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

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. (paper)