The UCL and LCL for non-detect data are given by the following:
\( UCL= \hat \mu + C \times SE, LCL=max(0, \hat \mu-C \times SE)\)
where \( \hat \mu \) is the Product Limit Estimator (PLE) estimate of the mean, \( SE \) is the standard error of the mean, and \( C \) is a critical value. The max operator sets the LCL to 0 if \( \hat \mu -C \times SE \) is negative. The critical value, \( C \), can be a quantile from the standard normal distribution or the \( t \) distribution, or it can be obtained from the Chebyshev inequality. As a conservative measure, only the critical values for the \( t \) distribution and the Chebyshev inequality are implemented in VSP.
For the UCL and LCL based on the \( t \) distribution, \( C=t_{1-\alpha,n_d -1} \) , the 100(1- \( \alpha \)) percentile of the \( t \) distribution with \(n_d -1\) degrees of freedom, where \( n_d \) is the total number of detects in the data. (Gilbert 1987, Table A2, page 255). For the UCL and LCL based on the Chebyshev inequality, \( C= \sqrt{ \frac{1}{\alpha} -1} \) .
UCLs based on the Chebyshev inequality often achieve confidence levels higher than 100(1- \( \alpha \))%. For this reason, they are conservative and are recommended for data that are asymmetric (skewed). (Symmetry can be informally ascertained using a box and whiskers plot). Note that the critical value for the Chebyshev inequality is the same no matter the sample size. Consequently, for large data sets ( \(n_d \) > 30), we recommend not using the Chebyshev inequality, as it may be overly conservative.
UCLs based on the \( t \) distribution are most likely to achieve 100(1- \( \alpha \))% confidence when the data are symmetric and/or when data sets are large ( \( n_d \) > 30). Additional details regarding the calculation of UCLs for non-detect data are given by Singh (2006).
Gilbert, R.O. 1987. Statistical Methods for Environmental Pollution Monitoring. John Wiley & Sons, Inc. New York, NY.
Singh, A., R. Maichle, and S.E. Lee. 2006. On the Computation of a 95% Upper Confidence Limit of the Unknown Population Mean Based Upon Data Sets with Below Detection Limit Observations. Prepared for the EPA. Report # EPA/600/R-06/022.
http://www.epa.gov/esd/tsc/images/EPA%20600%20R-06%20022.pdf