 Calibration residual

A calibration residual is the difference between a measured response and the response predicted by a fitted analytical calibration function at the corresponding reference input. With observed signals $y_i$, reference concentrations $c_i$ and fitted function $\hat{f}$, choose the conventionA calibration residual is an observable diagnostic of a fit; it is not the unobserved measurement error relative to a true response. Plotting regression residuals against reference input, fitted response and measurement order can reveal different model failures.^[ref]

For an original example in chosen consistent concentration and signal units, take reference inputs $c=(0,1,2)$ and responses $y=(1,3.2,5)$. Ordinary least squares givesThe calibration residuals sum to zero, as expected for an ordinary least squares fit with an intercept, but they are not individually zero. The sum of their squares is $2/75$; with three observations and two fitted coefficients, the residual estimate of noise standard deviation is $\sqrt{(2/75)/(3-2)}\approx 0.1633$signal units. One residual degree of freedom provides a weak estimate, rather than strong evidence for the assumed noise law.

A curved residual pattern suggests a missing response term, while changing scatter can suggest nonconstant noise variance. These diagnoses interact: an incorrect response curve itself can create apparent changes in scatter. Replicate standards and measurements across the intended range help distinguish them.^[ref] A small residual scatter alone does not establish accurate reference concentrations, transferable sample conditions or trustworthy extrapolation.