Calibration

Relates instrument indications to reference values using models, corrections, and uncertainties.

Conceptual context

Calibration does not simply “fix” an instrument. It compares its $y$ with a traceable $x_ref$ to reveal how the instrument actually responds. That comparison identifies systematic deviations, supports a $C$, and defines the interval over which the result is defensible. Calibration turns an isolated display value into physically interpretable information because it links the response of the instrument to known standards, operating conditions, and a stated $u_c$. The goal is not perfect truth, but a quantified relationship between indication and the best available estimate of the measurand.

🟢 Essential level

The central idea is to compare several instrument readings with known values. Each $x_ref$ provides a trusted point, and each $y$ shows how the instrument responds. If the points follow a regular trend, that response can be represented by a straight line. The $a$ tells how strongly the reading changes when the reference changes, while the $b$ reveals a possible zero offset. The difference between reading and reference is the $E$, and the $C$ has the opposite sign because it must compensate for that deviation. A useful calibration also states the interval from $x_min$ to $x_max$ and a $u_c$ describing the remaining doubt. Calibration therefore does not remove uncertainty; it converts uncertainty into explicit, usable information.

The procedure is meaningful only when the reference is trustworthy and the instrument is used under controlled conditions. A good result therefore includes a corrected number together with its conditions and limitations. Repeating the calibration periodically also reveals instrument drift caused by time, use, or wear, confirming whether the $C$ obtained remains valid.

🔵 Formal level

Basic formulas

Within the calibrated interval, the response is approximated by the linear model:

Linear calibration model

The $y$ is expressed as a function of $x_ref$, with $a$ and $b$ obtained from the fit. The $a$ gives response per unit reference and, for a linear instrument, matches the mean $S$. Sensitivity may also be estimated from increments:

Calibration sensitivity

where $\delta_{y}$ and delta_x_ref must refer to the same pair of calibration points. The signed deviation is defined by:

Indication error

Real student questions

Is calibration the same as adjustment? No. Calibration characterizes performance; adjustment changes the instrument.

Why does $C$ have the opposite sign to $E$? Because the $C$ must cancel the observed deviation when added to the reading.

Does a perfect straight line guarantee correct measurement? No. A response can be highly linear but systematically displaced from the reference.

Is $u_c$ the same as error? No. Error is an estimated deviation; uncertainty expresses incomplete knowledge of the assigned result.

May I use the calibration outside $x_min$ and $x_max$? Only as unsupported extrapolation, not with the same metrological claim.

Are more points always better? They help only when they cover the interval well and provide independent information.

Final synthesis

Calibration builds a physical relationship between $x_ref$ and $y$, identifies $E$, applies $C$, and states $u_c$ over a valid interval. The graph reveals whether the linear model is adequate and whether its parameters remain physically meaningful.