English: By the Law of Propagation of Uncertainty (see section 5.1.2 of the ISO Guide to the Expression of Uncertainty in Measurement) and assuming no covariance, we have:  (ucombined(mt))2 = (∂f/∂mR(u(mR)))2 + (∂f/∂at (u(at)))2 + (∂f/∂vt (u(vt)))2 + (∂f/∂vR (u(vR)))2 + (∂f/∂Dt(u(Dt)))2  Taking the partial derivatives of the function f(mR, at, vt, vR, Dt), we have:  ∂f/∂mR ≈ 1  ∂f/∂vR ≈ at  ∂f/∂vt ≈ at ∂f/∂at ≈ (vt - vR)  ∂f/∂Dt ≈ 1  Please note that using the equation: mt(1-(at/t)) = (mR - mR at/R + Dt), we get:  mt = {1/(1-(at/t))} {mR - mRat/R + Dt} instead of equation "f" above, we have the following derivatives without approximation:  ∂f/∂mR = {t/(t - at)} χ {(R -at)/R}  ∂f/∂Dt = {t /(t - at) χ1}  ∂f/∂R = {t /(t - at)χ - at mR/ R2}  ∂f/ ∂t χ {t /(t - at)}{- at mt/t2}  ∂f/∂at = {t /(t - at)χ(mR/ R)((R - t)/(t - at) + Dt /(t - at)}  Hence:  (ucombined(mt))2 ≈ [1(u(mR))]2+[(vt-vR)(u( at))]2+ [at(u(vt))]2 + [- at(u(vR))]2 + [1(u(Dt))]2  (ucombined(mt))2 ≈ [u(mR)]2+[(mt/t - mR/R)(u(at))]2+[at(u(vt))]2+[-at(u(vR))]2+[u(Dt)]2  Note that, again by the Law of Propagation of Uncertainty, we obtain   (u(vt))2 = u2(mt/t) = ( ∂(vt)/ ∂mt)2 u2(mt) + ( (∂vt)/ ∂t)2 u2(t) = (1/t)2 u2(mt) +(-mt/ t2)2 u2(t).  However, if the density determination is performed independently from the mass determination, we have:  (u(vt))2 ≈ ( ∂(vt)/t)2 u2(t) (-mt/t2)2 u2(t) and similarly (u(vR))2 ≈ (-mR/ R2)2 u2(R).  Therefore:  (ucombined(mt))2 ≈ [1(u(mR))]2+[(vt-vR)(u(at))]2+ [at(u(vt))]2 + [-at(u(vR))]2 + [1(u(Dt))]2 ≈ [u(mR)]2 + [(mt/t - mR/R)× u(at)]2 + [at(-mt/t2)× u(t)]2 + [-at(-mR/R2)× u(R)]2+[u(Dt)]2  u2 (mt) ≈ u2(mR) + (mt/t-mR/R)2u2(at) + at2(-mt/t2)2u2(t) + at2 (-mR/R2)2 u2(R) + u2(Dt)  Since at2 (-mR/R2)2 u2(R) should already be included in u2(mR), the combined standard uncertainty for the absolute mass mt is the square root of the following: u2 (mt) ≈ u2(mR) + (mt/t-mR/R)2u2(at) + at2(-mt/t2)2u2(t) + u2(Dt) where all terms are expressed in the same unit of mass  Please note that the above equation without approximation is: u2(mt) = [t /(t-at)]2 { u2(Dt) + [(R-at)/R]2 u2(mR) + [mRat/(R)2]2 u2(R) + [-mtat/(t)2]2 u2(t) + [(mR/R)(R-t)/(t-at) + Dt where all terms are expressed in the same unit of mass  3.2 Combined Uncertainty Equation for Conventional Mass  The equation below is the equation for the combined uncertainty associated with a calibration or tolerance test in terms of conventional mass.
French: Selon la loi de propagation d'incertitude (voir section 5.1.2 du Guide ISO pour l'expression de l'incertitude de mesure) et en supposant l'absence de covariance, on obtient :  (ucombined(mt))2 = (∂f/∂mR(u(mR)))2 + (∂f/∂at (u(at)))2 + (∂f/∂vt (u(vt)))2 + (∂f/∂vR (u(vR)))2 + (∂f/∂Dt(u(Dt)))2  Les dérivées partielles de la fonction f(mR, at, vt, vR, Dt), donnent : ∂f/∂mR ≈ 1 ∂f/∂vR ≈ -at  ∂f/∂vt ≈ at ∂f/∂at ≈ (vt - vR)  ∂f/∂Dt ≈ 1  Il convient de noter qu'en utilisant l'équation: mt (1-(at/t)) = (mR - mRat/R + Dt), on obtient :  mt = {1/(1-(at/t))} {mR - mRat/R + Dt} au lieu de l'équation « f » ci-dessus, ce qui donne les dérivées suivantes sans approximation :  ∂f/∂mR = {t/(t - at)}{(R -at)/R}  ∂f/∂Dt = {t /(t - at)}{1}  ∂f/∂R = {t /(t - at)}{- at mR/ R2}  ∂f/ ∂t = {t /(t - at)}{- at mt/t2}  ∂f/∂at = {t /(t - at)}{(mR/ R)((R - t)/(t - at) + Dt /(t - at)}  Donc : (ucombined(mt))2 ≈ [1(u(mR))]2+[(vt-vR)(u(at))]2+ [at(u(vt))]2 + [- at(u(vR))]2 + [1(u(Dt))]2  (ucombined(mt))2 ≈ [u(mR)]2+[(mt/t - mR/R)(u(at))]2+[at(u(vt))]2+[-at(u(vR))]2+[u(Dt)]2  Encore une fois, en utilisant la loi de propagation d’incertitude, on obtient : (u(vt))2 = u2(mt/t) = ( ∂(vt)/ ∂mt)2 u2(mt) + ( (∂vt)/ ∂t)2 u2(t) = (1/t)2 u2(mt) +(-mt/t2)2 u2(t).  Toutefois, si la détermination de la masse volumique est effectuée indépendamment de la détermination de la masse, on obtient :  (u(vt))2 ≈ ( ∂(vt)/t)2 u2(t) (-mt/t2)2 u2(t) and similarly (u(vR))2 ≈ (-mR/ R2)2 u2(R).  En conséquence : (ucombined(mt))2 ≈ [1(u(mR))]2+[(vt-vR)(u(at))]2+ [at(u(vt))]2 + [-at(u(vR))]2 + [1(u(Dt))]2 ≈ [u(mR)]2 + [(mt/t - mR/R)× u(at)]2 + [at(-mt/t2)× u(t)]2 + [-at(-mR/R2)× u(R)]2+[u(Dt)]2  u2 (mt) ≈ u2(mR) + (mt/t-mR/R)2u2(at) + at2(-mt/t2)2u2(t) + at2 (-mR/R2)2 u2(R) + u2(Dt)  Étant donné que at2 (-mR/R2)2 u2(R) devrait déjà être compris dans u2(mR), l'incertitude-type combinée pour la masse absolue mt est la racine carrée de ce qui suit :  u2 (mt) ≈ u2(mR) + (mt/t-mR/R)2u2(at) + at2(-mt/t2)2u2(t) + u2(Dt) où tous les termes sont exprimés dans la même unité de masse  Il convient de noter que l'équation susmentionnée sans approximation est ; u2(mt) = [t /(t-at)]2 { u2(Dt) + [(R-at)/R]2 u2(mR) + [mRat/(R)2]2 u2(R) + [-mtat/(t)2]2 u2(t) + [(mR/R)(R-t)/(t-at) + Dt où tous les termes sont exprimés dans la même unité de masse