The Theory of Uncertainty for Derived Results: Properties of Equations Representing Physicochemical Evaluation Systems

TL;DR

This paper develops a theoretical framework to assess and predict the uncertainty in derived physicochemical variables based on measurement uncertainties and system-specific relationships, aiding experiment design.

Contribution

It introduces a general method to evaluate and classify the uncertainty behavior of different physicochemical evaluation systems regardless of their specific relationships.

Findings

Uncertainty in derived variables is governed by system-specific parameters.

SSR classification into two groups based on their impact on limiting error.

A general relationship between measurement uncertainties and derived variable uncertainty.

Abstract

Any physiochemical variable (Ym) is always determined from certain measured variables {Xi}. The uncertainties {ui} of measuring {Xi} are generally a priori ensured as acceptable. However, there is no general method for assessing uncertainty (em) in the desired Ym, i.e. irrespective of whatever might be its system-specific-relationship (SSR) with {Xi}, and/ or be the causes of {ui}. We here therefore study the behaviors of different typical SSRs. The study shows that any SSR is characterized by a set of parameters, which govern em. That is, em is shown to represent a net SSR-driven (purely systematic) change in ui(s); and it cannot vary for whether ui(s) be caused by either or both statistical and systematic reasons. We thus present the general relationship of em with ui(s), and discuss how it can be used to predict a priori the requirements for an evaluated Ym to be representative, and hence to set the guidelines for designing experiments and also really appropriate evaluation models. Say: Y_m= f_m ({X_i}_(i=1)^N), then, although: e_m= g_m ({u_i}_(i=1)^N), "N" is not a key factor in governing em. However, simply for varying "fm", the em is demonstrated to be either equaling a ui, or >ui, or even <ui. Further, the limiting error (d_m^(Lim.)) in determining an Ym is also shown to be decided by "fm" (SSR). Thus, all SSRs are classified into two groups: (I) the SSRs that can never lead "d_m^(Lim.)" to be zero; and (II) the SSRs that enable "d_m^(Lim.)" to be zero. In fact, the theoretical-tool (SSR) is by pros and cons no different from any discrete experimental-means of a study, and has resemblance with chemical reactions as well.