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For small measurement errors, write $f=L/4-d^2/(4L)$. Its partial derivatives and first-order propagation of uncertainty areHere the $\sigma$ values are standard deviations used to describe uncertainty, and $\mathrm{Cov}(L,d)$ is the covariance. Independent carrier-reading errors give ${\sigma }_d^2={\sigma }_{r_{-}}^2+{\sigma }_{r_{+}}^2$; correlated readings require the term $-2\mathrm{Cov}(r_{-},r_{+})$. Averaging repeated independent pairs reduces the random component of the uncertainty of the mean, but does not remove calibration or model errors.

As a stipulated uncertainty calculation, let $L=45\mathrm{cm}$, $d=15\mathrm{cm}$, ${\sigma }_L=0.10\mathrm{cm}$ and ${\sigma }_{r_{-}}={\sigma }_{r_{+}}=0.05\mathrm{cm}$, with all three input errors independent. Then ${\sigma }_d=0.0707\mathrm{cm}$, $A=5/18$, $B=-1/6$ and ${\sigma }_f\simeq 0.0302\mathrm{cm}$. The result would be $f=10.000\mathrm{cm}$ with this standard uncertainty in the stipulated thin lens model; these numbers are not a measured uncertainty budget. A small neglected principal-plane separation gives an additional first-order bias $f_{\mathrm{thin}}-f\simeq Ah$, distinct from that random budget.

The two kinds of sensitivity must be distinguished. At fixed focal length,These derivatives diverge as $L$ approaches $4f$ from above: a small change in object-screen spacing strongly changes the separation of the two settings. Yet the inferred focal length's displacement sensitivity $|B|=d/(2L)$ tends to zero. That does not make coalescing settings easy to measure. For fixed $L$, define the axial mismatch between the predicted real image and the screen as $g(x)=fx/(x-f)-(L-x)$, for $x>f$. At a focused setting,Near $u=v=2f$, moving the optical lens changes this mismatch only weakly, making visual discrimination of the two settings difficult. At the example's settings $g^{\prime }(15\mathrm{cm})=-3$ and $g^{\prime }(30\mathrm{cm})=3/4$, so equal carrier motion produces different axial defocus responses. This is an axial mismatch calculation, not a full prediction of perceived image sharpness.

Choose $L$ sufficiently above $4f$ to separate the two accepted focusing ranges, while retaining both real images on the available screen and rail. There is no universal best $L$: focusing uncertainty itself depends on aperture, transverse magnification, optical aberrations and the target. Close to $d=0$, the nonnegative measured separation and the quadratic term in $d$ also make first-order propagation of uncertainty unreliable if its error is comparable to its value. Repeatability, controls and the principal plane correction determine whether the elegant two-distance formula is an adequate measurement model.

Suppose $N_0$ is independently calibrated and counts $y_i$ have predicted means ${\mu }_i=t_i{N}_0{e}^{-b{n}_i}$ under a specified candidate $b$. For independent sufficiently large Poisson counts, the Pearson chi-squared statistic is $Q={\sum }_i(y_i-{\mu }_i{)}^2/{\mu }_i$. With no parameters fitted to these test counts, its approximate reference chi-squared distribution has one degree of freedom per observation. Fitting a continuous $b$ to the same data instead removes one residual degree of freedom. Replacing the denominators by observed counts is a common estimated-error approximation but can behave badly at small counts.

Calibration uncertainty ${\sigma }_0$ also induces correlated prediction errors. First-order propagation of uncertainty givesA generalized quadratic form uses this covariance matrix. For sparse counts use the Poisson deviance or the full count likelihood, with simulation if a calibrated goodness-of-fit probability is required. Choosing a range after inspecting residuals also affects formal test calibration; a physically specified signal-dominated range is preferable when applying a pure attenuation model.

A laboratory determination of the gravitational constant must connect a sensor's reading to a gravitational signal from independently characterized masses. For an ideal two-point mass force measurement, invert the model:These are signed small changes obtained by taking the logarithmic derivative, not an instruction to add standard uncertainties with signs. For independent small input uncertainties, linear propagation of uncertainty instead givesCorrelated inputs require their covariances too. In particular, common geometric calibration errors must not be treated as independent simply because several distances were recorded.

For a concrete model check, hold the actual force and masses fixed but overestimate a true $r=0.100\mathrm{m}$ by $1\mu \mathrm{m}$. The exact inferred ratio isThe resulting bias is $20.0001$ parts per million, already comparable with the relative standard uncertainty of the recommended gravitational constant. A sensor can resolve very small changes while its inferred answer remains limited by source geometry. The example concerns one idealized separation; it is not an uncertainty estimate for any actual experiment.

Making the source mass larger or bringing it closer strengthens the signal, but also makes detailed mass density and geometry important. Nearby apparatus supplies its own Newtonian gravitational field; electromagnetic shielding does not remove that attraction. Sensor offsets, changing temperature, vibration and source motion can imitate or alter a signal. The Cavendish experiment illustrates calibration of a torsion constant; a different sensor requires its own calibrated response and a full source model.

Repeating readings can reduce independent measurement noise, but a fixed scale error survives averaging. A useful experiment deliberately changes the source configuration, checks whether the signal follows the predicted change, and characterizes what moves with the source. A systematic error that changes with that configuration also survives background subtraction. Comparing methods with different sensors and different dominant errors is therefore valuable.

An exponential decay rate $R=R_0{e}^{-t/\tau }$ becomes a straight line in $x=t$ and $y=\ln (R/R_{\ast })$, where $R_{\ast }$ is a fixed reference rate. Constant small fractional rate errors become approximately constant vertical errors $s\simeq {\sigma }_R/R$ by propagation of uncertainty. With independent Gaussian log-rate errors, linear least squares gives $\hat{m}=S_{xy}/S_{xx}$ and $\hat{c}=\bar{y}-\hat{m}\bar{x}$.

The covariance matrix is obtained by applying the variance rules to these linear estimators:Transforming back gives $\hat{\tau }=-1/\hat{m}$, ${\sigma }_{\tau }\simeq {\sigma }_m/{\hat{m}}^2$, ${\hat{R}}_0=R_{\ast }{e}^{\hat{c}}$ and ${\sigma }_{R_0}\simeq {\hat{R}}_0{\sigma }_c$. If a candidate lifetime fixes the slope $m_a$, the intercept must be refitted as $c_a=\bar{y}-m_a\bar{x}$. The profiled objective obeysThis follows by separating the centred slope residual from the unrestricted regression residual, whose inner product with the design columns is zero by the normal equations for linear least squares. Known supplied measurement errors are not rescaled by the observed reduced chi-squared. This log-error model is an approximation for proportional errors, not an exact replacement for every raw-count likelihood.

Low-temperature calorimetry measures a small sample's heat capacity by resolving a small temperature change after a known electrical energy input. The experimental challenge is that the support, heater and thermometer also warm up. Their calorimeter addenda must be removed before interpreting the sample's phonons or electronic heat capacity. The Debye T-cubed test turns the corrected response into a test of the low-temperature vibrational spectrum.

Manuel Brando's 2009 experiment used a sapphire platform suspended by four alloyed platinum wires from a copper ring fixed to a cryostat stage. Grease attached the sample; a chip heater and ruthenium-oxide resistance thermometer served platform PLS-2. A controller stabilized the bath; a lock-in amplifier supplied heating pulses and read the calibrated sensor. The steady rise calibrated thermal conductance as $G=P/\Delta T$. A separate $59.1\mathrm{mg}$, $99.99\%$ pure gold sample, compared with reference heat capacities, established the calorimeter addenda. Small pulses limited property variation; imperfect sample contact required a second relaxation time. The platform contribution rose below $1.5\mathrm{K}$, making simple extrapolation unsafe.^[ref]

For an original idealization, assume sample and platform share one temperature $T$, the bath remains at $T_b$, and both the total heat capacity and the weak-link thermal conductance $G$ are approximately constant across one small pulse. Ignore other heat transfer and any phase transition. The first law of thermodynamics becomesHere $C_s$ is sample heat capacity, $C_a$ is calorimeter addenda, $P$ is deposited electrical power from Joule heating, and $\tau$ is a time constant. This is the same balance as calorimeter heat-loss correction, used here with a deliberately weak cryogenic link and a separately subtracted platform. For a constant pulse $P_0$ beginning at bath temperature, then switched off at $t_h$,The cooling exponential decay determines $\tau$, while the heating plateau determines $G=P_0/(T_{\infty }-T_b)$. Together with calorimeter addenda, they give $C_s=G\tau -C_a$. Cooling alone determines a ratio, not an absolute heat capacity. With finite $t_h$, the observed end rise is smaller than the plateau; fit the full response or apply the displayed correction rather than identifying the two.

For the stipulated figure parameters, $G=5.00\times 10^{-8}\mathrm{W}{\mathrm{K}}^{-1}$, $\tau =20.0\mathrm{s}$, and $C_a=2.00\times 10^{-7}\mathrm{J}{\mathrm{K}}^{-1}$ give $C_s=8.00\times 10^{-7}\mathrm{J}{\mathrm{K}}^{-1}$. The asymptotic rise is $0.0400\mathrm{K}$, only $2\%$ of bath temperature. At $100\mathrm{s}$, the rise is $0.0397305\mathrm{K}$. Of $2.00\times 10^{-7}\mathrm{J}$ input, only $3.97305\times 10^{-8}\mathrm{J}$ remains stored: the other $1.60270\times 10^{-7}\mathrm{J}$ has flowed to the bath. Treating this long pulse as an isolated $Q/C$ experiment would overestimate total heat capacity by a factor of $5.034$.

Suppose independent standard uncertainties are $u(G)=0.05\times 10^{-8}\mathrm{W}{\mathrm{K}}^{-1}$, $u(\tau )=0.2\mathrm{s}$ and $u(C_a)=0.10\times 10^{-7}\mathrm{J}{\mathrm{K}}^{-1}$. First-order propagation of uncertainty givesThe uncertainty is one standard deviation under those assumptions, about $2.2\%$ of sample heat capacity. Shared thermometer calibration errors can correlate fitted $G$ and $\tau$; then their covariance contributes too. Smaller samples amplify the relative influence of calorimeter addenda. Proposed checks include varying pulse amplitude, repeating with a reference sample, checking sensor self-heating, and comparing single- and two-time-constant fits. A drifting bath, slow internal equilibration or a transition invalidates this simple reduction even if a cooling curve looks smooth.

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]

Mixing calorimetry determines heat capacities by bringing bodies at different temperatures into thermal contact. The striking feature is that the final temperature is a weighted arithmetic mean with heat capacities as weights, rather than weights given by mass. The vessel participates in the exchange even though it is not the sample being studied.

For a hot nonreacting sample of mass $m_x$, initial temperature $T_x$ and unknown specific heat capacity at constant pressure $c_x$, let receiving water have mass $m_w$, initial temperature $T_w$ and known specific heat capacity at constant pressure $c_w$. Let the vessel and immersed parts start at $T_w$ and have combined heat capacity $C_{\mathrm{cal}}$, excluding that water. Assume a common final temperature $T_f$, approximately constant pressure and heat capacities, no phase transition, no chemical reaction and negligible external nonexpansion work. For signed outward heat loss $Q_{\mathrm{loss},\mathrm{out}}$, the combined enthalpy balance isHenceThe loss is positive if the room receives net heat, negative if the room supplies it. If the vessel starts at a different temperature $T_{\mathrm{cal},i}$, replace its warming term by $C_{\mathrm{cal}}(T_f-T_{\mathrm{cal},i})$; do not assume the vessel shared the water's initial temperature. For variable material properties, replace each $mc\Delta T$ by the corresponding integral $m\int c_p(T)dT$ and each vessel term by $\int C_{\mathrm{cal}}(T)dT$. The inferred constant $c_x$ then represents an average over the sample's cooling interval.^[ref]

For a synthetic no-loss example, mix $0.100\mathrm{kg}$ of hot water at $80{}^{\circ }\mathrm{C}$ with $0.200\mathrm{kg}$ of cold water at $20{}^{\circ }\mathrm{C}$ in a vessel initially at $20{}^{\circ }\mathrm{C}$. Stipulate $c_w=4180\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ and $C_{\mathrm{cal}}=100\mathrm{J}{\mathrm{K}}^{-1}$. The enthalpy balance givesThe hot water releases $17.337\mathrm{kJ}$: the cold water receives $15.485\mathrm{kJ}$ and the vessel $1.852\mathrm{kJ}$. Omitting the vessel predicts $40{}^{\circ }\mathrm{C}$, despite perfect insulation. These are computed numbers from stipulated properties, not laboratory observations.

For an unknown-sample example using the same cold water and vessel, stipulate $m_x=0.100\mathrm{kg}$, $T_x=80{}^{\circ }\mathrm{C}$ and $c_x=450\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. With no loss, $T_f\approx 22.752{}^{\circ }\mathrm{C}$. Inverting the full balance returns $450\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$; ignoring vessel storage returns about $402\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. With a positive outward loss, ignoring that loss also biases the inferred $c_x$ downward under these hot-sample conditions. A small water temperature rise makes sensor offsets and loss estimates relatively influential; propagation of uncertainty must include the measured temperatures, masses and calibrated vessel term.

For positive measured inputs and $f=C{\prod }_i{x}_i^{a_i}$, logarithmic differentiation gives $df/f={\sum }_i{a}_{i}d{x}_i/x_i$. Applying propagation of uncertainty yields the displayed covariance formula. For independent input errors it reduces to ${\sigma }_f/|f|\simeq [{\sum }_i(a_i{\sigma }_i/x_i{)}^2{]}^{1/2}$. If the input uncertainties are strict bounds rather than standard deviations, a first-order conservative bound is $|\Delta f|/|f|\lesssim {\sum }_i|a_i|\Delta x_i/|x_i|$. These are different statistical interpretations; a product formula alone does not choose between them.

The source labels $L$ as the full chord from $P_1$ to $P_3$, with $L/2$ on each side of its midpoint. The Right triangle between the Circle centre, chord midpoint and endpoint givesFor fixed geometry and field, the inverse transverse momentum isIf the uncertainty is dominated by the central point, ${\sigma }_s\simeq {\sigma }_x$. Linear propagation of uncertainty givesUsing $|q/e|=1$, $B=2\mathrm{T}$, $L=0.6\mathrm{m}$ and ${\sigma }_x=28.87\mu \mathrm{m}$,The inverse-momentum resolution is approximately momentum independent in this spatial-error-only model; the relative momentum resolution grows as $p_T\sigma (1/p_T)$. Endpoint errors and multiple scattering are excluded by the stated assumptions.

For two photons, four-momentum conservation and their vanishing rest masses give the diphoton invariant massTaking a logarithmic differential,For small mutually independent measurement errors, propagation of uncertainty consequently givesPrinted-formula correction. The source's left-hand side, if it means $(\Delta M/M{)}^2$ as its description suggests, is missing the factor $1/4$ on the right. Its right-hand side instead equals $\mathrm{Var}(M^2)/M^4$ to first order. A concrete counterexample is a fixed opening angle and an exactly known $E_2$: because $M\propto \sqrt{E_1}$, a small $1\%$ fractional error in $E_1$ produces a $0.5\%$ fractional error in $M$, not $1\%$. Likewise, interpreting $\Delta M^2$ literally as a small change in $M^2$ would make the unsquared left-hand side incompatible with the squared terms. Correlated errors require the covariance terms as well.

The space angle is determined by the two polar angles and azimuthal separation:At ${\theta }_1={\theta }_2=45^{\circ }$ and $\Delta \varphi =180^{\circ }$, $\cos \alpha =0$ and $\alpha =90^{\circ }$. The angular coefficient is therefore one. Equal contribution of opening-angle uncertainty and the combined two independent $1\%$ photon energy errors requiresFor independent equally precise single-photon polar directions, ${\sigma }_{\alpha }^2={\sigma }_{{\theta }_1}^2+{\sigma }_{{\theta }_2}^2$ at this geometry, soAn azimuthal error contributes only at second order about $\Delta \varphi =\pi$. The corresponding invariant-mass resolution from the two energy errors alone is $0.707\%$; with an equally large angular contribution it is $1.00\%$. The common missing factor of four cancels when equating the two contributions, so the angular estimates remain valid despite the printed error.

The long-lived, electrically neutral kaon often traverses the particle detector without decaying into a reconstructable charged final state. Identifying it through a hadronic interaction or missing four-momentum is much less efficient than reconstructing the short-lived kaon through its nearby two-pion decay vertex. The efficiencies must be applied to the observed counts before taking their asymmetry:Using the leading aligned-amplitude approximation from c,The exact aligned formula instead gives $12.43^{\circ }$, an immaterial correction here. Treating the observed counts as independent Poisson random variables and the efficiencies as known, propagation of uncertainty givesThe separate contributions to ${\sigma }_{\mathcal{R}}$ are $0.0149$ and $0.0708$; the small $K_L$ sample dominates. Linear propagation of uncertainty yields $\sigma ({\theta }_C)\simeq 4.49^{\circ }$, although the nonlinear extraction and physical boundary make a symmetric confidence interval a rough summary. Unknown efficiency uncertainties, especially that of $K_L$, would be additional systematic contributions.

The supplied CKM matrix gives $\sin {\theta }_C\simeq 0.224$, hence ${\theta }_C\simeq 12.94^{\circ }$ and $\mathcal{R}\simeq 2(0.224/0.975{)}^2=0.1056$. The asymmetry difference is only about $0.12$ statistical standard deviations, so the measurement is compatible. Without the phase assumption, these data measure ${\tan }^2{\theta }_C\cos \delta$ to leading order and do not uniquely measure the Cabibbo angle.

The sagitta is the maximum distance between the circular track and its chord. The perpendicular bisector of a chord of length $L$ gives a right triangle with hypotenuse $R$ and sides $L/2$ and $R-s$. HenceAt fixed electric charge, magnetic field and lever arm, $p=|q|B{L}^2/(8s)$, so differentiation and propagation of uncertainty giveThus a fixed spatial measurement uncertainty produces a relative momentum uncertainty growing linearly with $p$. If momentum is expressed in $\mathrm{GeV}/c$, $B$ in tesla, $L,s$ in metres and $z=q/e$, restore the usual conversion $p=0.2998|z|BR$.

The magnetic bending angle is $\varphi \simeq |q|BL/p$. For a momentum estimate based on this angle, propagation of uncertainty gives $\sigma (p)/p=\sigma (\varphi )/\varphi$. Comparing the total Multiple Coulomb scattering angle with the bend therefore gives the commonly intended characteristic estimateIts independence of $p$ is the key result for relativistic particles: the scattering angle and magnetic bend both scale as $1/p$.

A precise standard deviation requires specifying how tracks are measured. If $\varphi$ is obtained from independently measured entrance and exit directions and all the material lies between them, the bending-plane projected root mean square is $\mathrm{RMS}(\theta )/\sqrt{2}$. For that estimator the equality isA three-point sagitta fit weights scatter positions differently again. The paper does not specify these measurement details, so it fixes the scaling and characteristic estimate, but not one universal geometric coefficient for all track fits.

The central estimate is $z=3.2743$. propagation of uncertainty must include the occurrence of $x$ in both numerator and denominator. With angular derivatives measured per radian,Interpreting the two quoted errors as independent standard deviations, convert them to radians before calculatingIf the quoted errors are instead absolute bounds, the first-order worst-case estimate is $|\delta z|\lesssim |z_x|\delta x+|z_y|\delta y=0.60$. Correlated errors add $2z_x{z}_y\mathrm{Cov}(x,y)$ to the variance. Treating $x+y$ and $x$ as independent would lose their shared measurement uncertainty.

From Snell's law, $n=\sin {\theta }_i/\sin {\theta }_r=1.5204$. Treating the quoted angle errors as independent one-standard-deviation uncertainties and using radians in derivatives,This refractive-index uncertainty from Snell's law is first-order propagation of uncertainty, with air's index approximated by one. Correlated angular calibration errors add $-2n^2\cot {\theta }_i\cot {\theta }_r\mathrm{Cov}({\theta }_i,{\theta }_r)$. If the quoted errors are hard bounds instead, their conservative first-order sum gives about $0.19$, rather than the statistical quadrature value.

For a fixed magnitude zero point, photometric magnitude uncertainty follows from propagation of uncertaintyThis is the absolute standard uncertainty of the apparent magnitude. The requested fractional uncertainty is insteadThis is the requested fractional standard uncertainty as a function of $m$; there is no single numerical fractional error independent of the magnitude. At $m=0$ it is undefined. Moreover, its value changes when the magnitude zero point changes; the absolute magnitude uncertainty is the useful invariant of such a zero-point shift. Treating the stated percentage as a finite perturbation rather than a standard uncertainty gives changes $-0.1035$ mag for $f\mapsto 1.1f$ and $+0.1144$ mag for $f\mapsto 0.9f$, consistent with the linear estimate.

Substitution in the moment of inertia givesFor independent errors interpreted as standard uncertainties, logarithmic differentiation and propagation of uncertainty giveConsequentlyThis is a $0.515\%$ statistical precision requirement. If “accuracy” instead means hard bounds whose worst cases must be guaranteed, first-order absolute errors add: $0.01+0.01+4{ϵ}_R\le 0.025$, giving $\boxed{{ϵ}_R\le 0.125\%,|\delta R|\le 1.25\mathrm{mm}}$ to first order. For an exact symmetric hard bound about the nominal values, the upper extreme is slightly more restrictive: $(1.01{)}^2(1+{ϵ}_R{)}^4\le 1.025$ gives ${ϵ}_R\le 0.1199\%$, or $1.20\mathrm{mm}$. The question does not distinguish a statistical uncertainty from a worst-case tolerance, so the convention matters.

For independent small uncertainties interpreted as standard deviations, propagation of uncertainty givesThe angular uncertainty is in radians. Since ${\sigma }_n/n=0.075/1.35=1/18$, the allowed angular contribution to the fractional uncertainty of the numerical aperture is only $\sqrt{0.06^2-(1/18{)}^2}$. ThereforeThe uncertainty in the refractive index already contributes $5.56\%$. If the quoted errors instead mean strict worst-case bounds, first-order absolute errors must be added rather than added in quadrature; that convention would require $|\delta \alpha |\lesssim \tan 60^{\circ }(0.06-1/18)=0.441^{\circ }$. The $2.25^{\circ }$ result uses the usual independent statistical-error convention.

Use $x=t$ in picoseconds and $y=\ln (R/R_{\ast })$. Since the time uncertainty is negligible, only the vertical errors enter. First-order propagation of uncertainty givesThe transformed measurements areThus the logarithmic transformation converts constant fractional rate uncertainties into approximately equal absolute log-rate uncertainties. The equality is the linear error approximation; finite error bars would become slightly asymmetric under an exact logarithm.

Assume independent approximately Gaussian log-rate errors with known common standard deviation $s=0.15$. The weighted least squares objective isSince all weights agree, the normal equations for linear least squares reduce toFor these data, $\bar{x}=1.5\mathrm{ps}$, $\bar{y}=4.1596533$, $S_{xx}=4.375{\mathrm{ps}}^2$ and $S_{xy}=-10.3782409\mathrm{ps}$. Hence $\hat{m}=-2.3721693{\mathrm{ps}}^{-1}$ and $\hat{c}=7.7179073$.

The covariance matrix of the fitted slope and intercept follows from the inverse weighted design matrix product, or directly by writing both estimates as linear combinations of the independent $y_i$:This givesThe parameter conversion and first-order propagation of uncertainty areConsequently,These are one-standard deviation errors. The estimates are correlated, since $\mathrm{Cov}(\hat{m},\hat{c})=-0.0077143{\mathrm{ps}}^{-1}$. The minimum is ${\chi }_{\min }^2=4.6537$ for four residual degrees of freedom. Because $s$ is supplied as a known measurement uncertainty, it should not be rescaled by the observed reduced ${\chi }^2$. This is a log-linear exponential-decay fit.