Statistics for Environmental Engineers

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Core butts


1


0.054


0.051


0.0525


2


0.050


0.050


0.0500


3


0.047


0.050


0.0485


4


0.092


0.091


0.0915


0.0606


System sand


1


0.052


0.050


0.0510


2


0.084


0.080


0.0820


3


0.044


0.041


0.0425


4


0.050


0.044


0.0470


0.0556

Source: Krueger, R. C. (1985). Characterization of Three Types of Foundry Waste and Estimation of Variance Components, Research Report, Madison, WI, Department of Civil and Environmental Engineering, The University of Wisconsin-Madison.


Batch 1    Batch 2    Batch 3


j1234 12341234


I .    .    .    . I .    _    I


Specimen Test


FIGURE 25.2 Nested design to estimate components of variance.

Variance Components Analysis


Variance components analysis is a method for learning what fraction of the total variance in a measurement process is caused by different components (factors) that contribute random variation into the sampling and testing process. If we have n measurements, denoted by yb i = 1,…,n, the sample variance for the entire data set is:


Ifa-y)2


n — 1


One design that allows the variance of each of these factors to be estimated independently of the other factors is the nested (or hierarchical) design shown in Figure 25.2.

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