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Land, C. E. (1975). “Tables of Confidence Limits for Linear Functions of the Normal Mean and Variance,”
in Selected Tables in Mathematical Statistics, Vol. III, Am. Math. Soc., Providence, RI, 358–419.
Landwehr, J. M. (1978). “Some Properties of the Geometric Mean and its Use in Water Quality Standards,”
Water Resources Res., 14, 467–473.

Exercises

7.1 Plankton Counts. Transform the plankton data in Table 7.2 using a square root transformation
x = sqrt( y) and a logarithmic transformation x = log( y) and compare the results with those
shown in Figure 7.3.
7.2 Lead in Soil. Examine the distribution of the 36 measurements of lead (mg/kg) in soil and
recommend a transformation that will make the data nearly symmetrical and normal.

7.6 32 5 4.2 14 18 2.3 52 10 3.3 38 3.4 4.3 0.10 5.7 0.10 0.10 4.4
0.42 0.10 16 2.0 1.2 0.10 3.2 0.43 1.4 5.9 0.23 0.10 0.10 0.23 0.29 5.3 2.0 1.0

7.3 Box-Cox Transformation. Use the Box-Cox power function to find a suitable value of λ to
transform the 48 lead measurements given below. Note: All < MDL values were replaced by
0.05.

7.6 32 5.0 4.2 14 18 2.3 52 10 3.3 38 3.4 4.3 0.05 0.05 0.10
0.10 0.05 0.05 0.05 0.0 0.05 1.2 0.10 0.10 0.10 0.10 0.10 0.23 4.4 0.42 0.10
16. 2.0 2.0 1.0 3.2 0.43 1.4 0.10 5.9 0.10 0.10 0.23 0.29 5.3 5.7 0.10

7.4 Are Transformations Necessary? Which of the following are correct reasons for transforming
data? (a) Facilitate interpretation in a natural way. (b) Promote symmetry in a data sample.
(c) Promote constant variance in several sets of data. (d) Promote a straight-line relationship
between two variables. (e) Simplify the structure so that a simple additive model can help us
understand the data.
7.5 Power Transformations. Which of the following statements about power transformations are
correct? (a) The order of the data in the sample is preserved. (b) Medians are transformed to
medians, and quartiles are transformed to quartiles. (c) They are continuous functions. (d)
Points very close together in the raw data will be close together in the transformed data, at
least relative to the scale being used. (e) They are smooth functions. (f) They are elementary
functions so the calculations of re-expression are quick and easy.

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