# Statistics for Environmental Engineers

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The average of the 24 observations is y = 0.06673 mg/L. Table 25.2 also gives the averages for the duplicate measurements on each specimen, ybs, and the average of the eight measurements made on each batch of waste (yb).

Analysis of Variance Table for the Foundry Waste Example

 Source of Variation Sum of Squares df MS F Average 0.10693 1 Batches 0.00367 2 0.001835 367 Specimens 0.00624 9 0.000693 139 Tests 0.00006 12 0.000005 Total 0.11690 24

The sum of squares for each variance component is:

nb ns nt

Total SSt=    = 0.082 + 0.0842+•••+ 0.0442= 0.11690

b=1 s=1 t=1

Average SSave= nbnsn,y2 = 3 (4)(2)(0.06675)2= 0.106934

nb

Batch SSb = ^n^(yb — y)2

b=1

= (4)(2)[(0.0840 — 0.06675)2+ (0.0606 — 0.06675)2+ (0.0556 — 0.06675)2]

= 0.003671

nb ns

Specimen SSs = nt^^(ybs — Уь)2

b=1 s= 1

= 2[(0.083 — 0.0840)2+ (0.1085 — 0.0840)2+ ••• + (0.047 — 0.0556)2]

= 0.000624

nb ns nt

Test SSt = XXX(ybst — ybs)2

b=1 s=1 t=1

= (0.082 — 0.083)2+ (0.084 — 0.083)2+ •••+( 0.044 — 0.0470)2= 0.000057

Table 25.4 gives the full analysis of variance table, with sums of squares, degrees of freedom, and mean square values. The mean squares (the sums of squares divided by the respective degrees of freedom) are MSb= 0.001835, MSs= 0.000693, and MSt= 0.000005. The mean squares are used to estimate the variances. Table 25.3 shows that MStestimates a,2, MSsestimates nta2s + a2, and MSbestimates nsntal + n,a2s + a2. Using these relations with the computed mean square values gives the following estimates:

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