Statistics for Environmental Engineers

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Geo. mean


= 0.01195

Note: Concentrations in mg/kg.


Transformed Values, Means, and Variances as a Function of X


X =1


IT,


о


1


II


О


II


IT,


о


II


<4


X = 1


-0.0061


-0.0146


-0.0451


-0.1855


-0.9770


-0.0070


-0.0159


-0.0467


-0.1877


-0.9800


-0.0141


-0.0235


-0.0550


-0.1967


-0.9900


-0.0284


-0.0343


-0.0633


-0.2032


-0.9950


0.0108


-0.0203


-0.0519


-0.1937


-0.9870


yX


= -0.01496


-0.0228


-0.0529


-0.1930


-0.9839


Var( yX)


= 0.000093


0.000076


0.000078


0.000113


0.000255


0.3


о


о


о


x



я


я


>



о


я

-1.0 -0.5    0.0 0.5 1.0


X


0.2


0.1


0.0


FIGURE 7.4 Variance of the transformed cadmium data as a function of X.


values of X. The geometric mean of the untransformed data is yg = exp(-4.42723) = 0.01195. The denominator for X = 0.5, for example, is 0.5(0.01195) ‘    = 4.5744; the denominator for


X = -0.5 is -765.747.


Table 7.6 gives some of the power-transformed data for X = —1, -0.5, 0, 0.5, and 1. X = 1 is no transformation (= yt 1) except for scaling to be comparable to the other transformations. X = 0 is the log transformation, calculated from Y-0) = yg ln (yj), which is again scaled so the variance can be compared directly with variances of other power-transformed values.

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