# 4.2.2 UCL Calculation Method

The concept of variability in estimates applies to UCLs as well as to the estimates of the means themselves. Several methods exist for calculating a UCL for estimates of the mean for a set of data. These methods often yield different answers for the same set of data. For example, if a 95% UCL is estimated for a population 100 times, the 95% UCL will, on average, be greater than or equal to the true mean of the population 95 times. The ability of different methods to produce a value that meets the definition of a 95% UCL depends in part on the number of samples used to estimate the mean, as well as the distribution (e.g., normal, lognormal, gamma) and dispersion of the data. One method might generate 95% UCLs greater than or equal to the true mean for a population 95% of the time, while another 95% UCL method might generate estimates greater than the true mean only 80% of the time. In the latter case, although a 95% UCL method was used, that method did not perform up to the specified level for that population. Had more samples been taken to estimate the mean or if the concentrations were distributed differently, the second method might have performed satisfactorily while the first method was deficient.

In practice, the true mean is unknown, but with simulation we can define the mean. Simulation studies help guide the selection of a UCL method based on simulation-specific information, assumptions, and decision error criteria. In practice, we cannot compare the performance of any UCL calculated at a site because the true mean within the DU is unknown. Similarly, there are no statistical calculations or diagnostics that can be used to compare the individual replicates or UCL to the unknown mean. These are limitations that apply to both discrete and ISM sampling. However, the likely performance of alternative UCL methods can be explored using simulation studies. Such studies have already been conducted by USEPA (2010b) to guide in the calculation of 95% UCLs for discrete sampling protocols. This type of performance evaluation has not been previously conducted for ISM sampling, so initial simulation studies were conducted in the development of this guidance, as summarized in Section 4.4.

- Student's t UCL
- Chebyshev UCL

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Three or more ISM samples are needed to calculate a 95% UCL. In theory, all of the UCL methods that are applied to discrete sampling results can also be applied to ISM. In practice, however, because fewer than eight replicate ISM samples are likely to be collected for a DU, fewer options are typically available to calculate a UCL compared with discrete sampling data. The small number of replicates precludes GOF evaluations as well as the use of methods that require more samples than typically collected in ISM sampling (USEPA 2010a). Therefore, the options for UCL calculations reduce to the set of methods that require only the parameter estimates themselves: mean and SD. Two candidate UCL equations that can accommodate ISM data sets and which are expected to "bracket" the range of UCLs that may be calculated from a data set are the Student's-t (representing the low end of the range) and Chebyshev (representing the high end of the range) UCLs as discussed in Section 4.2.2.1.

^{4} Note that throughout this document a UCL on a mean estimate is presented as 95% UCL. It is important to note that this is only an example of a UCL. It is possible to use a 90% UCL, 98% UCL, 99% UCL, etc. The specific UCL used should be determined by the project team during systematic planning.