the gray region. To reduce costs (i. e., decrease the number of samples required), several options are available:

° Decrease the confidence level for the test ° Increase the width of the "gray region" (not recommended if the parameter of interest is near the Action Level)

° Divide the population into smaller less heterogeneous decision units, or use a stratified sampling design in which the population is broken down into parts that are internally less heterogeneous ° Employ composite sampling (if non volatile constituents are of interest and if allowed by the regulations).

Note that seemingly minor modifications to the sampling design using one or more of the above strategies may result in major increases or decreases in the number of samples needed.

When estimating costs, be sure to include the costs for labor, travel and lodging (if necessary), expendable items (such as personal protective gear, sample containers, preservatives, etc.), preparation of a health and safety plan, sample and equipment shipping, sample analysis, assessment, and reporting. Some sampling plans (such as composite sampling) may require fewer analyses and associated analytical costs, but might require more time to implement and not achieve the project objectives. EPA's Data Quality Objectives Decision Error Feasibility Trials Software (DEFT) (USEPA 2001a) is one tool available that makes the process of selecting the most resource effective design easier.