*Article* **The GUM Tree Calculator: A Python Package for Measurement Modelling and Data Processing with Automatic Evaluation of Uncertainty**

**Blair D. Hall**

Measurement Standards Laboratory of New Zealand, Lower Hutt 5010, New Zealand; blair.hall@measurement.govt.nz

**Abstract:** There is currently interest in the digitalisation of metrology because technologies that can measure, analyse, and make critical decisions autonomously are beginning to emerge. The notions of metrological traceability and measurement uncertainty should be supported, following the recommendations in the *Guide to the Expression of Uncertainty in Measurement* (GUM). However, GUM offers no specific guidance. Here, we report on a Python package that implements algorithmic data processing using 'uncertain numbers', which satisfy the general criteria in GUM for an ideal format to express uncertainty. An uncertain number can represent a physical quantity that has not been determined exactly. Using uncertain numbers, measurement models can be expressed clearly and succinctly in terms of the quantities involved. The algorithms and simple data structures we use provide an example of how metrological traceability can be supported in digital systems. In particular, uncertain numbers provide a format to capture and propagate detailed information about quantities that influence a measurement along the various stages of a traceability chain. More detailed information about influence quantities can be exploited to extract more value from results for users at the end of a traceability chain.

**Keywords:** measurement uncertainty; guide to the expression of uncertainty in measurement; measurement modelling; uncertainty propagation; metrological traceability; uncertain number
