Statistics for Environmental Engineers

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Error Suppression and Magnification

A nonlinear function can either suppress or magnify error in measured quantities. This is especially true of the quadratic, cubic, and exponential functions that are used to calculate areas, volumes, and reaction rates in environmental engineering work. Figure 10.1 shows that the variance in the final result depends on the variance and the level of the inputs, according to the slope of the curve in the range of interest.

Example 10.5

Particle diameters are to be measured and used to calculate particle volumes. Assuming that the particles are spheres, V = nD3/6, the variance of the volume is:

Var( V) = (^D2) d = 2.467D4dD


d = 1.571 D1 aD

The precision of the estimated volumes will depend upon the measured diameter of the particles. Suppose that aD = 0.02 for all diameters of interest in a particular application. Table 10.2 shows the relation between the diameter and variance of the computed volumes.

At D = 0.798, the variance and standard deviation of volume equal those of the diameter. For small D (<0.798), errors are suppressed. For larger diameters, errors in D are magnified. The distribution of V will be stretched or compressed according to the slope of the curve that covers the range of values of D.

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