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 CaC0_{3}) = 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

**(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.*

**LOQ***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:** https://mechem.rd.ciencias.ulisboa.pt/wp-content/uploads/sites/99/2022/05/Managing_parameters_determined_by_measured_values_difference_VerC.xlsx