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.^[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 $c\ge 0$, the analytical evaluation function is $g(y)=\sqrt{y}$. Response4gives estimate2. Without the nonnegative restriction, the same response would leave the two possibilities $c=\pm 2$, so there would be no single-valued inverse function.

The inverse function rule gives a local sensitivity coefficient:For the example at $y=4$, $g^{\prime }(4)=1/4$, 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, yetThus 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.