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At the central values, $z=\sqrt{\tan 45^{\circ }}+0.10^2=1.010$. For independent input standard errors, linear propagation givesThe angular derivative here is with respect to radians, so ${\sigma }_x=2\pi /180=0.03491$. At $x=\pi /4$, the derivative is one, andThus $\boxed{z=1.010\pm 0.035}$, with the stated measurement uncertainty interpreted as one standard error. Correlated inputs would require a covariance term; their correlation is not supplied.

A systematic error is a reproducible bias in the measurement model or procedure, such as a zero offset, incorrect gain, background contamination, misalignment, loading or temperature-dependent calibration. Repeating the same biased procedure more often reduces independent random scatter but does not remove the bias. An unknown correction also has a measurement uncertainty, even when the mechanism is systematic.

A simple instrument model is $y=(1+g)x+a+d(t)+ϵ$, with gain error $g$, offset $a$, drift $d(t)$ and random noise $ϵ$. Comparison with calibrated standards at several input values separates an offset from a scale error and can test nonlinearity. Use standards that cover the operating range, verify units and reference values, and propagate uncertainty in the standards into the corrected result. Calibration changes the model; it does not make the corrected measurement exact.

A blank or source-off measurement estimates additive background. For independent source-plus-background count $K_s$ in time $t_s$ and background count $K_b$ in time $t_b$, the net-rate estimate and its counting variance areSubtracting a measured background removes its estimated mean, not its fluctuations or its uncertainty. A long background exposure can reduce the latter, while drifting backgrounds require monitoring or a joint model.

Reversal can distinguish a wanted signal from an unchanged offset. For example, $V(+I)=IR+V_{\mathrm{th}}$ and $V(-I)=-IR+V_{\mathrm{th}}$ give $R=[V(+I)-V(-I)]/(2I)$ when the thermoelectric offset is stable between readings. Interchanging sample and reference, reversing field direction or reversing an apparatus orientation can similarly expose sign-dependent biases, but only if the wanted and unwanted contributions transform differently.

Track slow instrumental drift using frequent references, interleaved readings or an ABBA measurement sequence. Equal-time readings in the order A, B, B, A cancel a common linear drift in the difference estimator $(y_1+y_2-y_0-y_3)/2$. Nonlinear drift or unequal sample conditions can survive. Randomizing run order helps prevent a time drift from imitating dependence on a controlled experimental variable.

Control physical couplings directly: stabilize temperature, minimize gradients, shield unwanted fields, align geometry, and use high input impedance when loading is the concern. Differential and bridge measurements suppress common changes when their channels are well matched. Varying an apparatus parameter or using an independent method can reveal an unsuspected bias. Report residual corrections, their uncertainties and correlations; adding more identical readings cannot justify discarding a shared calibration uncertainty.

For a simple pendulum in the small-angle approximation, the oscillation period is $T=2\pi \sqrt{L/g}$, soInterpret the quoted random timing error and length error as standard uncertainties, with independent errors and no additional systematic error. For $N$ independent measurements, the standard error of the sample mean of the period is ${\sigma }_{\overline{T}}/T=0.02/\sqrt{N}$. The length has relative standard uncertainty ${\sigma }_L/L=0.1/25.0=0.004$. By propagation of uncertainty,Requiring a relative standard uncertainty no greater than $0.01$ givesFor $N=20$, the combined relative standard uncertainty is $0.009798$, or $0.980\%$; $N=19$ gives $1.001\%$ and just fails. Repeating the timing does not reduce the fixed length measurement uncertainty. The stated “accuracy” is consequently a statistical uncertainty target, not a guarantee for every experimental outcome. This is an example of repeated-timing uncertainty in a pendulum gravity measurement.

A proposed modern teaching measurement puts a pipette-measured portion of an iron ${\mathrm{Fe}}^{2+}$ chemical solution in a conical flask, acidifies with excess dilute sulfuric acid, and adds standardized potassium permanganate from a clamped burette while swirling. The documented endpoint principle is that added purple permanganate is consumed until the first small excess leaves a pale purple colour; no separate colour indicator is required. Insufficient acidity can produce brown manganese dioxide, invalidating the assumed five-electron half-reaction.^[ref]

Record initial and final burette readings; their difference is the delivered volume $V$. The following controls are proposed additions to test the inference. Run a reagent blank with the same water, sulfuric acid and endpoint criterion but no sample, obtaining $V_b$. Use a separately established potassium permanganate concentration $c$, clear the burette tip of air, and repeat sample portions. A known added amount of substance of ${\mathrm{Fe}}^{2+}$ should increase consumption by one fifth of that amount. A blank estimates reagent demand and the small endpoint excess; it cannot remove an interfering sample reactant.

Assuming complete, selective reaction according to the acidic permanganate-iron balance, the corrected inference isWith $c$ in $\mathrm{mol}{\mathrm{L}}^{-1}$, use $V$ and $V_b$ in litres. As an original stipulated calculation, not an experimental result, take $c=0.02000\mathrm{mol}{\mathrm{L}}^{-1}$, $V=12.60\mathrm{mL}$ and $V_b=0.10\mathrm{mL}$. Then the permanganate consumption is $0.2500\mathrm{mmol}$ and the inferred ${\mathrm{Fe}}^{2+}$ amount is $1.250\mathrm{mmol}$. In a $25.00\mathrm{mL}$ sample portion this corresponds to $0.05000\mathrm{mol}{\mathrm{L}}^{-1}$.

In this example the corrected volume is $12.50\mathrm{mL}=0.01250\mathrm{L}$. The acidic permanganate-iron balance gives $z_A=1$ for ${\mathrm{Fe}}^{2+}$ and $z_T=5$ for permanganate. Thus $0.2500\mathrm{mmol}$ of permanganate accepts $1.250\mathrm{mmol}$ of formal electrons, exactly what $1.250\mathrm{mmol}$ of ${\mathrm{Fe}}^{2+}$ supplies. The amount of substance ratio is five to one, despite equality of the exchanged electron amounts.

The denominator is the original $25.00\mathrm{mL}$ aliquot, not its final volume after adding sulfuric acid, water and titrant. Such dilution changes concentrations in the flask without changing the initial sampled amount of substance of ${\mathrm{Fe}}^{2+}$. Omitting the $0.10\mathrm{mL}$ titration blank would instead give $1.260\mathrm{mmol}$ and $0.05040\mathrm{mol}{\mathrm{L}}^{-1}$, an overestimate of $0.80\%$ relative to the corrected constructed value. The displayed digits express the stipulated calculation, rather than establish its measurement uncertainty.

The endpoint detects residual permanganate, not the iron identity directly. Other sample reactants that consume permanganate raise the apparent ${\mathrm{Fe}}^{2+}$ amount; prior oxidation of sample ${\mathrm{Fe}}^{2+}$ to ${\mathrm{Fe}}^{3+}$ lowers it. A recovery check probes response to the added iron but does not prove absence of every interference. The readout determines the reacting ${\mathrm{Fe}}^{2+}$ inventory under the stated conditions, rather than total iron in an arbitrary mixture. No measured titres are asserted here.

Photovoltaic mode uses no external reverse bias: a transimpedance amplifier can hold zero terminal electric potential difference, while an open electrical circuit develops a photovoltage. Photoconductive mode biases the diode cathode positive relative to the diode anode, generally reducing junction capacitance and increasing speed and dark current.^[ref]

A simple calculation separates the readout from the conversion. If a linear transimpedance amplifier has feedback electrical resistance $R_f=1.00\mathrm{M\Omega }$, the $650\mathrm{nm}$ model above produces an output change of magnitude $R_f{I}_{\mathrm{ph}}=0.393195\mathrm{V}$. The output sign depends on the connection. Increasing $R_f$ increases that electric potential difference, but does not create more collected charge carriers.

For a reverse-biased photodiode feeding an electrical load $R_L$, Ohm's law gives a readout magnitude $V_L\simeq I_{\mathrm{ph}}{R}_L$ while the response remains linear. With negligible dark current and other series drops, the remaining reverse bias is $V_R\simeq V_b-I_{\mathrm{ph}}{R}_L$, where $V_b$ is the supply magnitude. Thus the chosen electrical load can consume the available bias: a larger readout does not automatically reduce measurement uncertainty. For the same constructed example, $V_b=5\mathrm{V}$ and $R_L=1\mathrm{M\Omega }$ leave about $4.60680\mathrm{V}$; this algebra alone does not establish a real device's linear range.

Thorlabs' silicon FDS100 experiment sends a collimated, fibre-coupled laser beam through a beam splitter to the photodiode and a reference optical power meter. Varying reverse bias and electrical load tests linearity. The measured readout approaches saturation when its electric potential difference consumes the bias.^[ref] A separate low-signal setup uses a $637\mathrm{nm}$ laser, polarizers, an optical chopper, optical attenuators and a lock-in amplifier inside a light-tight enclosure. These are actual apparatus choices, not the parameters of the constructed example above.^[ref]

For an independent dark current check, Thorlabs enclosed a photodiode in nested metal boxes to exclude light and suppress interference, used a Keithley 6487 ammeter supplying reverse bias, and monitored temperature. Only one specimen per type was tested; the nearby temperature sensor did not directly contact the semiconductor. Such a dark control measures the baseline but cannot certify spectral responsivity.^[ref]

To infer spectral responsivity at a selected wavelength, measure the radiant flux actually reaching the test photodiode, rather than just the reference arm. A reference-arm measurement calibration must include the beam splitter ratio and path losses. With signed readouts in one fixed orientation, subtract a beam-blocked baseline and usefor direct electric current readout or a calibrated linear transimpedance amplifier, respectively. Repeating at several incident powers tests whether this ratio is constant. Blocking only the source estimates ambient light plus electronic baseline; covering the detector excludes ambient light too. Their difference distinguishes optical background from dark current and readout offset. For a broad optical spectrum, the ratio is a spectrum-weighted response, not generally $R$ at one wavelength.

For independent Poisson distribution collection, the root mean square shot noise in a one-sided rectangular frequency band of signal bandwidth $B>0$ isHere $I_d$ is reverse dark current; thermal electrical noise and amplifier noise also contribute.^[ref]

Subtracting a mean dark current removes its offset, not its fluctuations: subtracting a constant leaves the variance unchanged. Consequently $I_{\mathrm{ph}}\gg I_d$ makes dark current shot noise small relative to signal shot noise; it does not by itself establish high signal-to-noise ratio. With an independently measured baseline, the variance of that estimate must also be propagated. Increasing the measurement signal bandwidth raises the admitted noise. Calibration of wavelength, reference radiant flux, readout gain and collection geometry, together with drift and linearity, determines what measurement uncertainty can be claimed.

Cecil Frank Powell's 1950 lecture illustrates the successive decays of pions and muons through their tracks and explains why the nearly fixed secondary muon range, allowing for range straggling, supports a single neutral decay partner. He describes its identification as a neutrino tentatively. The neutral particle itself leaves no recorded track. He contrasts this decay with the nuclear reactions caused by stopped negative pions, connecting the latter with the nuclear force.^[ref]

Here is a modern special relativity derivation for the dominant leptonic pion decay, assuming a free pion at rest, a massless muon neutrino, and no emitted photon. Write $m_{\pi }$ and $m_{\mu }$ for the pion and muon masses. By conservation of momentum the daughter momenta are opposite with equal magnitude $p$. By conservation of energy and the energy–momentum relation,Substitute $E_{\mu }=m_{\pi }{c}^2-pc$, square, and cancel $p^2{c}^2$. This givesThe resulting $p$ is fixed, although the muon's direction is not set by the incoming stopping track. Thus the kink angle relative to that earlier track is not the angle between the two daughters: in the pion rest frame, the daughters are back to back. A neutral-partner arrow in a diagram represents inferred momentum, not an observed string of grains.

For an original calculation, adopt rounded input values $m_{\pi }{c}^2=139.57\mathrm{MeV}$ and $m_{\mu }{c}^2=105.66\mathrm{MeV}$. They give $pc=29.79\mathrm{MeV}$, $T_{\mu }=4.119\mathrm{MeV}$ and $E_{\nu }=29.79\mathrm{MeV}$. Their sum satisfies $T_{\mu }+E_{\nu }=(m_{\pi }-m_{\mu })c^2=33.91\mathrm{MeV}$. These are rounded calculation inputs and outputs, not a fit to the 1947 plates.

At fixed material and particle identity, a fixed initial kinetic energy suggests a narrow particle range distribution broadened by range straggling and measurement uncertainty. The synthetic example convolves a single chosen range with a normal distribution; a second example superposes two such chosen ranges. It explains why repeated comparable stopping distances can be informative, without pretending to extract a historical spectrum.

The decay equations do not identify the neutral partner as a neutrino on their own: a massless photon would satisfy the same energy and momentum equations. Nor does any isolated kink establish the pion's identity. The inference gains force from calibrated particle range, track grain density, repeated decay patterns and independent particle properties. Historical particle names should not obscure the distinction: a muon is a lepton, whereas a pion is a meson.

A Michelson interferometer turns displacement into a count of interference fringes: a moving mirror changes both the outward and return paths. This lets a wavelength act as a ruler whose divisions need no engraving. Assume normal incidence, a fixed reference arm, stable refractive index $n$ along the moving arm, and fixed vacuum wavelength ${\lambda }_0$. Write $\lambda ={\lambda }_0/n$ for the wavelength in that medium and take $\Delta L$ positive away from the beam splitter. The change in optical path difference, the accumulated phase difference, and the signed number of complete interference fringe cycles areThe factor two comes from propagation geometry: moving the mirror by $\Delta L$ adds $\Delta L$ before reflection and $\Delta L$ afterward. One bright-to-bright cycle means mirror travel $\lambda /2$; bright-to-dark means $\lambda /4$, half a cycle. All lengths must use compatible units, and $\Delta N$ is dimensionless. A constant reflection phase difference changes the initial fringe position but cancels from this change formula.

For an original hypothetical example, suppose $\lambda =600\mathrm{nm}$ in the actual surrounding medium and a continuous scan gives $40.25$ cycles, including the final fractional cycle. Then $\Delta L=(600\mathrm{nm})(40.25)/2=12075\mathrm{nm}=12.075\mu \mathrm{m}$. With the wavelength treated as exact, a fringe-reading measurement uncertainty of $0.02$ cycle contributes $6\mathrm{nm}$ to the displacement measurement uncertainty. An uncertainty of $0.01\mathrm{rad}$ in a local phase difference instead corresponds to $600(0.01)/(4\pi )\mathrm{nm}\approx 0.477\mathrm{nm}$. This illustrates measurement resolution, not a complete claim of accuracy: wavelength calibration, air changes, mirror tilt and missed cycles can dominate.

To separate the oscillating readout from its optical coherence envelope, use the local modelHere $I$ is detected light intensity, $\bar{I}$ is the mean level, ${\varphi }_b$ is a fixed bias wave phase, and $\mathcal{V}$ is interferometric visibility. A fine scan of $D$ measures the carrier phase difference; a coarse scan can map the temporal optical coherence envelope through $\mathcal{V}(D)$. The envelope maximum locates matched delays only under the source and apparatus assumptions; it is not a bright fringe identifying an arbitrary mirror position. Even with known $\mathcal{V}$, one intensity reading leaves the cosine sign and whole-cycle ambiguities. Continuous motion of known direction, or an independently obtained pair of signals in phase quadrature, is needed to recover signed travel through reversals.

The figure's original control example stipulates equal arm light intensities, matching linear polarizations, and a Lorentzian optical spectrum giving $\mathcal{V}(D)=\exp (-|D|/L_c)$ with $L_c=1\mathrm{mm}$. In vacuum, arm mismatches $|L_2-L_1|=0.60\mathrm{mm}$ and $0.10\mathrm{mm}$ give $\mathcal{V}=e^{-1.2}\approx 0.301$ and $e^{-0.2}\approx 0.819$. Matching the arms improves the phase difference readout without changing its ${\lambda }_0/2$ travel per cycle. A fine mirror scan of one cycle changes $D$ by only $600\mathrm{nm}$ for the stipulated $600\mathrm{nm}$ source, much less than $L_c$. These are calculated values, not historical data.

A proposed measurement sequence follows from this model. Overlap the returning beams, match their linear polarizations, find a high-contrast delay and use a small calibrated mirror scan to determine the sign and fringe period. Read a fixed spatial sample of the interference fringes with a photodiode; record each arm separately and the dark signal, avoid detector saturation, and retain enough time resolution to count every cycle. Approach mechanical settings consistently and repeat the return scan to expose lost motion or drift. For very small changes, bias near phase quadrature, where $|dI/d\varphi |=\bar{I}\mathcal{V}$ is largest; an intensity extremum has zero first-order sensitivity. A known drive voltage alone does not provide measurement calibration for mirror travel: compare the actual travel with the fringe count using an independently known wavelength.

Control temperature, pressure, vibration and air currents because the instrument measures optical path difference, not mirror motion alone. For a fixed reference arm and fixed ${\lambda }_0$, the first-order change of the moving arm's round-trip path is $\Delta D\approx 2n\Delta L+2L\Delta n$. Thus an unrecognized $\Delta n$ mimics $\Delta L_{\mathrm{app}}=L\Delta n/n$. A hypothetical $L=0.10\mathrm{m}$, $n\approx 1$, and $\Delta n=10^{-7}$ give $10\mathrm{nm}$ apparent travel. Holding the mirror still while varying or monitoring the air conditions tests this confounder. Separate checks of alignment, wavelength stability and thermal expansion are necessary before sub-fringe measurement resolution becomes a defensible measurement uncertainty.

A pressure measurement determines pressure relative to a specified reference using an instrument such as a manometer or pressure transducer. Gauge pressure, absolute pressure and differential pressure require different references; measurement calibration and measurement uncertainty determine how the reading can be interpreted.

A prism minimum-deviation index measurement infers a transparent sample's relative refractive index from two angles: its known apex angle $A$ and the smallest deviation ${\delta }_{\min }$ of a selected spectral line. A spectrometer combines a source and slit, a collimator, a rotating optical prism table, and a viewing optical telescope with an angular readout. The collimator supplies a nearly parallel beam; the optical telescope locates its emergent direction. This turns an otherwise difficult measurement of the phase velocity of light into a measurement of angles.

For the symmetric path through a uniform optical prism, with the same external medium at both faces and positive refractive indices,The angles $i,e,r_1,r_2$ are measured from face normals. This is the minimum deviation relation, not a formula for an arbitrary prism orientation. An absolute refractive index additionally requires $n_{\mathrm{ext}}$.

For example, stipulate $A=60^{\circ }$ and $n_{\mathrm{rel}}=3/2$. Then $i=e=\arcsin (3/4)\simeq 48.5904^{\circ }$ and ${\delta }_{\min }\simeq 37.1808^{\circ }$. Substitution recovers $n_{\mathrm{rel}}=1.5$. These synthetic values illustrate the geometry; they are not readings for the prism in the laboratory protocol below.

A Michigan State University teaching experiment uses a mercury lamp, slit, collimator, prism table and optical telescope, with a subsidiary angular scale read to one arcminute. First focus the optical telescope on a distant object; with the prism removed, align its crosshair to a narrow vertical slit image, focus the collimator, and establish the direct-beam zero. Insert and centre the prism. Select the green mercury spectral line, tabulated at $546.1\mathrm{nm}$, then rotate the prism while tracking its image with the optical telescope. Locate the turning point where continued prism rotation reverses the image motion, and read the deviation. Re-establish the minimum separately for each other spectral line. The protocol assumes $A=60^{\circ }$; that assumed apex angle is not an independently measured result. ^[ref]

For an uncertainty-aware implementation, repeat the turning-point search from both prism-rotation directions, recheck the direct-beam reference, and record individual readings before averaging. Independently determine $A$ or include its calibration uncertainty; record the chosen spectral line, sample identity, temperature and surrounding-air conditions. These are proposed controls deduced from the inference, not claimed additional instructions or results of that manual. Finite slit width, unresolved spectral lines, imperfect collimation, table tilt, angular calibration and sample nonuniformity can limit the inferred refractive index. One arcminute readout resolution alone does not establish one arcminute total measurement uncertainty. A single selected wavelength gives one index, not a measured optical dispersion curve.

A short geometric derivation uses $r_1+r_2=A$ and $\delta =i+e-A$. Reversibility exchanges entrance and exit. At the symmetric stationary path, $i=e$ and $r_1=r_2$; applying Snell's law gives the displayed index relation. The plotted calculation checks that this stationary point is a minimum on the chosen transmitting branch. To assess local sensitivity, differentiate the inferred index using angles in radians:This first-order relation exposes both angular contributions; it is not an instruction to add signed systematic errors as independent random errors. In the synthetic example, a deviation error of $+1$ arcminute at fixed $A$ changes the inferred relative refractive index by approximately $+1.92\times 10^{-4}$.

A 2013 experiment by Paul A. Williams and collaborators mounted a 25-mm dielectric mirror on a commercial force-restoration balance. Removing its pan-weight compensation spring allowed horizontal force sensing with a horizontal laser path. They measured mirror optical reflectance and varied incidence from about $7^{\circ }$ to $45^{\circ }$. A 1071-nm laser, capable of 530 W, was switched at 0.1 Hz. Digitized mass-equivalent readings were converted to force using $g$; quadrature demodulation isolated the switched signal from drift. ^[ref]

Measurement calibration used a 20-g mass, with checks down to 1 mg. The optical forces lay below that checked range, limiting the accuracy claim. An angle scan tested the predicted cosine dependence at fixed intercepted power. Replacing the mirror with a carbon nanotube coated absorber increased absorbed power while reducing the predicted optical force; no modulation peak was detected. This bounded gas-heating artifacts under those conditions, rather than proving their universal absence. The fitted slope was about 7% below $2/c$. Inferring laser power requires calibrated force, optical reflectance, radiometric absorptance and angle of incidence, with measurement uncertainty for each. ^[ref]

Range straggling is the event-to-event spread in particle range for otherwise similar incident particles, caused by fluctuations in their energy loss and transport through matter. It must be distinguished from measurement uncertainty and from the incomplete recording of particles that escape.

A regulated bath holds its temperature near a chosen setting through feedback control, while stirring reduces internal temperature gradients. Its finite heat capacity buffers disturbances; external heating and cooling replace the ideal heat bath's unlimited capacity. This is useful for thermometer calibration and for maintaining a blackbody calibration source at a known temperature.

Joel B. Fowler's A Third Generation Water Bath Based Blackbody Source reports a NIST test of a stirred water bath containing a copper blackbody cavity. A bottom plate supplied heating and cooling; a double stirrer circulated water. Electronic feedback control regulated heating. An electronic thermometer with a thermistor probe provided the readout. The preparation was to approach a selected setting, wait for stabilization, adjust against the measured temperature and then log continuously.^[ref]

To test spatial uniformity, one thermistor stayed near the cavity tip while another visited ten locations at three heights. The simultaneous reference distinguished spatial differences from temporal drift. An insulated dome reduced thermal conduction along probe casings. Table 1 reports nonzero departures at several settings: circulation did not make every location identical. The measurement uncertainty analysis separately included uniformity, stability, setting, immersion and readout contributions. These tests supported a bounded temperature assignment around the cavity, rather than infinite heat capacity or perfect thermal equilibrium throughout the apparatus.^[ref]

A resistance pressure gauge is a pressure sensor whose electrical resistance is calibrated against pressure. Even a simple local response needs both pressure and temperature. For an original illustrative model definewhere $R_0$ is the reference electrical resistance at $p=0$, $T=T_0$, and zero history offset. The derivatives $\alpha$ and $\beta$ are local sensitivities with units $\mathrm{P}{\mathrm{a}}^{-1}$ and ${\mathrm{K}}^{-1}$, respectively; $h$ is a dimensionless history offset. Constant sensitivities are an approximation over a chosen range, not material constants valid at every pressure.

Choose synthetic coefficients $\alpha =0.020\mathrm{GP}{\mathrm{a}}^{-1}$ and $\beta =2.0\times 10^{-4}{\mathrm{K}}^{-1}$, with $T-T_0=10\mathrm{K}$ and $h=0$. A measured ratio $R/R_0=1.022$ then gives $p=(0.022-0.002)/0.020=1.00\mathrm{GPa}$. Ignoring the temperature change gives $1.10\mathrm{GPa}$, a $0.10\mathrm{GPa}$ error. This model is deliberately stipulated; these numbers are not measured manganin coefficients or Bridgman's data.

If two histories differ by $\Delta h=0.0004$, their outputs at the same pressure differ by $0.0004R_0$; interpreting both with a single history-free curve gives an apparent pressure separation $\Delta h/\alpha =0.020\mathrm{GPa}$. The half separation is $0.010\mathrm{GPa}$. Reversing the drive can additionally cause mechanical friction hysteresis, independently of this sensor model. Record both the electrical calibration history and the loading direction.

For independent small standard measurement uncertainties $u_y,u_T,u_h$ and exact stipulated $\alpha ,\beta$, first-order propagation givesFor $u_y=10^{-4}$, $u_T=0.5\mathrm{K}$, and $u_h=10^{-4}$, this is $u_p=0.00866\mathrm{GPa}$. Real calibration also includes coefficient measurement uncertainties, correlations, reference-point measurement uncertainty, and any unresolved nonhydrostatic loading. A sensor calibrates its local environment; it cannot by itself certify the absence of deviatoric stress throughout the sample.

Resistor tolerance specifies the permissible manufacturing deviation of a resistor's electrical resistance from its nominal value under stated conditions. For nominal electrical resistance $R_{\mathrm{nom}}>0$ and fractional resistor tolerance $0\le \tau <1$, the specified range is $R_{\mathrm{nom}}(1-\tau )\le R\le R_{\mathrm{nom}}(1+\tau )$. This differs from measurement uncertainty: resistor tolerance describes a component specification, while measurement uncertainty describes knowledge of a measured value; the tolerance alone is not a probability distribution for that value.

Two nominally equal resistors can have different actual electrical resistances, changing their voltage divider ratio even without an external electrical load. In a loaded voltage divider, use the actual electrical resistances to calculate both the unloaded output and the Thévenin resistance governing voltage divider loading error. A mismatch from resistor tolerance and a change from voltmeter loading are distinct effects that can occur together.

The 1975 experiment mixed powdered sodium chloride and 0.5%-chromium ruby in a 4:1 volume ratio. An Inconel gasket, initially $0.15\mathrm{mm}$ thick with a $0.3\mathrm{mm}$ hole, held the powder and 4:1 methanol–ethanol. At $25{}^{\circ }\mathrm{C}$, X-ray diffraction rings on film $12\mathrm{cm}$ away gave salt compression; a reference equation of state assigned pressure. After decreasing load, they waited at least five hours and bracketed diffraction exposures with $R_1$ readings to check drift.^[ref]

The recommended linear slope was $2.740\mathrm{GPa}\mathrm{n}{\mathrm{m}}^{-1}$, with a $95\%$ fit confidence interval of $\pm 0.016\mathrm{GPa}\mathrm{n}{\mathrm{m}}^{-1}$, to $19.5\mathrm{GPa}$. Reference-scale systematic measurement uncertainty remained separate. Above about $10.4\mathrm{GPa}$ the medium was only quasihydrostatic; separate range fits checked consistency.^[ref]

The logical chain is compression to reference pressure, paired with optical shift, then a fitted response. X-ray diffraction supplies the reference comparison during calibration; subsequent ruby readings transfer that scale to other samples. The reference equation of state is therefore part of the meaning of the reported pressure. Changing it can move all calibrated readings together, even if the optical measurements repeat perfectly. A regression residual tests agreement with the chosen reference within the sampled conditions; it cannot independently certify that reference.

Construct a synthetic local calibration to show the calculation independently of the historical data. Choose reference pairs $(\Delta {\lambda }_i,p_i)=(1.00,2.69),(2.00,5.42),(3.00,8.09)$, with shifts in $\mathrm{nm}$ and pressures in $\mathrm{GPa}$. The reference zero is already subtracted. Treat the shifts as exact for this fit and give the three reference pressures equal weights. Minimizing the ordinary least squares objectiveand setting its derivative to zero givesThe regression residuals are $(-0.01,0.02,-0.01)\mathrm{GPa}$. Their weighted moment ${\sum }_i\Delta {\lambda }_i(p_i-a\Delta {\lambda }_i)$ vanishes, as required by the least-squares normal equations. An unknown reference offset requires fitting an intercept instead; forcing it to zero can bias the slope. If shift errors matter, this elementary fit is insufficient.

Use the local measurement modelHere $p$ denotes pressure relative to the reference, $a$ is calibrated sensitivity, and $k_T$ is a local wavelength sensitivity to temperature at fixed pressure. Choose $a=2.70\mathrm{GPa}\mathrm{n}{\mathrm{m}}^{-1}$, $k_T=0.010\mathrm{nm}{\mathrm{K}}^{-1}$ and a reading $\lambda -{\lambda }_0=2.00\mathrm{nm}$ at $T=T_0$. The inferred pressure is $5.40\mathrm{GPa}$. These coefficients and readings are stipulated, not measured ruby constants. With the same pressure, a $10\mathrm{K}$ increase in temperature would add $0.100\mathrm{nm}$ to the shift in this model; neglecting it would overestimate pressure by $0.270\mathrm{GPa}$.

Assign independent standard uncertainties $u_a=0.020\mathrm{GPa}\mathrm{n}{\mathrm{m}}^{-1}$, $u_{\lambda }=u_0=0.010\mathrm{nm}$ and $u_T=0.50\mathrm{K}$ for the temperature difference. Here $u_a$ is an independently assigned calibration uncertainty, not an estimate from the regression residuals above; treat $k_T$ as exact in this example. First-order propagation of uncertainty follows by differentiating the measurement model:The four standard contributions are $0.0400,0.0270,0.0270,0.0135\mathrm{GPa}$, giving $u_p=0.0569\mathrm{GPa}$. This is a standard uncertainty, not automatically a $95\%$ interval. Correlated reference and loaded wavelength readings require the corresponding covariance term; common wavelength-axis drift may cancel in their difference. Uncertain $k_T$, nonzero $T-T_0$, model inadequacy and unresolved deviatoric stress require a broader budget. Taking more optical readings reduces some random errors while leaving a common reference-scale error.

For a short circular track of chord length $L$, sagitta $s$ and fixed magnetic field $B$, $p=|q|B{L}^2/(8s)$. Differentiating this inverse relation gives $\sigma (p)/p=\sigma (s)/s$. A fixed spatial measurement uncertainty therefore makes the relative momentum uncertainty grow linearly with momentum.