Characteristic functions

AQS-BW - Measurement uncertainty

from German PT data

This web page is intended to show the results of the calculation of characteristic functions from German water PT data. Data from German organizers of water PTs from 2000 to 2017 (mainly AQS Baden-Württemberg) are used to show the concentration dependence of the relative reproducibility standard deviation [1]. A characteristic function [2-4] of the following form is used to describe this dependence.

where y(hat) is the relative standard deviation and x is the concentration of the analyte.

The function is determined from several, different data sets by weighted regression (least squares regression).

For small concentrations, this function implies that the absolute standard deviation is constant and equal to α. For high concentrations, the relative standard deviation is assumed to be constant and equal to β. The concentration describing the transition from constant absolute to constant relative standard deviation (the break point) can be calculated from α/β.

α and β are estimated from PT data. Due to evaluation problems, concentrations close to the limit of quantification are usually avoided in proficiency tests. Therefore, in most cases the estimation of the parameter β is possible, but the estimation of α from PT data is often not reliable enough. In the table linked below, α-values are given only if the breakpoint is above the 1st quartile of the data set, i.e. 25% of the data are below the breakpoint. In the same way, β-values are only given if the breakpoint is below the 3rd quartile of the data set, so that 25% of the data are above the breakpoint.

The breakpoint

For many instrumental methods, it is common practice to specify relative measurement uncertainties for high concentrations and absolute uncertainties for small concentrations. Many laboratories wonder how to select the limiting concentration. The "breakpoint" calculated from the characteristic function can be used for this purpose, assuming that the concentration dependence of intra-laboratory precision is comparable to that of inter-laboratory precision.

Constant absolute standard deviation α at low concentrations

A common way to calculate the limit of quantification is LOQ = 10 · sr,0, where sr,0 is the repeatability standard deviation of a blank. If we assume, as is often done, that the repeatability standard deviation is approximately half the reproducibility standard deviation, then we can estimate a mean limit of quantitation of participating laboratories fro LoQ = 10·sR,0/2 = 5·α. Where a valid α-value could be calculated from the interlaboratory data, a BG is indicated in the tabulation. In all other cases it is indicated that the BG is smaller than 5 times the absolute standard deviation at the lowest concentration.

Please note that the calculated limits of quantification depend strongly on the concentrations of the samples analyzed in the proficiency test. High concentrations also lead to high limits of quantification. Conversely, α-values for the characteristic function can also be calculated from laboratory-specific limits of quantification.

Constant relative standard deviation β at high concentrations

In ISO 11352 and in NORDTEST technical report TR537 a method for measurement uncertainty estimation is described that quantifies the precision component and the component of uniqueness due to systematic deviations separately and then combines them. For the precision component, the reproducibility within the laboratory has to be estimated and quantified as standard deviation uR,w. If we assume that the repeatability standard deviation is half of the reproducibility standard deviation, then the mean standard deviation describing within-laboratory reproducibility will be approximately 0.8·sR, and thus uR,w=0.8·β. This value is also given in the table below. If no valid β-value could be estimated, a value of < 0.8 times the function value at the highest concentration is given instead. This value for uR,w can be used by laboratories to check the plausibility of their own estimates.


According to ISO 21748, the expanded uncertainty, U, is equal to 2·sR. With the estimated values α and β, the uncertainty can now be determined over the entire concentration range using the characteristic function.

Legal regulations, such as the European Drinking Water Directive or the European Water Framework Directive, require that the analytical methods used meet certain uncertainty requirements. The characteristic function - with its parameters estimated as described above - shows which measurement uncertainty can be achieved on average for a certain content in reality.

[1] Data is from the following sources: 

  • AQS Baden-Württemberg, Stuttgart
  • Institut für Hygiene und Umwelt der Freien und Hansestadt Hamburg
  • Niedersächsiches Landesgesundheitsamt, Standort Aurich
  • Landesamt für Natur, Umwelt und Verbraucherschutz Nordrhein-Westfalen
  • Bayerisches Landesamt für Umwelt
  • Landesbetrieb Hessisches Landeslabor
  • Niedersächsischer Landesbetrieb für Wasserwirtschaft, Küsten- und Naturschutz
  • Landesamt für Umwelt- und Arbeitsschutz
  • Staatliche Betriebsgesellschaft für Umwelt und Landwirtschaft Sachsen

[2] Thompson, M., Mathieson, K., Damant, A.P., and Wood, R.: A general model for interlaboratory precision accounts for statistics from proficiency testing in food analysis. Accred. Qual. Assur. (2008) 13:223-230).
[3] Thompson, M, and Coles, B.J.: Examples of the ‘characteristic’ function applied to instrumental precision in chemical measurement. Accred. Qual. Assur. (2009) 14:147-150.
[4] Thompson, M, and Coles, B.J.: Use of ‘characteristic function’ for modelling repeatability precision. Accred. Qual. Assur. (2011) 16:13-19.

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