UCL of the mean based on Chebyshev inequality

The following equation is used to calculate the one-sided (1–α) 100% UCL using the Chebyshev approach:


= arithmetic mean of all ISM samples
= standard deviation of all ISM samples
r = number of ISM samples

The Chebyshev is generally considered to be a conservative estimate of the UCL because it generally achieves or exceeds the desired coverage rates, even for nonnormal distributions. However, with small numbers of samples, the estimates of the mean and SD can be unstable, and the coverage is not guaranteed, especially when the contaminant distribution in the DU is highly heterogeneous. Each simulation discussed in Section 4.3 includes performance metrics for both the Student's-t and Chebyshev UCLs to illustrate conditions in which each may be favored.