This paper presents a unified theory of measurement errors and uncertainties. The proposed theory unifies the theory of errors with the theory of measurement uncertainties and merges some of the best practices in error and uncertainty analysis. The unified theory is based entirely on the frequentist statistics. It recovers the traditional classification of random and systematic errors (as the primary classification) and retains the GUM’s Type A and Type B classification (as the secondary classification). The unified theory employs two new estimators (presented in this paper) for estimating the expanded uncertainty: one for direct measurements and the other for indirect measurements. The Welch–Satterthwaite formula is no longer needed because these two new estimators do not need the effective degrees of freedom (DOF) of the total (i.e. combined) standard uncertainty. This significantly simplifies uncertainty analysis and avoids the difficulty and subjectivity in determining the DOF of a Type B component and the limitations of the Welch–Satterthwaite formula. Jensen’s inequality theoretically guarantees that the two new estimators are conservative. Numerical analyses and test examples suggest that the two new estimators are only slightly conservative. This paper demonstrates the validity and effectiveness of these two new estimators and provides a comprehensive comparison with existing methods. The unified theory is easy to understand and easy to use; it provides reliable and realistic estimates of uncertainties.

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