# 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 1 = nbisnt y Batches SSb = n,n,Eb=i(Уь — y)2 nb — 1 MSb = ns що1 + UtO2 + ot nb — 1 Specimens SSs = nb ns 2 nb (ns — 1) SSs що) + ot ntLb=i±s= 1 (ybs — ys) MSs = —— Пь(Is — 1) Tests SSt = nnn nb ns(n, — 1) MS = SS« ot Xb=1 Xs=lXt=1( ybst — ybs) t nbns(nt — 1) Total II n, n n b s t 2 ^b = 1^s=1 ^t=1 ybst E- C/2 CO

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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