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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.