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Using the Characteristics of the Sample Range of Repeated Observations of a Measurand to Estimate Its Numerical Value and Type A Standard Uncertainty Cover

Using the Characteristics of the Sample Range of Repeated Observations of a Measurand to Estimate Its Numerical Value and Type A Standard Uncertainty

Measurement Science Review
Volume 25 (2025): Issue 6 (December 2025)
By: Igor Zakharov and  Olesia Botsiura  
Open Access
|Dec 2025

Figures & Tables

Fig. 1.

Dependence of the efficiency Var of various estimators of the expected value for different PDFs of the observed dispersion of IMI readings: (a) Arcsine; (b) Uniform; (c) Triangular; (d) Normal.

Fig. 2.

Dependence of the ratio of the variances of the sample mid-range and the sample average on the number of repeated measurements n with their uniform distribution law.

Fig. 3.

Dependence of the coefficient α on n for different distribution laws: o – arcsine; ◊ – normal; □ – uniform.

Fig. 4.

Standard hygrometer Testo 400 (a) and humidity generator type “Huminator” (b).

Fig. 5.

Histogram of standard hygrometer readings scatter.

Fig. 6.

Dependence of estimates of the expected value on n: –– arithmetic mean; – – mid-range

Fig. 7.

Experimental dependence (▲) of the conversion coefficient α of the sample range of readings of the standard hygrometer into the sample standard deviation on the sample size n.

Reference hygrometer readings_

№. of observationsWsi [% RH]№. of observationsWsi [% RH]
126.13626.10
226.11726.11
326.12826.13
426.10926.14
526.141026.12

Expressions for various estimates of the measurand_

EstimatorsFormulas
Sample mid-range – – (1) Mn=ymax+ymin2 {M_n} = {{{y_{max }} + {y_{min }}} \over 2}
Sample mean –– (2) y¯n=1n∑q=1nyq {\bar y_n} = {1 \over n}\sum\limits_{q = 1}^n {{y_q}}
Median – ⸳ – (3) Medn=yn+12,n−oddyn2+yn2+12,n−even Me{d_n} = \left\{ {\matrix{ {{y_{{{\left( {n + 1} \right)} \over 2}}},} \hfill & {n - {\rm{odd}}} \hfill \cr {{{{y_{{n \over 2}}} + {y_{{n \over 2} + 1}}} \over 2},} \hfill & {n - {\rm{even}}} \hfill \cr } } \right.
DOI: https://doi.org/10.2478/msr-2025-0040 | Journal eISSN: 1335-8871
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Language: English
Page range: 366 - 370
Submitted on: Aug 25, 2025
|
Accepted on: Nov 4, 2025
|
Published on: Dec 23, 2025
Published by: Slovak Academy of Sciences, Institute of Measurement Science
In partnership with: Paradigm Publishing Services
Publication frequency: Volume open
Keywords:
repeated measurements,
extreme values of a sample,
sample range,
midrange,
Monte Carlo method,
estimates of the measurand,
type A measurement uncertainties
Related subjects:
Engineering,
Electrical engineering,
Control engineering, metrology and testing

© 2025 Igor Zakharov, Olesia Botsiura, published by Slovak Academy of Sciences, Institute of Measurement Science
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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