# 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.

Two UCL calculation methods were evaluated for use with ISM samples:
• Student's t UCL
• Chebyshev UCL