Statistics for Environmental Engineers

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LSI = 0.25 ± 2(0.18) -0.11 < LSI < 0.61 RSI = 6.5 ± 2(0.32)    5.86 < RSI < 7.14


These ranges may seem surprisingly large given the reasonably accurate pH measurements and pHs estimates. Both indices will falsely indicate scaling or corrosive tendencies in roughly one out of ten calculations even when the water quality is exactly on target. A water utility that had this much variation in calculated values would find it difficult to tell whether water is scaling, stable, or corrosive until after many measurements have been made. Of course, in practice, real variations in water chemistry add to the “analytical uncertainty” we have just estimated.


In the example, we used a standard deviation of 0.15 pH units for pHs. Let us apply the same error propagation technique to see whether this was reasonable. To keep the calculations simple, assume that A, Ks, K2, and are known exactly (in reality, they are not). Then:


Var(pHs) = (log10e)2([Ca]-2Var[Ca] + [Alk]-2 Var[Alk]}


The variance of pHs depends on the level of the calcium and alkalinity as well as on their variances. Assuming [Ca] = 36 mg/L, o[Ca] = 3 mg/L, [Alk] = 50 mg/L, and 0[Alk] = 3 mg/L gives:


Var(pHs) = 0.1886([36]-2(3)2 + [50]-2(3)2} = 0.002


which converts to a standard deviation of 0.045, much smaller than the value used in the earlier example. Using this estimate of Var(pHs) gives approximate 95% confidence intervals of:


0.03 < LSI < 0.47


6.23 < RSI < 6.77

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