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 Hardness – Calcium Hardness
Chromium (III) = Total Chromium – Chromium (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. T‘d = 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 d, T‘d – Cell D11
The output of the spreadsheet, the LOQ for d determinations, is on cell D14