Inverse calibration and fitted-parameter uncertainty
For the valid linear calibration y=a+bx, an unknown input is estimated as x̂=(y₀−â)/b̂.
Key knowledge
For the valid linear calibration y=a+bx, an unknown input is estimated as x̂=(y₀−â)/b̂. Its uncertainty includes the future indication, fitted intercept/slope, their covariance, reference standards and applicable extra influences. First-order propagation uses gradient (1/b̂,−1/b̂,−x̂/b̂) with respect to (y₀,â,b̂).
Values and equations
x̂=(y₀−â)/b̂; u²(x̂)≈gᵀΣg; g=(1/b̂,−1/b̂,−x̂/b̂)
- x̂
- Estimated input; measurand unit
- y₀,â
- Future indication and fitted intercept; indication unit
- b̂
- Fitted slope; indication unit per measurand unit
- g
- Gradient; appropriate output/input units
- Σ
- Joint covariance matrix of (y₀,â,b̂); entry-specific units
- u(x̂)
- Input standard uncertainty; measurand unit
Conditions and scope
- Nonzero adequately determined slope, future indication within validated range and a locally linearizable joint uncertainty model.
- Use the covariance of all fitted parameters and include any dependence between new indications and calibration data. Uncertain near-zero slope or nonlinear inversion may require distribution propagation and asymmetric intervals.
Sources · 3
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
National Institute of Standards and Technology (NIST)3 sources
Evidence and review
NIST/SEMATECH e-Handbook: Calibration of future measurements: Inverse linear indication formula and nonlinear root/range choices; excludes inferring equivalence from failure to reject a=0,b=1. NIST/SEMATECH e-Handbook: Uncertainty of calibrated values: Parameter uncertainty plus future measurement and applicable additional contributions. NIST/SEMATECH e-Handbook: Propagation of calibration errors: Partial derivatives and parameter covariance; the displayed quadratic example's general propagation method specialized here to a linear inverse.
2026-10-10 · Independent AI editorial review: Independent root AI editorial review of all29 authored Measurement claims/conditions, displayed equations and37 origin relations. The original26 claims were read against41 primary source sections; all5 requested corrections and the3 affected source bindings were subsequently rechecked. Three actual scalar-nonlinear/vector/dynamic applications were read, independently differentiated or integrated, and checked against the specific additional source material described below. This is reference-content review, not human specialist certification, instrument certification or experimental replication. Full-field sign-off waits for the combined final conceptual matrix. Receipt: measurement-independent-content-review.1.
Entry details
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- measurement-inverse-calibration-uncertainty
- Revision
- university-core.1
- Structured reference record
- University core expansion