Statistics for Environmental Engineers

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Case Study Solution


Figure 24.1 is a dot diagram showing the location and spread of the data from each laboratory. It appears that the variability in the results is about the same in each lab, but laboratories 4 and 5 may be giving low readings. The data are replotted in Figure 24.2 as deviations about their respective means. An analysis of variance will tell us if the means of these laboratories are statistically different.


Lab 5 Lab 4 Lab 3 Lab 2 Lab 1


1Ю 2’o    30    40    510


Lead Concentration (pg/L)


FIGURE 24.1 Dot plots comparing the results from five laboratories.


ANOVA Table for the Comparison of Five Laboratories


Source of Variation


Sum of


Squares


Degrees of Freedom


Mean


Square


F


Ratio


Between laboratories


13.88


4


3.47


6.81


Within laboratories


22.94


45


0.51


Total


36.82


49

FIGURE 24.2 Plot of the data from the five laboratories and the distributions of within-laboratory and between-laboratory variation.


The variance within a single laboratory should be due to random errors arising in handling and analyzing the specimens. The variation between laboratories might be due to a real difference in performance, or it might also be due to random variation. If the variation between laboratories is random, the five observed laboratory means will vary randomly about the grand mean of all 50 measured concentrations ( y = 3.84 ^g/L) and, furthermore, the variance of the five laboratories’ means with respect to the grand mean will be the same as the variance within laboratories.

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