Statistics for Environmental Engineers

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UCL,-a = exp (x + 0.5Sx2 + ^^-a)


n-1


LCLa = exp fx + 0.5s2 + -S^O v    7n — V


The quantities H1-a and Ha depend on sx, n, and the confidence level a. Land (1975) provides the necessary tables; a subset of these may be found in Gilbert (1987).


A power transformation model developed by Box and Cox (1964) can, so far as possible, satisfy the conditions of normality and constant variance simultaneously. The method is applicable for almost any kind of statistical model and any kind of transformation. The transformed value Y^X) of the original variable yt is:


X


Y


yt 1


Xy


X-1 g


where yg is the geometric mean of the original data series, and X expresses the power of the transformation. The geometric mean is obtained by averaging ln(y) and taking the exponential (antilog) of the result. The special case when X = 0 is the log transformation: Ya) = ygln(yt). X = -1 is a reciprocal transformation, X = 1/2 is a square root transformation, and X = 1 is no transformation. Example applications of this transformation are given in Box et al. (1978).


Example 7.5


Table 7.5 lists 36 measurements on cadmium (Cd) in soil, and their logarithms. The Cd concentrations range from 0.005 to 0.094 mg/kg. The limit of detection was 0.01. Values below this were arbitrarily replaced with 0.005. Comparisons must be made with other sets of similar data and some transformation is needed before this can be done. Experience with environmental data suggests that a log transformation may be useful, but something better might be discovered if we make the Box-Cox transformation for several values of X and compare the variances of the transformed data.

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