Statistics for Environmental Engineers

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0.0    0.1    0.2    0.3    0.4


■    •    •    Я= 1


:    . in (у)


-6    -5    -4    -3    -2


FIGURE 7.5 Dot diagrams of the data and the square root, log, and reciprocal square root transformed values. The eyecatching spike of 11 points are “non-detects” that were arbitrarily assigned values of 0.005 mg/kg.

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Transformations are not tricks to reduce variation or to convert a complicated nonlinear model into a simple linear form. There are often statistical justifications for making transformations and then analyzing the transformed data. They may be needed to stabilize the variance or to make the distribution of the errors normal. The most common and important use is to stabilize (make uniform) the variance.


It can be tempting to use a transformation to make a nonlinear function linear so that it can be fitted using simple linear regression methods. Sometimes the transformation that gives a linear model will coincidentally produce uniform variance. Beware, however, because linearization can also produce the opposite effect of making constant variance become nonconstant (see Chapter 45).


When the analysis has been done on transformed data, the analyst must consider carefully whether to report the final results on the transformed or the original scale of measurement. Confidence intervals that are symmetrical on the transformed scale will not be symmetric when transformed back to the original scale. Care must also be taken when converting the mean on the transformed scale back to the original scale. A simple back-transformation typically does not give an unbiased estimate, as was demonstrated in the case of the logarithmic transformation.

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