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An analytical calibration function relates the amount or concentration of an analyte to the expected response of a measurement process under specified conditions. Writing analyte quantity as $c$ and observed signal as $Y$, a forward model is $\mathbb{E}[Y\mid c]=f(c)$. IUPAC defines the calibration function through expected gross signal, rather than through an exact noise-free value for each individual reading.^[ref]

For an original linear response model, choosein consistent chosen signal/concentration units. A measured signal $y=3.20$ gives estimate $\hat{c}=(y-a)/b=1.50$. The inverse conversion is an analytical evaluation function when the forward function is invertible on the calibrated range.^[ref] A calibration plot and its inverse computation have different input/output roles.

One way to estimate the relation is to prepare standards with reference concentrations, measure replicate signals, fit a response model and examine calibration residuals. If noise has different variance at different concentrations, an unweighted straight-line fit need not represent the measurement errors adequately. The reference concentrations and preparation procedure also carry measurement uncertainty.

For the linear inverse, first-order error propagation with independent uncertainties in $y$, $a$ and $b$ givesAt the example estimate, choosing $u_y=0.10$, $u_a=0.04$ and $u_b=0.10$ gives $u_c\simeq 0.0923$ in concentration units. Treating the fitted coefficients as exact would give only $u_y/b=0.05$. If the fitted slope and intercept are correlated, their covariance terms must also be included; the displayed formula assumes independence solely for this calculation.

The response need not be linear. An inverse must respect the fitted range and any monotonicity restrictions. Nor does inverting an expected-response curve generally commute with taking an expectation for a nonlinear curve. Measurement calibration supplies a relation and uncertainty model; it does not guarantee that extrapolation, another sample matrix or another instrument state obeys the same 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.