Statistics for Environmental Engineers

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These calculations can be organized as an analysis of variance table. Table 25.3 shows the algebra of the analysis of variance for the three-factor nested design discussed here. Some additional nomenclature


General Analysis of Variance Table for Estimating the Variance Components from a Three-Factor Nested Experimental Design


Source of


Degrees of


Mean Square


Variation


Sum of Squares


Freedom


Mean Square


Estimates


Average


SSave =


2


= nbisnt y


1


Batches


SSb =


n,n,Eb=i(Уь — y)2


nb 1


MSb =


nb — 1


ns що1 + UtO2 + ot


Specimens


SSs =


nb ns 2


ntLb=i±s= 1 (ybs — ys)


nb (ns 1)


SSs


MSs = ——


Пь(Is — 1)


що) + ot


Tests


SSt =


nnn


Xb=1 Xs=lXt=1( ybst — ybs)


nb ns(n, — 1)


MS = SS«


ot


t


nbns(nt 1)


Total


II


E-


C/2


CO


n, n n b s t^b = 1^s=1 ^t=1 ybst

FIGURE 25.3 Error components contributing to variance in the measured quantity.


is also given. In practice, the calculations usually are done by computer and many software packages have this capability.

Case Study: Sampling Waste Foundry Sand


The data in Table 25.2 showed that three batches of foundry solid waste (baghouse dust, core butts, system sand) were collected. Each batch was subdivided in the laboratory into four specimens, and each specimen was analyzed in duplicate. This gives nb = 3, ns = 4, and nt = 2, for a total of 3 x 4 x 2 = 24 observations.

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