1.1. Introduction

UncertRadio allows for the evaluation of a measurement in the field of measuring activity by inserting defining equations the full calculation of the output quantity and its combined uncertainty (according to ISO GUM) and to calculate the values of the Decision threshold and the Detection limit (according to ISO 11929:2019), which are closely related to uncertainty.

The program assumes the ISO GUM interpretation which is based on the Bayesian theory of the measurement uncertainty (Bayesian statistics). The Bayesian measurement uncertainty does not have a statistical uncertainty, nor does it consider degrees of freedom, which makes it different from conventional (frequentist) statistics. Therefore, degrees of freedom need not to be treated in the program.

One restriction with respect to its usage referred to the circumstance that only those measurement problems could be treated in which the measurements of counts or of counting rates are terminated if the counting time reaches its pre-set value. The statistically different case where the measurement is terminated by reaching a pre-set value of counts, where the counting time is a random variable, is also treated by the program, since program version 2.4.04 (see Treatment of numbers of counts and count rates).

The present state of the program allows calculations for up to three output quantities.

The sequence of steps being treated is:

  1. Input of a short text description of the measurement problem.

  2. Input of the equations defining the output quantity y of the measurement procedure; these define the “evaluation model”; the first of the equations must define the output quantity/quantities.

    Note: If more than one output quantity is to be treated, the calculation of the values of output quantity, uncertainty, uncertainty budget, Decision threshold and Detection limit, respectively, refers only to a single output quantity. By starting with a project, by default the first of the output quantities is “activated”; at a later stage another one can be selected in the main menu to be the “active one”.

  3. Automatic extraction of the formula symbols and input of their meaning (unit, meaning); automatic classification as independent and dependent symbols; manual addition of other symbols, not used explicitly in the equations.

  4. Selection of symbols which define the net counting rate (Rn) and the gross counting rate, only for the purpose of the calculation of Decision threshold and Detection limit; the net counting rate in this case must be the “procedure dependent net counting rate” in which the counting rate contribution due to interference from other radionuclides, usually obtained by calculation, is taken into account.

  5. Input of measured values and their associated standard uncertainties of input quantities (independent symbols) in table form.

  6. For the input of measurement uncertainties, their associated distribution can be chosen from: a) normal distribution, b) uniform distribution, c) triangular distribution and d) gamma distribution, as well as a few others. Half-widths of the latter two are converted by the program according to ISO GUM to standard uncertainties; for a complete list see other distributions.

    Note: In the case of low-level applications with very low count numbers, the so-called “(N+x) rule may be selected to improve the results, from which the associated counting rate variables are considered as being Gamma distributed.

  7. For the input one can choose between absolute and relative uncertainties.

  8. For counting rates or the number of counts uncertainties can also be defined by formulae.

  9. The standard deviation of the gross count rate must be defined by a formula; this is the “uncertainty function” (standard uncertainty) of the gross counting rate, which is the basis for deriving values of Decision threshold and Detection limit.

  10. Input of covariances between input quantities which can be given as formulae or as values of correlation coefficients in tabular form.

  11. Numerical calculation of values and combined standard uncertainties for the quantities classified as dependent quantities and for the output quantity and of the uncertainty budget; consideration of covariances/correlations between input quantities is possible; calculation of the best estimates and confidence limits based on a Bayesian method characterizing the value and standard uncertainty of the output quantity.

  12. Iterative numerical calculation of Decision threshold and Detection limit for the output quantity based on the numerical evaluation of its combined uncertainty taking covariances into account.

  13. A Monte Carlo simulation can be started for an examination of the above-mentioned calculations. This allows an independent calculation of the value of the output quantity and its combined uncertainty. Partial derivations are not necessary in this case: for every input quantity classified as independent, a value is drawn from its associated distribution from which a value of the output quantity is calculated using the defined equations. From the many-fold repetition of this step, a statistical distribution is obtained for the output quantity. This is used to derive the “best estimate” from its mean and the combined uncertainty from its standard deviation. Confidence limits are estimated as Quantiles of that distribution, whereas Decision threshold and Detection limits require creating separate MC distributions with modified values of the output quantity and their estimation by corresponding Quantiles.

  14. Finally, a complete report can be created as a text file containing all the equations, input values, uncertainty budget table and the final results - including those from the Monte Carlo simulation, the PDF file also contains the MC graphs.

UncertRadio is well suited for calculation comparison for such solutions, which one may have already developed with spreadsheet calculations. The latter may get quite complex and sometimes are not so easily manageable especially with respect to the correct uncertainty propagation.