Statistics for Environmental Engineers

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36


179


178


72


124


119


108


154


141


144


125


113

6.2 Wastewater Effluent TSS. The histogram shows one year’s total effluent suspended solids data (n = 365) for a wastewater treatment plant (data from Exercise 3.5). The average TSS concentration is 21.5 mg/L. (a) Assuming the plant performance will continue to follow this pattern, indicate on the histogram the upper 5% and upper 10% levels for out-of-control performance. (b) Calculate (approximately) the annual average effluent TSS concentration if the plant could eliminate all days with TSS > upper 10% level specified in (b).

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75


Final Effluent Total Susp. Solids



c


a)


з


C


a)


80 60 40 20 0

7

Using Transformations


KEY WORDS antilog, arcsin, bacterial counts, Box-Cox transformation, cadmium, confidence interval, geometric mean, transformations, linearization, logarithm, nonconstant variance, plankton counts, power function, reciprocal, square root, variance stabilization.


There is usually no scientific reason why we should insist on analyzing data in their original scale of measurement. Instead of doing our analysis on y it may be more appropriate to look at log(y), Jy, 1 /y, or some other function of y. These re-expressions of y are called transformations. Properly used transformations eliminate distortions and give each observation equal power to inform.


Making a transformation is not cheating. It is a common scientific practice for presenting and interpreting data. A pH meter reads in logarithmic units, pH = -log10 [ H+] and not in hydrogen ion concentration units. The instrument makes a data transformation that we accept as natural. Light absorbency is measured on a logarithmic scale by a spectrophotometer and converted to a concentration with the aid of a calibration curve. The calibration curve makes a transformation that is accepted without hesitation. If we are dealing with bacterial counts, N, we think just as well in terms of log(N) as N itself.

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