# A.4.3 Results Using Discrete Sampling

Table A-4 shows a few of the 2000 iterations from the UCL calculations based on using the mean and standard error calculated from nine systematic grid discrete samples (see upper left plot in Figure A-8) from a DU. These values represent absolute concentrations (e.g., mg/kg). The values from the UCL column are then compared to the true mean. A sampling design achieves the desired statistical coverage if, for example, the UCL values underestimate the true mean in fewer than 100 of the 2000 iterations (i.e., 5%). Figure A-9 shows a histogram of 2000 UCL values from one simulation scenario where the y-axis represents the percentage of 2000 in each bin (note that the y-axis is distorted to show the low bin counts). The red line identifies the location of the true mean. This UCL histogram shows that the coverage was only 76%, which is a significant departure from the theoretical design of 95%. The simulation results provide an example demonstrating how one of the performance metrics (coverage of the UCL) may indicate that an ISM sampling design is unlikely to yield reliable results.

Table A-4. Example of mean and 95% UCL calculations for each iteration of a simulation

Mean |
UCL |
---|---|

0.61 | 0.76 |

0.72 | 0.94 |

1.01 | 1.46 |

0.79 | 1.18 |

0.81 | 1.02 |

F**igure A-9. Histogram of the calculated UCL values using a simulated data set with 2000 iterations. ^{7}**

The discrete sampling examples were restricted to calculations using Student’s-*t* UCL and Chebyshev UCL. Other methods for UCL calculations are typically considered to attain appropriate coverage by implementing USEPA’s ProUCL or comparable software. For sites with heavy right-tailed distributions and spatial heterogeneity, discrete sampling methods with up to 100 samples taken are not sufficient to use a *t*-statistic to calculate a reliable UCL. However, the Chebyshev UCL does provide adequate coverage for many of the DUs at multiple sample sizes. Additional discrete sampling results are discussed in the subsequent sections.

^{7}For display purposes the y-axis is in terms of percentage of 2000 and is distorted (not evenly spaced between ticks) to highlight the low count bins. The red line identifies the true mean of 0.776.