Statistics for Environmental Engineers

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Relative Errors


The coefficient of variation (CV), also known as the relative standard deviation (RSD), is defined by siy . The CV or RSD, expressed as a decimal fraction or as a percent, is a relative error. A relative error implies a proportional error; that is, random errors that are proportional to the magnitude of the measured values. Errors of this kind are common in environmental data. Coliform bacterial counts are one example.

Example 9.1


Total coliform bacterial counts at two locations on the Mystic River were measured on triplicate water samples, with the results shown below. The variation in the bacterial density is large when the coliform count is large. This happens because the high density samples must be diluted before the laboratory bacterial count is done. The counts in the laboratory cultures from locations A and B are about the same, but the error is distorted when these counts are multiplied by the dilution factor. Whatever variation there may be in the counts of the diluted water samples is multiplied when these counts are multiplied by the dilution factor. The result is proportional errors: the higher the count, the larger the dilution factor, and the greater the magnification of error in the final result.


Location


A


B


Total coliform (cfu/100 mL)


13, 22, 14


1250, 1583, 1749


Averages


yA = 16.3


Ув = 1527


Standard deviation (s)


ON


II


sB = 254


Coefficient of variation (CV)


0.30


0.17

We leave this example with a note that the standard deviations will be nearly equal if the calculations are done with the logarithms of the counts. Doing the calculations on logarithms is equivalent to taking the geometric mean. Most water quality standards on coliforms recommend reporting the geometric mean. The geometric mean of a sample y1, y2,…,yn is yg = Jy1 x y2 x ••• x yn, or yg = antilog[1 Xlog(yt)].

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