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

 

RBSilva, 2022/05/18

 

Link to the spreadsheet: http://mechem.rd.ciencias.ulisboa.pt/wp-content/uploads/sites/99/2022/05/Managing_parameters_determined_by_measured_values_difference_VerC.xlsx