4.4.2 Extrapolating from DUs
As discussed in Section 4.2, the motivation for collecting replicate ISM samples within a DU is to obtain an estimate of the variance in the mean, from which a UCL can be calculated. When a site includes many DUs, it may be tempting to extrapolate the estimate of the variance (or the CV) from one DU to another. However, we must first consider the extent to which the distributions may be comparable across DUs. Two related questions about the distribution should be considered:
- Identically distributed: Does our knowledge of potential sources suggest that similar contaminant distributions can be expected at the spatial scales represented by each DU? In effect, we would like to be able to assume that the distributions are approximately the same.
- Normally distributed: Estimates of the means and SDs will vary by random chance across DUs even if the distributions are the same and the same number of increments are used. Is it preferable to extrapolate estimates of the SD or CV?
Both questions require that we understand factors that might influence the relationship between the mean and SD of ISM replicate results within a given DU. Statistical theory suggests that we can expect the estimated mean and SD to be independent for normal distributions but positively correlated for positively skewed distributions (Goovaerts 1997). If the ISM mean and variance estimates are independent, this notion presents a challenge because we would have no reason to assume that the ratio of the SD to the mean (as represented by the CV) is the same. DUs with relatively high estimated means may have low SDs and vice versa. Instead of extrapolating the average CV across DUs, we would introduce less uncertainty by extrapolating the average SD. By contrast, if the parameters are correlated because of some asymmetry in the distribution of mean concentrations (despite the CLT, as described in Section 4.2), then it would be preferable to extrapolate the average CV. A priori knowledge about the distribution shape is unlikely, and this source of uncertainty cannot be fully addressed through simulation studies. Therefore, one must be very cautious in how information is extrapolated between DUs and how an extrapolation may ultimately introduce decision errors.