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<p>The article considers the theoretical framework underlying the mathematical processing of indirect measurement results expressed as a&#xa0;quotient. In practice, measurement results and accuracy measures are represented by approximate formulas obtained using the linearization method. In this case, the proper presentation of the result involves determining its systematic error through an additional assessment of the degree of formula approximation. It is shown that the systematic error of an indirect measurement result can be determined at known arithmetic means and standard deviations of measurement results. The value of variance is practically impossible to adjust without considering the probability distributions of random measurement errors. It is established that an analytical formula for the quotient of random variables, derived as a&#xa0;necessary and sufficient condition for the Taylor series expansion of the quotient of random variables, can be represented as a&#xa0;linear function of random measurement errors. For the specified linear function, on the basis of theorems on the descriptive statistics of functions of random arguments, exact formulas were obtained, which describe the expected value and variance and coincide with the formulas that are used in practice as approximations. Formulas representing the result of indirect measurements as a&#xa0;quotient were obtained using a&#xa0;method other than the linearization method; thus, these formulas can be considered exact without any accuracy assessment of their approximation. The study results can be useful for specialists involved in measurements in various fields of science and technology, for example, instrument makers and metrologists, as well as undergraduate and graduate students of relevant specialties.</p>

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Method for determining the result and accuracy of indirect measurements expressed as a quotient

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • Evgenii V. Eremin

摘要

The article considers the theoretical framework underlying the mathematical processing of indirect measurement results expressed as a quotient. In practice, measurement results and accuracy measures are represented by approximate formulas obtained using the linearization method. In this case, the proper presentation of the result involves determining its systematic error through an additional assessment of the degree of formula approximation. It is shown that the systematic error of an indirect measurement result can be determined at known arithmetic means and standard deviations of measurement results. The value of variance is practically impossible to adjust without considering the probability distributions of random measurement errors. It is established that an analytical formula for the quotient of random variables, derived as a necessary and sufficient condition for the Taylor series expansion of the quotient of random variables, can be represented as a linear function of random measurement errors. For the specified linear function, on the basis of theorems on the descriptive statistics of functions of random arguments, exact formulas were obtained, which describe the expected value and variance and coincide with the formulas that are used in practice as approximations. Formulas representing the result of indirect measurements as a quotient were obtained using a method other than the linearization method; thus, these formulas can be considered exact without any accuracy assessment of their approximation. The study results can be useful for specialists involved in measurements in various fields of science and technology, for example, instrument makers and metrologists, as well as undergraduate and graduate students of relevant specialties.