Page:NIOSH Manual of Analytical Methods - Chapter P.pdf/14

 Note 2: For statisticians only. Single-evaluation correction of bias: Details are given here illustrating the tolerance interval approach, bias correction, imprecise reference concentrations, and the use of the symmetric accuracy range function. The derivation is not entirely general, but is given here for guidance in handling the myriad possibilities in measurement uncertainty. Though the derivation is slightly complicated, the result obtained is simple. Suppose that estimates having an as-yet-unknown constant bias relative to true concentrations C (not necessarily constant) may be modeled as:

where the random variable

is approximately normally distributed about zero with variance

. For evaluating the method, assume that reference concentration measurements can be made simultaneously and modeled by:

where ratio

has variance

, assumed known accurately. Measure



and compute estimates

at υ = n − 1 degrees of freedom, where the

and

approximately normally distributed random variable

(to the order of

has variance

given by:

, neglecting Cauchy effects of reciprocals of random variables).

Future bias-corrected measurements terms of raw measured values as:

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values of the

of unknown concentration

221

can be defined in

NIOSH Manual of Analytical Methods