: pp. 5-11
Lviv Polytechnic National University
Lviv Polytechnic National University

An attention is considered to analysis of instrumental component of result’s uncertainty. While production control in industry, the mentioned component for concrete measuring instrument with the concrete measurement results has to be determined and analyzed. Important is the question of random impacts effect on to uncertainty of extreme (minimum or maximum) observations. Since in the practice of the test products qualifying there exist the cases when the result of such measurement becomes an extreme: the minimal value xmin that is the first one from ordered observations, or the maximal one that is the last one from ordered observations. To provide the reliable result of the experiment, it is important to take into account the impact of these random deviations in evaluating the uncertainty of processing the measurement results.

Research is fulfill for: (i) the normal distribution of the parameter of testing samples and the uniform distribution of the instrumental component (ii) the uniform distribution of the parameter of testing samples and uniform distribution of the instrumental component and (iii) the uniform distribution of the parameter of testing samples and normal distribution of the instrumental component. With the aim of quality comparison the change in the distribution shape, the histograms of the normalized relative deviation z1,y1 and minimal observation y1 are built.

If we measure xi the value of ith tested sample parameter, the random impacts ∆ri cause the changes of the considered observations yi = xi+∆ri. Then standard deviation of registered observation becomes bigger than the standard deviation σx of parameter x. In every measurement these changes are random, and their impact can be described by convolution of the distribution px(x) of the tested parameter values and the random effects distribution pr(Dr). If the distribution of observations and random effects is normal, the its density p1(z1) and other parameters including the expansion coefficient are immutable and remain such as for normal distribution. If the distribution differs from the normal, the resultant distribution is normalized in the next way; due to this the expansion coefficient even less differs from to the expansion coefficient calculated according to the distribution of the observations themselves. I.e., for the number of observations ≤ 10 the expansion coefficient does not exceed a few percent. Dependences of the expansion coefficients on the random deviation impacts of measuring instruments results at different distributions are investigated by Monte Carlo method.

Therefore, in calculation of the expanded uncertainty of extreme observation when distributions of both components in measurements are known, the random impacts can be taken into account directly. In case of shortage of information on distribution of random affects, the coefficients for the calculating the expanded uncertainty with sufficient accuracy (for practice equal to the few percent) are accepted such as for normal distribution.

[1] I. Bubela, “Processing of measurement results by deviation of their statistics properties from typical”, PhD Diss., Lviv, Ukraine: Lviv Polytechnic National University, 2016.

[2] I. Bubela, “Processing of measurement results by deviation of their statistics properties from typical”, PhD Thesis, Lviv, Ukraine: Lviv Polytechnic National University, 2016.

[3] M. Dorozhovets. I. Popovych, Z. Warsza, “Method of evaluation the measurement uncertainty of the minimal value of observations and its application in testing of plastic products”, Advanced Mechatronics Solutions. Advances in Intelligent Systems and Computing, Springer International Publishing, Switzerland, vol.393. p.421–430, 2016.

[4] M. Dorozhovets, I. Bubela, “Computing uncertainty of the extreme values in random samples”, International Journal of Computing, vol.15(2). p.127–135, 2016.

[5] M. Dorozhovets, Z. Warsza, I. Popovych, “Uncertainty evaluation of the minimal value measurements”, Measurement Automation Monitoring, vol. 61, no.08, p.395-398, 2015.

[6] ASM International. Tensile Testing, 2nd Edition, 2004, [On-line]. Available: search/-/journal_content/56/10192/05106G/PUBLICATION.

[7] D 638. Test Method for Tensile Properties of Plastics, Annual Book of ASTM Standards, vol.08.01. [On-line]. Available:

[8] GOST 11262-80. Standard of testing methods and conditions of plastic materials and products, 1980.

[9] JCGM 100:2008. Evaluation of measurement data — Guide to the Expression of Uncertainty in Measurement (GUM 1995 with minor corrections), 2008. [On-line]. Available: documentation/JCGM_100_2008_E.pdf.

[10] JCGM 101:2008. Evaluation of measurement data—Supplement 1 to the «Guide to the Expression of Uncertainty in Measurement»—Propagation of distributions using a Monte-Carlo method, 2008. [On-line]. Available:

[11] I. Bubela, “Processing of the observations results with the Flatten-Gaussian distribution by the order statistics method”, Scientific Papers of RUT. Electrotechnics. no.34. p.71-80. 2015.