Measurement Uncertainty - Propagation of Distributions

Propagation of Distributions

The true values of the input quantities ... are unknown. In the GUM approach, ... are characterized by probability distributions and treated mathematically as random variables. These distributions describe the respective probabilities of their true values lying in different intervals, and are assigned based on available knowledge concerning ... . Sometimes, some or all of ... are interrelated and the relevant distributions, which are known as joint, apply to these quantities taken together.

Consider estimates ..., respectively, of the input quantities ..., obtained from certificates and reports, manufacturers' specifications, the analysis of measurement data, and so on. The probability distributions characterizing ... are chosen such that the estimates ..., respectively, are the expectations of ... . Moreover, for the th input quantity, consider a so-called standard uncertainty, given the symbol, defined as the standard deviation of the input quantity . This standard uncertainty is said to be associated with the (corresponding) estimate .

The use of available knowledge to establish a probability distribution to characterize each quantity of interest applies to the and also to . In the latter case, the characterizing probability distribution for is determined by the measurement model together with the probability distributions for the . The determination of the probability distribution for from this information is known as the propagation of distributions.

The figure below depicts a measurement model in the case where and are each characterized by a (different) rectangular, or uniform, probability distribution. has a symmetric trapezoidal probability distribution in this case.

Once the input quantities ... have been characterized by appropriate probability distributions, and the measurement model has been developed, the probability distribution for the measurand is fully specified in terms of this information. In particular, the expectation of is used as the estimate of, and the standard deviation of as the standard uncertainty associated with this estimate.

Often an interval containing with a specified probability is required. Such an interval, a coverage interval, can be deduced from the probability distribution for . The specified probability is known as the coverage probability. For a given coverage probability, there is more than one coverage interval. The probabilistically symmetric coverage interval is an interval for which the probabilities (summing to one minus the coverage probability) of a value to the left and the right of the interval are equal. The shortest coverage interval is an interval for which the length is least over all coverage intervals having the same coverage probability.

Prior knowledge about the true value of the output quantity can also be considered. For the domestic bathroom scale, the fact that the person's mass is positive, and that it is the mass of a person, rather than that of a motor car, that is being measured, both constitute prior knowledge about the possible values of the measurand in this example. Such additional information can be used to provide a probability distribution for that can give a smaller standard deviation for and hence a smaller standard uncertainty associated with the estimate of .