image

The (1 - α)100 percent normal UCL (corrective action) for the mean of n measurements is computed as

image

When nondetects are present, several reasonable options are possible. If n < 8, nondetects are replaced by one-half of the detection limit (DL) since with fewer than eight measurements, more sophisticated statistical adjustments are typically not appropriate. Similarly, a normal UCL is typically used because seven or fewer samples are insufficient to confidently determine distributional form of the data. Because of a lognormal limit with small samples can result in extreme limit estimates, it is reasonable and conservative to default to normality for cases in which n < 8.

If n ≥ 8, a good choice is to use the method of Aitchison (1955) to adjust for nondetects and test for normality and lognormality of the data using the Shapiro-Wilk test. However, the ability of the Shapiro-Wilk test (and other distributional tests) to detect nonnormality is highly dependent on sample size. For most applications, 95 percent confidence is a reasonable choice. Note that alternatives such as the method of Cohen (1961) can be used; however, the DL must be constant.

LOGNORMAL CONFIDENCE LIMITS FOR THE MEDIAN

For a lognormally distributed constituent—that is, image (x) is distributed image—the (1 -α)100 percent LCL for the median or 50th percentile of the distribution is given by

image

where image and sy are the mean and standard deviation of the natural log transformed concentrations. Note that the exponentiated limit is, in fact, an LCL for the median and not the mean concentration. In general, the median and corresponding LCL will be low than the mean and its corresponding LCL. The (1 - α)100 percent UCL for the median 50th percentile of the distribution is given by

image



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