Statistics for Environmental Engineers

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The time series plot gives a good impression about variability and randomness. The probability plot shows how frequently any particular value has occurred. The probability plot can be used to estimate the median value. If the median is above the MDL, draw a smooth curve through the plotted points and estimate the median directly. If the median is below the MDL, extrapolation will often be justified on the basis of experience with similar data sets. If the data are distributed normally, the median is also the arithmetic mean. If the distribution is lognormal, the median is the geometric mean.


The precision of the estimated mean and variances becomes progressively worse as the fraction of observations censored increases. Comparative studies (Gilliom and Helsel, 1986; Haas and Scheff, 1990; Newman et al., 1989) on simulated data show that Cohen’s method works quite well for up to 20% censoring. Of the methods studied, none was always superior, but Cohen’s was always one of the best. As the extent of censoring reaches 20 to 50%, the estimates suffer increased bias and variability.


Historical records of environmental data often consist of information combined from several different studies that may be censored at different detection limits. Older data may be censored at 1 mg/L while the most recent are censored at 10 ^g/L. Cohen (1963), Helsel and Cohen (1988), and NCASI (1995) provide methods for estimating the mean and variance of progressively censored data sets.


The Cohen method is easy to use for data that have a normal or lognormal distribution. Many sets of environmental samples are lognormal, at least approximately, and a log transformation can be used. Failing to transform the data when they are skewed causes serious bias in the estimates of the mean.

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