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Sample preparation methods are usually developed following respective recommendations of the applicable sampling standards. Modern sampling theories allow designing and optimizing these methods. Random errors in sample preparation are calculated based on a theoretical description of the piecewise heterogeneity of the sample obtained using the formulas for the fundamental sampling error. The concept of a piecewise coefficient of variation is introduced and used to develop a formula for the relative error of the sample preparation method. Using a method compiled in accordance with GOST 14180-80 for copper ore as an example, the relative error is established for the preparation of an ore sample with the copper mass fraction of 1.3 %. It is shown that a change in the final preparation size from 0.1 to 0.08 mm affects the error only insignificantly, and sample size changes by stages allow designing a preparation method with the smallest error. It is advisable to analyze the method compiled and change its parameters on the basis of a structural assessment of the influence of individual preparation stages on the error. Sample preparation examples for copper and gold-bearing ore are used to demonstrate the analysis procedure and the parameter changes. Traditionally, the minimum sample masses are established for all stages based on the volumetric heterogeneity of the sample being tested and the size of the sample material. The minimum masses should be found depending on the grain size of the valuable mineral in the ore, the permissible relative error for the size reduction, and the material size for the sample reduced by a factor of 1.5 for nonferrous metal ores.

Introduction. Theoretical results have been obtained that allow calculating random sampling errors. One of the main calculation formulas for determining the sampling error is the sample reduction error formula. The sampling errors determined by this formula differ from the errors determined experimentally. The sample reduction error consists of several components, a separate quantitative determination of which is necessary when developing methodological support for testing processes. It is impossible to determine all these components separately from each other experimentally. It is necessary to determine the ratio of the components of the specified formula. Methods of research. The sample reduction error, determined analytically, is the minimum possible reduction error when this operation is ideally performed. To take into account the deviation from the ideal conditions for performing the reduction operation, it is necessary to experimentally estimate the amount of the actual deviation and link it with the theoretical result. As a result, the value of the correction factor can be obtained, which should be entered into the formula for calculating the reduction error. In order to eliminate the need for experimental determination of the error of the method of measuring the mass fraction, experiments to determine the correction factor should be performed on artificial samples with markers. Research procedure. Experiments were performed to reduce samples with markers. 480 reductions of the same sample were performed, which showed the coincidence of the theoretical and experimental distributions of the number of markers in the reduced samples. The correction factor in the experiment with markers of the correct shape was 1.3. The same coefficient in the experiment with markers whose granulometric composition match with that of the sample material was 2.0. The average value of the correction factor in reproducibility conditions was 2.13. Results and analysis. As a result of two experiments on multiple reduction of the sample, it was found that the correction factor under reproducibility conditions should be within 1.3 and 2.0. Similar information about the differences in reproducibility and repeatability errors in international and Russian standards shows that in order to move from a theoretical formula to a real reproducibility error, a correction factor from 2.0 to 3.0 should be introduced. Conclusions. The introduction of correction coefficients into the reduction error formula makes it possible to calculate the real errors of sample reduction, as well as quantify the results of mineral products testing based on the calculation.

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Introduction. An immediate estimate of the accidental error of sample reduction can only be made if the measurement method of execution is zero. This can be achieved by imitating the grains of a useful component with markers fully extracted from the reduced sample. The markers can be larger than 44 "Izvestiya vysshikh uchebnykh zavedenii. Gornyi zhurnal". No. 4. 2021 ISSN 0536-1028 the maximum size of the sample material and are extracted from the sample using screens. Markers whose granulometric composition matches the sample composition should be extracted completely from the reduced sample using a hand magnet in the case. Methodology. A small number of markers of the correct shape imitates forgeable gold grains or d99 platinum. A much larger number of free-form markers simulate the granulometric composition of a sample in the –1+0.5 mm class. It is necessary to find a form factor linking the particles real volume with the cube volume. For magnetite markers, the form factor is 0.4. Results and analysis. The samples have been reduced with regular shaped and free-form markers, which makes it possible to experimentally determine the error of reduction. Theoretical formulas found errors for the conditions of experiments. For experiments with the regular shaped markers, a 54.77– 58.43% relative reduction error has been obtained according to 480 parallel measurements. Theoretically determined 57.64% relative error falls into this range. Similar relative reduction errors with free-form markers are 8.82–10.00% and 9.15%. Conclusion. The fairness of the reduction error analytical formula has been directly evaluated for the first time. The reduction error analytical formula can be applied when analyzing the schemes of sample preparation.