Analytically determining of the relative inaccuracy (error) of indirectly measurable variable and dimensionless scale characterising quality of the experiment

Abstract

In the following paper we present an easily applicable new method for analytical representation of the maximum relative inaccuracy (error) of an indirectly measurable variable $f=f(x_1, x_2, ..., x_n)$ as a function of the maximum relative inaccuracies (errors) of the directly measurable variables $x_1, x_2, ..., x_n$. Our new approach is more adequate for the objective reality. The gist of it is that in order to find the analytical form of the maximum relative inaccuracy of the variable $f$ we take for being fixed variables statistical mean values $\displaystyle \overline{\left|\frac{x_1}{f}\cdot \frac{\partial f}{\partial x_1}\right|}, \overline{\left|\frac{x_2}{f}\cdot \frac{\partial f}{\partial x_2}\right|}, ..., \overline{\left|\frac{x_n}{f}\cdot \frac{\partial f}{\partial x_n}\right|}$ of the modules of the coefficients of influence of relative inaccuracies $\displaystyle \frac{\Delta x_1}{x_1}$, $\displaystyle \frac{\Delta x_2}{x_2}, ..., \frac{\Delta x_n}{x_n}$ in $f$. The numerical value of the maximum relative inaccuracy of the variable $f$ is found using the statistical mean values of the absolute values of the relative inaccuracies $\displaystyle \overline{\left|\frac{\Delta x_1}{x_1}\right|}, \overline{\left|\frac{\Delta x_2}{x_2}\right|}, ..., \overline{\left|\frac{\Delta x_n}{x_n}\right|}$. Moreover, we look into functions which are continuous but are not differentiable in respect to certain arguments in some points. Having this in mind we develop the theory of errors, which we will call with what we feel is a more precise term -- theory of inaccuracies. We introduce some new terms -- space of the relative inaccuracy and plane of the relative inaccuracy of $f$. We also define a sample plane of the ideal absolutely accurate experiment and using it we define a universal numerical characteristic -- a dimensionless scale for evaluation of the quality (accuracy) of the experiment.

Keywords

• Indirectly measurable variable
• maximum relative error
• dimensionless scale