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

Fig. 5.

Histogram of standard hygrometer readings scatter.
Histogram of standard hygrometer readings scatter.

Fig. 6.

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