# Statistics for Environmental Engineers

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Measurements must be taken at enough locations to test the measured world against the real world; that is, to check whether the conservation of mass has been satisfied. Redundancies in the set of data and the constraining relations are used to adjust the inconsistencies.

The example assumed that each measured value was equally reliable. This is not usually the case. Instead, the measurement errors might be proportional to the magnitude of the flow, or some measurements might have been repeated whereas others were done only once, or instruments (or personnel) with different precision might have been used at different locations. Such problem-specific information is included by using appropriate weights that could be estimated subjectively if replicate measurements are not available from which to estimate the variances.

In the case where N = M (no degrees of freedom), errors in the measured values will propagate into the calculated values. This is discussed in Chapter 49.

### References

Deming, W. E. (1943). Statistical Adjustment of Data, New York, John Wiley (Dover, Ed., 1964).

Madron, F (1992). Process Plant Performance: Measurement and Data Processing for Optimization and Retrofits, New York, Ellis Horwood.

Ripps, D. L. (1961). “Adjustment of Experimental Data,” Chem. Engr. Prog. Symp. Ser, 61(55), 8-13. Schellpfeffer, J. W. and P. M. Berthouex (1972). “Rational Adjustment of Imbalances in Plant Survey Data,” Proc. 29th Ind. Waste Conf., Purdue University.

Schrage, Linus (1999). Optimization Modeling with LINGO, Chicago, IL, Lindo Systems, Inc.

### Exercises

48.1    Flow Adjustment. Solve the flow adjustment problem assuming that flows 4 and 7 are measured four times more precisely than the other flows (i.e., w1 = 1, w2 = 1, w3 = 1, w4 = 4, w5 = 1, w6 = 1, and w7 = 4)

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