Statistics for Environmental Engineers

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The general propagation of error model that applies exactly to all linear models z = f(x1,x2,…,xn) and approximately to nonlinear models (provided the relative standard deviations of the measured variables are less than about 15%) is:

al ~ (dz/dxj)2of + (dz/dx2)2C2 + — + (dz/dxn)2ol

where the partial derivatives are evaluated at the expected value (or average) of the xi. This assumes that there is no correlation between the x’s. We shall look at this and some related ideas in Chapter 49.


Betz Laboratories (1980). Betz Handbook of Water Conditioning, 8th ed., Trevose, PA, Betz Laboratories. Langlier, W. F. (1936). “The Analytical Control of Anticorrosion in Water Treatment,” J. Am. Water Works Assoc., 28, 1500.

Miller, J. C. and J. N. Miller (1984). Statistics for Analytical Chemistry, Chichester, England, Ellis Horwood Ltd.

Ryznar, J. A. (1944). “A New Index for Determining the Amount of Calcium Carbonate Scale Formed by Water,” J. Am. Water Works Assoc., 36, 472.

Spencer, G. R. (1983). “Program for Cooling-Water Corrosion and Scaling,” Chem. Eng., Sept. 19, pp. 61-65.


10.1    Titration. A titration analysis has routinely been done with a titrant strength such that concentration is calculated from C = 20( y2 — y1), where (y2 — y1) is the difference between the final and initial burette readings. It is now proposed to change the titrant strength so that C = 40(y2 — y1). What effect will this have on the standard deviation of measured concentrations?

10.2 Flow Measurement. Two flows (Q1 = 7.5 and Q2 = 12.3) merge to form a larger flow. The standard deviation of measurement on flows 1 and 2 are 0.2 and 0.3, respectively. What is the standard deviation of the larger downstream flow? Does this standard deviation change when the upstream flows change?

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