Compliance of analytical results with regulatory or specification limits: a probabilistic approach

Abstract 

The practice of determining and quoting an expanded uncertainty with a measurement result allows interpretation of results in relation to compliance limits. Against a fixed limit, the probability that the population mean of the measurement is, in fact, over the limit may be readily calculated, and may be more useful than the observation that the probability is greater or smaller than an arbitrary value (e.g. 95%). When the limit itself is uncertain, it is possible to combine probabilities of the limit and measurement to determine the probability that the given measurement is over the limit. This probability, for an upper compliance limit, is derived from the cumulative probability that a possible measurement value (x _M ) is over a given value (X, i.e. Pr(x _M > X)) and the probability density function that the compliance limit (x _L ) is the same or lower value, f(X > x _L ). Integration over all values for X provides the probability that the measurement is over the true compliance limit. A MATLAB script is provided to calculate this probability. A number of examples are assessed using this probability.