# 4.2.2.1 UCL of the mean based on Student's-t distribution

The following equation is used to calculate the one-sided (1–α) 100% UCL using the Student's-t approach:

where

= arithmetic mean of all ISM samples | |

= standard deviation of all ISM samples | |

r |
= number of ISM samples |

t |
= (1–α)^{th} quantile of the Student’s-t distribution with (r–1) degrees of freedom |

The Student’s-*t *UCL is expected to provide valid 95% UCL values when the distribution of means is approximately normal. The central limit theorem (CLT, Casella and Berger 2001) provides support for the use of a Student’s-*t* UCL for composite sampling as well as ISM sampling. The CLT is useful because it defines the distribution of the mean of the samples without having to know the exact underlying distribution of the data. The number of samples, *n*, and the shape of the distribution of the data are the two factors that most influence the accuracy of the approximation of the distribution of the mean. For approximately symmetric or slightly skewed distributions, a relatively small number of samples (e.g., n = 15) may be sufficient for the estimates of the mean to be approximately normally distributed as theorized by the CLT. If the population distribution is moderately skewed, a larger number of samples (e.g., n ≥ 30) is required to reliably invoke the CLT (Casella and Berger 2001). More highly skewed distributions require even larger numbers of samples. When the distribution of replicate samples is right-skewed instead of normal, the consequence of using the Student’s-*t* UCL is that it will underestimate the true mean more often than desired.

In ISM sampling, the coverage of the Student’s-*t *UCL also depends on the SD of the ISM replicates. The influence of the combination of factors for different sampling regimes can be difficult to anticipate. The simulation results in Section 4.3 demonstrate various performance metrics associated with the use of the Student’s-*t* distribution for a wide range of plausible scenarios.