Calculation of the expanded uncertainty for large uncertainties using the lognormal distribution
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GENERAL PAPER
Calculation of the expanded uncertainty for large uncertainties using the lognormal distribution Alex Williams1 Received: 31 August 2019 / Accepted: 10 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract For large uncertainties, calculating the expanded uncertainty using a normal distribution for the values of the measurand can lead to negative values for the lower limit of the expanded uncertainty and unrealistic large values for the upper limit, when the relative uncertainty is constant over wide concentration range. Using the lognormal distribution overcomes these problems and is particularly important when the relative uncertainty is larger than 10%; below this value, both distributions give almost identical results. The use of the lognormal distribution can be appropriate when the model equation for the derivation of the value of the measurand consists of products of input quantities, with positive values. Most measurement results are given as a mean and a relative uncertainty, and the purpose of this paper is to show how, for a lognormal distribution, the expanded uncertainty can be derived directly from these two parameters. Keywords Lognormal distribution · Expanded uncertainty · Large uncertainties
Introduction It is currently a common practice to calculate the expanded uncertainty assuming that the PDF (Probability Density Function) of the measurand is approximately normal. This works well for small uncertainties, but for large uncertainties it can lead to negative values for the lower limit of the expanded uncertainty. Also, when the relative uncertainty is constant for a range of values of measurand then, as shown later, the upper limit becomes unrealistically large due to a singularity in the formula for the upper limit. First it is shown that for many analytical measurements, in order to calculate the expanded uncertainty, it is appropriate to assume that the PDF of the measurand is lognormal. For relative uncertainties of less than 10%, both the normal and lognormal distributions give virtually identical results. Above this value, the two distributions diverge, with the PDF of the lognormal agreeing with the PDF calculated using the Monte Carlo method for combining uncertainty components [1]. Then it is shown how to use the mean value and the
* Alex Williams [email protected] 1
relative uncertainty of measurement result to calculate the expanded uncertainty utilising the lognormal distribution.
Lognormal distribution There are a number of distributions that can be used when the value of measurand is known to have a positive value; for example, in [2] the beta distribution was chosen because the distribution tends towards normal as the value of the measurand increases, and in [3], the gamma function was used as an example to show an assessment of compliance close to a limit. When, as in the present case, the model equation used to calculate the value of the measurand is the product of the input quantities, in either the denominator or numerat
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