3.4.2 Uncertainty Due to Unaccounted Effects in Certification of Chemical Reference Materials

Raghu N. Kacker
Statistical Engineering Division, ITL The measurand of a chemical reference material is some property of the particular sample. Due to the limitations of cost, time, and available scientific knowledge, some potentially significant components of uncertainty frequently remain unaccounted for in the stated standard uncertainty concerning the measurand. The unaccounted effects may include unrecognized systematic effects and recognized but not corrected systematic effects. The unaccounted effects contribute to the differences in the results of measurement of a common measurand produced by two or more methods. This paper proposes a new statistical method, based on the Guide to the Expression of Uncertainty in Measurement, to evaluate the uncertainty due to unaccounted effects when two or more equally trustworthy and independent methods are used to measure a common measurand. Suppose x₁ and x2 are the best estimates from two equally trustworthy and independent methods and that u(x₁) and u(x₂) are the corresponding standard uncertainties representing uncertainty due to recognized random and systematic effects. Using the proposed Trapezoid Method, the consensus value of the measurand based on modeling the residual discrepancy due to unaccounted effects is y = (x₁ + x₂)/2, and the standard uncertainty concerning the measurand is $u(y) = \sqrt{(1/2)^2 u^2(x_2) + (1/2)^2 u^2(x_2) + (5/12)(x_{(2)} - x_{(1)})^2 }$. The last component in this expression is a measure of the uncertainty due to unaccounted effects in the measurement methods. The proposed approach can be extended to more than two methods. The figures show two examples that illustrate the proposed results. Amounts of cadmium and mercury in certain samples of soil containing lead paint were determined by two different methods for each element. The figures show for each element, the expanded uncertainty intervals for the amount of trace element based on the individual methods alone. The differences in the results from two methods are partly due to unaccounted effects. The figures also show the combined result (from two methods) and expanded uncertainty about it based on the Trapezoid Method (T-M) and the Schiller-Eberhardt Method (SE-M). The research is continuing to (1) extend the results to independent methods that are not equally trustworthy, and to (2) evaluate the uncertainty due to heterogeneity in the batch of material. The author gratefully acknowledges valuable inputs, and feedback he has received from his colleagues Keith Eberhardt, Tyler Estler, Robert Goldberg, Al Jones, Mark Levenson, Hung-kung Liu, Eric Shirley, Barry Taylor, James Yen, and members of the NIST (Washington) Editorial Review Board (WERB).

Figure 17: Expanded uncertainty intervals about the individual results from measurement methods and about the combined results based on the Trapezoid Method (T-M), and the Schiller-Eberhardt Method (SE-M).

       

Date created: 7/20/2001
Last updated: 7/20/2001
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