Accreditation and Quality Assurance

, Volume 21, Issue 1, pp 33–39 | Cite as

Problems and risks occurred during uncertainty evaluation of a quantity calculated from correlated parameters: a case study of pH measurement

  • Józef Wiora
Practitioner’s Report


Parameters of a model describing a measurement process obtained during a calibration experiment allow one to calculate a measurement result, but a simple estimation of measurement uncertainties of the parameters is not sufficient to assess the uncertainty of the result. In this paper, an example of a pH measurement conducted using an ion-selective electrode is presented, in which the uncertainty is evaluated taking into consideration the existing correlation between the parameters of the electrode. The calculations apply either covariances or correlation coefficients that have to be computed additionally. The example presented in this paper illustrates that there are some problems with rounding of variables which, because of the existing very strong correlations, significantly changes the sought uncertainty. This approach is compared with other approaches, that is, usage of uncorrelated variables and Monte Carlo simulations that are described in an earlier work. It is concluded that the approach of uncertainty evaluation, in which covariances or correlation coefficients are explicitly calculated, is work-consuming and may cause significant discrepancies between correct and obtained assessments if some roundings or approximations are done, or if the correlation coefficient is obtained experimentally based on data including random errors.


Correlation coefficient Covariance Uncertainty propagation pH measurement 



The author would like to thank the anonymous reviewers for their comments on an earlier draft of this manuscript and the Polish Ministry of Science and Higher Education [BK/265/RAu1/2014 (02/010/BK_14/0031)] for financial support.


  1. 1.
    JCGM 100 (2008) Evaluation of measurement data—guide to the expression of uncertainty in measurement.
  2. 2.
    Adams TM (2002) A2LA guide for the estimation of measurement uncertainty in testing. American Association for Laboratory Accreditation.
  3. 3.
    QUAM (2012) Guide CG 4 Quantifying uncertainty in analytical measurement. EURACHEM / CITAC.
  4. 4.
    EA-4/02 (2013) Expression of the uncertainty of measurement in calibration. European co-operation for Accreditation.
  5. 5.
    EA-4/16 (2003) EA guidelines on the expression of uncertainty in quantitative testing. European co-operation for Accreditation.
  6. 6.
    Bremser W, Hässelbarth W (1998) Shall we consider covariances? Accredit Qual Assur. doi: 10.1007/s007690050199 Google Scholar
  7. 7.
    Hyk W, Stojek Z (2013) Quantifying uncertainty of determination by standard additions and serial dilutions methods taking into account standard uncertainties in both axes. Anal Chem. doi: 10.1021/ac4007057 Google Scholar
  8. 8.
    Meija J, Pagliano E, Mester Z (2014) Coordinate swapping in standard addition graphs for analytical chemistry: a simplified path for uncertainty calculation in linear and nonlinear plots. Anal Chem. doi: 10.1021/ac5014749 Google Scholar
  9. 9.
    Wulff SS, Weitz MA (2005) Measurement uncertainty in the calibration of low-flow ambient air samplers. Qual Reliab Eng Int. doi: 10.1002/qre.678 Google Scholar
  10. 10.
    Wiora J (2015) Uncertainty evaluation of pH measured using potentiometric method. Adv Intell Syst Comput. doi: 10.1007/978-3-319-11310-4_45 Google Scholar
  11. 11.
    Morf WE (1981) The principles of ion-selective electrodes and of membrane transport. Akadémiai Kiadó, BudapestGoogle Scholar
  12. 12.
    Buck RP, Rondinini S, Covington AK, Baucke FGK, Brett CMA, Camoes MF, Milton MJT, Mussini T, Naumann R, Pratt KW, Spitzer P, Wilson GS (2002) Measurement of pH. Definition, standards, and procedures (IUPAC recommendations 2002). Pure Appl Chem. doi: 10.1351/pac200274112169 Google Scholar
  13. 13.
    Malengo A, Pennecchi F (2013) A weighted total least-squares algorithm for any fitting model with correlated variables. Metrologia. doi: 10.1088/0026-1394/50/6/654 Google Scholar
  14. 14.
    Leito I, Strauss L, Koort E, Pihl V (2002) Estimation of uncertainty in routine pH measurement. Accredit Qual Assur. doi: 10.1007/s00769-002-0470-2 Google Scholar
  15. 15.
    JCGM 101 (2008) Propagation of distributions using a Monte Carlo method.

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of Automatic ControlSilesian University of TechnologyGliwicePoland

Personalised recommendations