Tutorial review: expressing the agreement between the measured value and the certificate value-a closer look at the...

Examination of the unsatisfactory ways that authors of articles included in the 2026 Atomic Spectrometry Update concerned with the analysis of foods and clinical materials presented the results of the analysis of a certified reference material (CRM) shows that little has changed in the past 25 years. A survey of the publications by ten expert metrological organizations shows that the use of CRMs to evaluate the bias of a newly developed method in a single laboratory gets little attention compared with other uses. One organization, IUPAC, specifically recommended using significance testing, and several organizations suggested procedures that strongly resemble the t-test for the comparison of the means of two data sets. An examination of the general performance of the unequal variance t-test in which the number of degrees of freedom are calculated by the Welch-Satterthwaite equation, revealed that when using Excel, the outcome is significantly influenced by the rounding down of the number of degrees of freedom to an integer value. The anomalies were removed by the calculation of t-values for non-integer numbers of degrees of freedom. Application of this version of the test to the situation where one of the data sets is a CRM, requires that the standard uncertainty and the degrees of freedom for the reference value are known. Language added relatively recently to NIST certificates indicates that the standard uncertainty may be obtained from the expanded uncertainty by dividing by the coverage factor. If the standard uncertainty is half the expanded uncertainty, which is equivalent to a 95% confidence interval, then the associated number of degrees of freedom is 60.4. The validity of this approach and of the numbers of degrees of freedom is confirmed in the 2020 NIST Special Publication 260-202. To illustrate the application of this version of the t-test, an example is presented based on the determination of the total arsenic content of NIST SRM 1568c (rice flour). Recommendations are made for how to report such results in publications, emphasizing the importance of quoting p-values, and an Excel worksheet is provided in the SI to facilitate the calculations.