Metrology

Metrology asks what a measurement means and how closely it can be reproduced. An instrument reading becomes useful when it is related to a reference through measurement calibration, with a stated measurement uncertainty. Measuring a changing object against a changing ruler makes the problem especially clear: an observed difference contains the responses of both.

A scientific instrument maker designs, constructs or improves scientific instruments, turning a proposed measurement principle into a working apparatus. Geometry, material choice, adjustment and readable scales can determine whether the intended measurement resolution and measurement calibration are achievable. The Fortin barometer illustrates why the design of a reference surface and a reading method belongs to the science of a measurement, as well as to its manufacture.

Instrument maker associated with the Fortin barometer.

A process for determining the value of a physical quantity relative to a reference and a stated model. A reported result must distinguish its numerical value, units and measurement uncertainty.

measurement accuracy concerns closeness to the value of the specified measurand; high repeatability alone does not establish it. calibration, an adequate response model and assessment of residual systematic effects are required alongside statistical precision.

The measurand is the quantity intended to be measured, specified with enough detail about the system, conditions and averaging convention to distinguish it from instrument output or a different derived quantity.

An organization publishing internationally coordinated formulations and reference data for the physical properties of water and steam.

Official website: iapws.org

Measurement of length and geometrical dimensions. A length standard supplies a reference; a differential length comparator measures the difference between two lengths rather than treating its own structure as perfectly invariant.

A reference realizing a specified length, under specified conditions. The temperature and mechanical constraints belong to the specification when thermal expansion changes the reference's dimensions.

An instrument for comparing two lengths through their difference. Suppose a test rod and reference both have length $L_0$ at temperature $T_0$, and have constant linear thermal expansion coefficients ${\alpha }_s$ and ${\alpha }_r$ over a small interval. Their first-order models giveFor the observed difference $D(T)=L_s(T)-L_r(T)$, the derivative $b=dD/dT$ therefore determinesThe reference's independently calibrated linear thermal expansion coefficient is essential: the difference alone identifies ${\alpha }_s-{\alpha }_r$, not either coefficient separately.

For an original numerical example, stipulate $L_0=1.00\mathrm{m}$, ${\alpha }_s=1.0\times 10^{-6}{\mathrm{K}}^{-1}$ and ${\alpha }_r=9.0\times 10^{-6}{\mathrm{K}}^{-1}$. Heating both by $10\mathrm{K}$ gives sample and reference increases of $10\mu \mathrm{m}$ and $90\mu \mathrm{m}$. Thus the sample expands even though its measured difference decreases by $80\mu \mathrm{m}$. These are chosen model parameters, not measured values for a historical Invar specimen or reference.

If ${\alpha }_r$ and $b$ have independent standard uncertainties $0.2\times 10^{-6}{\mathrm{K}}^{-1}$ and $0.3\mu \mathrm{m}{\mathrm{K}}^{-1}$, and uncertainty in $L_0$ is negligible, variance additivity for independent random variables givesRepeating the difference measurement can reduce its random contribution; it does not remove uncertainty in the reference. A temperature gradient or unequal rod temperatures also breaks the common-temperature model and can introduce a systematic error.

Alternating sample and reference readings can suppress instrument drift. In this original four-reading model, measure objects $A,B,B,A$ at equally spaced times $0,\Delta t,2\Delta t,3\Delta t$. Assume the objects are stable during that short sequence and each reading is its object's length plus the same affine function of time, $a+bt$. Thenbecause both pairs have mean time $3\Delta t/2$. The constant offset and linear drift cancel. With a further drift term $c{t}^2$, direct substitution instead leaves bias $-2c\Delta t^2$. A shorter sequence reduces that term, but neither this ordering nor averaging corrects unequal temperatures, sample changes or reference measurement uncertainty. This is a modern illustrative protocol; the historical account below establishes alternation, without attributing this particular four-reading sequence to Charles-Édouard Guillaume.

Charles-Édouard Guillaume sought affordable, stable length standards. His 1920 lecture describes two micrometer microscopes on stone supports sighting marks on a rod immersed in water. Their separation was not fixed: one method alternated the sample with a reference in a second trough at nearly constant temperature. His relative method compared sample and platinum–iridium reference in the same bath, adding the reference's known expansion to the measured difference. Composition and treatment affected the nickel–iron alloys; low expansion held over a limited temperature range.^[ref]

The differential length comparator calculation explains why a calibrated reference is needed even when its expansion is larger than the sample's. Keeping both rods in the same bath targets one source of systematic error; allowing for changes in the instrument baseline targets another. Discovering a low-expansion alloy and demonstrating its dimensional stability are connected experimental tasks.

Measurement uncertainty describes the spread of values reasonably attributable to a specified measurand, using the available information. It differs from measurement error, which is a difference from a reference value: the uncertainty can be quantified even when that difference is unknown. A useful result therefore states what was measured, the unit of measurement, the value, and how its uncertainty was evaluated.^[ref]

For a wire's electrical resistance obtained from $R=V/I$, uncertainty in the voltage and electric current propagates through that measurement model. Shared measurement calibration effects can make the inputs correlated, so adding percentages or treating every contribution as independent can be wrong. The measurement uncertainty budget records those assumptions. Report a standard uncertainty when a standard-deviation scale is intended, or an expanded uncertainty with its stated coverage factor; a bare plus-or-minus number does not specify which was used.

A record of uncertainty contributions in a specified measurement model, their evaluation methods, sensitivity coefficients and relevant correlations. For $y=g(x_1,\dots ,x_n)$, first-order propagation of uncertainty gives $u_y^2={\sum }_{ij}({\partial }_{i}g)({\partial }_{j}g)\mathrm{Cov}(x_i,x_j)$. A correction for a known systematic error can itself have uncertainty; applying the correction does not remove that contribution.^[ref]

A systematic error is a component of measurement error that remains constant or changes predictably when measurements are repeated under specified conditions. A ruler with an incorrect scale, a thermometer with a measurement zero offset, and an unaccounted instrumental drift can all produce such effects. Closely agreeing readings can therefore have poor measurement accuracy: repetition tests their scatter, not the correctness of the shared reference.

An estimated systematic error should be corrected when possible, with the correction's measurement uncertainty included in the measurement uncertainty budget. Suppose a balance reads $m+b$ and a reference gives an estimate $\hat{b}$ of its offset. The corrected result is $m_{\mathrm{read}}-\hat{b}$; uncertainty in $\hat{b}$ remains even if the same unknown offset affected every weighing. Changing the reference, using a blank, or testing systematic-error cancellation by reversal can reveal effects that a long sequence of identical readings misses. The Type A evaluation of uncertainty and Type B evaluation of uncertainty distinction concerns how uncertainty is evaluated, so neither label is synonymous with systematic error.^[ref]

Uncertainty associated with imperfect knowledge of calibration, efficiency, backgrounds or other persistent model inputs. Repeating the same measurement does not necessarily average it away. Correlated nuisance parameters can propagate it through a likelihood; estimating such an uncertainty is distinct from asserting that the systematic error has a known sign and magnitude.

If a controlled reversal changes the desired signal's sign while leaving an additive offset unchanged, readings $y_{+}=S+b$ and $y_{-}=-S+b$ separate them by half-difference and half-sum. Offset drift, gain changes, hysteresis or a background that also reverses violate the simple cancellation. Interleaving polarities and independent blank measurements test those assumptions; signal reversal does not cancel an arbitrary systematic error.

If a wanted response changes sign on reversing a controlled setting while an offset remains unchanged, the difference of the reversed readings isolates the response. For $V_{\pm }=\pm RI+V_0$, the estimator $(V_{+}-V_{-})/(2I)$ cancels $V_0$. The method requires offset stability between readings. It does not cancel a common current-scale calibration error, and it does not remove even-in-current Joule heating. A useful reversal test specifies which physical effects change sign, then tests the remaining effects independently.

Measurement calibration establishes a relation between an instrument's indications and reference values, with the relevant measurement uncertainty. It is distinct from changing the instrument until it reads correctly: an instrument can be calibrated by determining a response curve and applying a correction afterward.

A practical voltage experiment connects a meter to several reference values across its working range and repeats the sequence in both directions. Fitting $V_{\mathrm{indicated}}=a{V}_{\mathrm{reference}}+b$ estimates a scale factor and measurement zero offset; departures from that relation test the response model. Repetition at later times checks instrumental drift. Uncertainty in the reference, the fitted parameters, and unresolved departures belongs in the measurement uncertainty budget, even if the displayed readings appear very stable.^[ref]

A constant additive contribution $b$ to an instrument reading is a zero offset. A blank or zero-signal reading can estimate it, and a systematic-error cancellation by reversal can cancel it if it remains unchanged while the desired signal changes sign. The residual correction uncertainty remains part of the measurement uncertainty.

A change in an instrument's response or zero point with time. A reference measured alongside the target can track the component shared by reference and target light.

A force probe measures an applied force through a calibrated response. A bending beam with strain gauges is one realization; orientation and measurement calibration matter.

A balance that applies a compensating force to hold its load support near a reference position; the compensation signal measures the applied force.

A metrologist studies measurements and the references, methods and uncertainties that make them comparable. The work connects a specified measurand to measurement calibration, measurement uncertainty and reproducible units of measurement. An experiment that improves the stability of a length standard or checks the response of a force probe is therefore investigating how confidently other experiments can compare their results.

An author credited on Mercury Barometers and Manometers by W. G. Brombacher and others.

An author credited on Mercury Barometers and Manometers by W. G. Brombacher and others.

An author credited on Mercury Barometers and Manometers by W. G. Brombacher and others.

Coauthor of Maintenance and Validation of Liquid-in-Glass Thermometers by Christina D. Cross and others.

Identity: www.nist.gov/people/paul-williams

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Identity: tsapps.nist.gov/publication/get_pdf.cfm?pub_id=914101

Identity: www.nist.gov/people/john-h-lehman

Homepage: www.nist.gov/people/gregory-f-strouse

Homepage: www.nist.gov/publications/new-nist-automated-calibration-system-industrial-grade-platinum-resistance-thermometers

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Homepage: tsapps.nist.gov/publication/get_pdf.cfm?pub_id=830609

Homepage: www.npl.co.uk/our-people/jonathan-pearce

Homepage: www.npl.co.uk/our-people/richard-rusby

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Homepage: www.ptb.de/cms/ptb/fachabteilungen/abt7/abteilungsbericht2021.html

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Homepage: www.inrim.it/en/contacts/giuseppina-lopardo

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A metrologist.^[ref]

A metrologist.^[ref]

A calibrated vessel used to infer a liquid's mass density from its mass and filled volume; see pycnometer density measurement for an inspected experiment and the accounting formula.

Measurement resolution is the smallest change in an instrument indication that its reading method can distinguish. It contributes to measurement uncertainty, but does not by itself establish accuracy or remove systematic error. The manometer height-to-pressure calibration shows how a height scale affects the inferred pressure.

A unit of measurement is a chosen reference for comparing magnitudes of a physical quantity. A value written as a number times a measurement unit changes its numerical part when the measurement unit changes, while representing the same physical quantity.

One thousandth of a litre, equal to one cubic centimetre.

A unit of volume equal to one cubic decimetre.

The British pharmaceutical volume of 60 minims; 8 make an Imperial fluid ounce.

A unit of volume; the Imperial and United States measures differ. An Imperial fluid ounce contains 8 fluid drachms.

One eighth of an Imperial gallon, containing 20 fluid ounces.

A unit of volume containing 8 Imperial pints.

One sixteenth of an avoirdupois pound, containing 437.5 grains.

A unit of mass containing 16 avoirdupois ounces or 7000 grains.

A unit of time equal to $86400$ seconds.

A unit of area used for scattering cross-sections.

The International System of Units base unit of measurement of time.

An attosecond is one quintillionth of a second.