Uncertainty estimation with a small number of measurements, part I: new insights on the t -interval method and its limitations

Abstract

Abstract The conventional approach to estimating measurement uncertainty employs the t -interval when the population standard deviation is unknown and the sample size is small (<30). This is because the t -interval, or t -based uncertainty, is considered to be the ‘exact’ solution to estimating measurement uncertainty for small samples. However, three paradoxes have been found to be attributable to the t -interval. This paper is the first one (Part I) in a series of two papers (Part I and Part II). It presents some new insights on the t -interval and explores its true underlying meaning. This paper reveals that the t -interval is a result from a distorted statistical inference in the transformed sample space. The transformation distortion is the root cause of extremely high t -scores when the sample size is very small (<5), resulting in unrealistic estimates of uncertainty. This is the fundamental limitation of the t -interval method for uncertainty estimation. Part II will propose a redefinition of uncertainty and a modification of the conventional approach to estimating measurement uncertainty.

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