Use of Welch-Satterthwaite approximation in calibration of voltage standards

In certifying banks of solid state voltage standards, the customer is provided with predictive equations that are used to predict mean voltage and uncertainty in voltage. These equations are provided for both the bank average and the individual standards constituting the bank over the period of certification, which is typically a year. Use of the Welch-Satterthwaite Approximation (W-SA) can significantly reduce the number of calibrations required to provide satisfactory levels of uncertainty for the bank average. We first review the standard approach in metrology for expressing the uncertainty in a measurement system. We provide an intuitive discussion of the W-SA along with examples of its use and information on its usefulness when there are only a few degrees of freedom. A case study involving a bank of four solid state voltage standards is presented in detail. We also propose an analysis technique that can be used to investigate the usefulness and potential limitations of the W-SA when estimating measurement uncertainties. The analysis technique can be used to estimate approximation error, sampling error, and the coverage probabilities of confidence intervals computed using the W-SA.