Managing the uncertainty and limit of quantification of parameters determined by the difference between independently measured values

Many chemical parameters are determined as the difference, d, between two measured values, designated as larger (g) and lower (w) inputs, (d = g – w; where g > w) determined from independent measurements such as:


Magnesium Hardness (mg/L of CaC03) = Total HardnessCalcium Hardness

Chromium (III) = Total ChromiumChromium (VI)


The uncertainty associated with d depends on the uncertainty on g and w. It is known from the uncertainty propagation law that the square of the standard uncertainty of d, u^2(d), is equal to the sums of the squares of the standard uncertainties of g and w (u^2(d) = u^2(g)+u^2(w)).

As the difference decreases, their relative standard uncertainty (u‘(d) = u(d)/d) raises. Therefore, it is crucial to determine the minimum difference that can be quantified with an acceptable uncertainty to set a limit of quantification, LOQ, for the difference.

For cases where the LOQ is estimated as ten times the intermediate precision standard deviation and precision is a major uncertainty component at the LOQ (i.e. trueness/bias and additional uncertainty components are negligible), the LOQ should be associated with a relative standard uncertainty of about 10% (i.e. Td = 10%).

The enclosed spreadsheet (Link) allows determining the LOQ of d, by specifying the values of the following variables:


g – Cell D7

Relative standard uncertainty of g, u‘(g) = u(g)/g – Cell D9

Relative standard uncertainty of w, u‘(w) = u(w)/w  – Cell D10

Target relative standard uncertainty of dTd – Cell D11


The output of the spreadsheet, the LOQ for d determinations, is on cell D14


RBSilva, 2022/05/18


Link to the spreadsheet: