# 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 IT, X = 1 о II о 1 II II <4 -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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