**1. Introduction**

The national and international infrastructures that disseminate critical measurement information throughout society are due for renovation. Designed to be operated and supervised by skilled people, there is now a call to digitalize these essentially paper-based systems. The best way to proceed is by no means clear. However, a coordinated international effort will be needed to reap real benefits from digitalization. A recent paper reviewed work performed so far to develop a common digital format for reporting measurement data, which is generically referred to as a "digital calibration certificate" (DCC) [1]. The DCC will be a fundamental component of digital measurement infrastructures, but many decisions still need to be made about its structure. Among these is how best to represent measurement uncertainty.

The International Committee for Weights and Measures (CIPM), which directs metrological activities carried out by parties to the Metre Convention [2], recognized the need to

**Citation:** Hall, B.D.; Koo, A. Digital Representation of Measurement Uncertainty: A Case Study Linking an RMO Key Comparison with a CIPM Key Comparison. *Metrology* **2021**, *1*, 166–181. https://doi.org/ 10.3390/metrology1020011

Academic Editor: Simona Salicone

Received: 16 September 2021 Accepted: 1 December 2021 Published: 6 December 2021

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coordinate digitalization of the international measurement system and has established a CIPM Task Group on the Digital SI (CIPM-TG-DSI) [3], supported by a team of experts. In early 2021, this team made a public request for use cases to identify situations where the digitalization of existing metrological infrastructure might improve outcomes or address difficulties. This paper presents a preliminary analysis of one of those use cases.

The study relates to the analysis and linking of international measurement comparisons. Although comparison analysis is a specialized topic, the limitations of current reporting in calibration certificates become clear in this context, and so, the benefits of digitalization are easily recognized. Comparison linking is an interesting case, because correlations in the data can significantly affect the results. Handling the correlations complicates data analysis, especially the evaluation of uncertainty. Our study shows that digitalization can help; it can enhance the information available in the final results while hiding and largely automating the more laborious aspects of processing data. The challenge posed by correlated data arises in many other measurement scenarios as well, so the representation of measurement uncertainty described in this work would offer advantages in other digital systems.

### *1.1. Measurement Comparisons in the CIPM MRA*

The CIPM Mutual Recognition Arrangement (MRA) [4] is a framework used to establish the equivalence of measurement standards in different economies. Specific calibration and measurement capability (CMC) claims are approved by expert Consultative Committees of the CIPM and then published in a database by the International Bureau of Weights and Measures (BIPM). To maintain or extend CMC entries, national metrology institutes (NMIs) must provide evidence in support of their claims. This evidence is often obtained by participating in international measurement comparisons [5].

Our case study involves two kinds of comparison: a CIPM key comparison and a subsequent RMO key comparison (organized by a regional metrology organization). In a CIPM comparison, a group of NMIs submit measurements of a particular quantity associated with an artifact. The data are used to determine a comparison reference value, and then, for each participant, a degree of equivalence (DoE) is calculated, which characterizes the difference between the participant's result and the comparison reference value.

After an initial CIPM comparison has been completed, a number of other RMO comparisons may be carried out. This provides a way to assess the equivalence of NMIs that did not participate in the initial comparison. The results of an RMO comparison must be linked to those of the initial CIPM comparison, which means that several participants from the initial comparison must participate again in the RMO comparison.

A DoE is considered to reflect the level of consistency of one participant's measurement standard with those of other participants. An uncertainty is evaluated for each DoE, which allows the significance of each result to be assessed: if the magnitude of a DoE is greater than its expanded uncertainty (typically at a 95% level of confidence), then the evidence for equivalence is considered weak. DoEs evaluated during an RMO comparison have equal standing to DoEs obtained from the initial CIPM comparison.
