# Statistics for Environmental Engineers

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= E^-d)2 S    2(n — 1)

where dj is the difference between the two samples for each laboratory, d is the average of all laboratory sample pair differences, and n is number of participating laboratories (i.e., number of sample pairs).

It captures the pairs that would be expected from random error alone. Laboratories falling inside the circle are considered to be doing good work; those falling outside have poor precision and perhaps also a problem with bias. For these data:

+ (4.6 — 4.54)2

0.55 ig/L

(4.3 — 4.54 ) 2 + ( 3 .0 — 4.54 )2 + 2(8-1)

The radius of the circle is s times Student’s t for a two-tailed 95% confidence interval with v = 8 — 1=7 degrees of freedom. Thus, t70025 = 2.365 and the radius is 0.55(2.365) = 1.3 lg/L.

A 45° diagonal can be added to the plots to help assess bias. A lab that measures high will produce results that fall in the upper-right quadrant of the plot. Figure 9.2 shows four possible Youden plots. The upper panels show laboratories that have no bias. The lower. left panel shows two labs that are biased high and one lab that is biased low. The lower-right panel shows all labs with high bias, presumably

Measured on the    Measured on the