= Analytical evaluation function
An <analytical evaluation function> converts an observed analytical response into an estimated <analyte> amount or <concentration> by inverting an <analytical calibration function>. If the forward function is $y=f(c)$ on a specified calibrated interval, the <analytical evaluation function> is $g=f^{-1}$ on the corresponding response interval. IUPAC defines the evaluation function as this inverse.<Evaluation function in the IUPAC Gold Book>[E02226, Definition]{ref}
The direction matters. A forward model describes an expected response at a given <concentration>; its <inverse function> is used with the response observed for an unknown sample. For an original dimensionless model $f(c)=c^2$ restricted to $c\geq0$, the <analytical evaluation function> is $g(y)=\sqrt y$. Response4gives estimate2. Without the nonnegative restriction, the same response would leave the two possibilities $c=\pm2$, so there would be no single-valued <inverse function>.
The <inverse function rule> gives a local <sensitivity coefficient>:
$$
g'(y)=\frac1{f'(g(y))},\qquad u_c\simeq|g'(y)|u_y.
$$
For the example at $y=4$, $g'(4)=1/4$, so a response <standard uncertainty> of0.2would contribute approximately0.05to the <concentration> <standard uncertainty>. Uncertainty in the fitted <analytical calibration function> adds further contributions and may introduce <covariances>.
Inversion also changes the effect of averaging. Equal-probability responses3and5have mean4, yet
$$
\frac{\sqrt3+\sqrt5}{2}\approx1.9841\ne\sqrt4=2.
$$
Thus inverting the mean response and averaging individually inverted responses need not agree. This original calculation illustrates a nonlinear transformation, not a universal correction for analytical bias. The calibrated range, response noise and uncertainty in the forward fit must be specified before interpreting the resulting estimate.
Back to article page
