Statistics for Environmental Engineers

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The two bottom rows of Table 7.6 give the mean and the variance of the power-transformed values. The suitable transformations give small variances. Rather than pick the smallest value from the table, make a plot (Figure 7.4) that shows how the variance changes with X. The smooth curve is drawn as a reminder that these variances are estimates and that small differences between them should not be taken seriously. Do not seek an optimal value of X that minimizes the variance. Such a value is likely to be awkward, like X = 0.23. The data do not justify such detail, especially because the censored values (y < 0.01) were arbitrarily replaced with 0.005. (This inflates the variance from whatever it would be if the censored values were known.) Values of X = —0.5, X = 0, or X = 0.5 are almost equally effective transformations. Any of these will be better than no transformation (X = 1). The log transformation (X = 0) is very satisfactory and is our choice as a matter of convenience.


Figure 7.5 shows dot diagrams for the original data, the square root (X = 0.5), the logarithms (X = 0), and reciprocal square root (X = —0.5). The log transformation is most symmetric, but it is not normal because of the 11 non-detect data that were replaced with 0.005 (i.e., 1/2 the MDL).


:    .    :    :    я=-о.б


I—^• т ■————I 1/Vу 2    4    6    8    10    12    14    16


:    :    Я=0


•    М И* ••••••••    •    у


0.00 ‘ 0.02 ‘ 0.0^    0.0^    0.0^ 0.10


:    :_:    Я= 0.5


I-,-.-т=-.-, V у

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