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:
3/15/03
values of the
of unknown concentration
221
can be defined in
NIOSH Manual of Analytical Methods
