Two Theories of Experimental Error

The reader … will have seized my meaning if he perceives that the different situations in which uncertain inferences may be attempted admit of logical distinctions which should guide our procedure. Sir Ronald Fisher [1] Following the widespread adoption of new approaches to the combination of experimental uncertainties, two theories of error are identified and their possible justifications assessed. They are the “orthodox theory” based on the familiar distinction between random and systematic errors and the “randomatic theory” which dispenses with the distinction and treats all errors as the orthodox theory treats random errors. The orthodox theory suffers from a number of important confusions about the nature of its central distinction, about the combination of uncertainties, and about which populations of results can correctly be said to contain random errors. These confusions are clarified and the central distinction is argued to be objective. Three justifications are developed for the randomatic theory: 1) that it is implied by the generally accepted law of error propagation, 2) that all so-called systematic errors belong to populations characterized by hitherto unnoticed frequency-based distributions, and 3) that they belong to subjectivist prior distributions. But, upon examination, the argument in terms of the law of error propagation is found to beg key controversial questions, the frequency-based distributions are found not always to be of suitable form, and the subjectivist distributions are found to be unrealistic. Thus the randomatic theory remains unjustified by objective standards. Moreover, its use could lead to the underestimation of uncertainties in the usual sense of the maximum possible or conceivable error in the result of a particular specified experiment. The concept of systematic error is argued to be indispensable and new recommendations are formulated which are orthodox in general character.