The propagation of distributions offers a more general capability than the law of propagation of uncertainty. It is relevant when the conditions for that law to apply do not hold or when there is doubt over whether they hold.

Introduction

It is important to realise that it is not just for 'complicated' models that guidance is needed. There are many problems where the model is (superficially) 'simple,' e.g., when it contains just one or two variables. For such 'small' models, it cannot generally be expected that the central limit theorem will kick in to provide a normal distribution for the value of the measurand. The reason is that the theorem requires there to be a 'sufficiently large' number of quantities and other conditions to apply. In practice, the theorem often kicks in very early, but this cannot always be assumed.

An Example

A good example arises when converting the Cartesian co-ordinates of a complex-valued quantity, as occurs frequently in electrical, acoustical and optical metrology, into polar co-ordinates.

One of the operations involves forming the magnitude R from the real and imaginary parts X and Y using R = (X² + Y²)^1/2. Even if X and Y are normally distributed, R will not be so. However, if the uncertainties associated with X and Y are small compared with the magnitudes of X and Y, R will be close to normal. If this is not the case, as arises when X and Y are close to zero, the distribution of R will be far from normal. As a consequence, a 95% coverage interval calculated for R based on the application of the law of propagation of uncertainty may not be realistic. Indeed, it may even include negative values, which cannot of course physically arise for a magnitude.

The figures below indicate what might happen. On the left is the distribution for R that results from the application of the law of propagation of uncertainty and the right that from the use of the propagation of distributions. The shaded regions correspond to the shortest 95% coverage intervals for R.

The SSfM Best-Practice Guide No. 6, Uncertainties and Statistical Modelling, available from the SSfM download page, gives further information.

This tutorial is an abridgement of an article that first appeared in Counting on IT Issue 11.