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The original apparatus has two small masses at distance $L/2$ from the suspension axis. Neglecting the rod's moment of inertia, their total moment of inertia is $I=2m(L/2{)}^2=m{L}^2/2$, consistent with the supplied torsion pendulum period. Treat $m$ and $L$ as exact because no uncertainties are specified. Rearranging the period relation givesThe torsion constant is a restoring torque per unit angle. In dimensional calculations the radian is dimensionless.

Using first-order propagation of uncertainty, with the quoted period uncertainty interpreted as a standard uncertainty,Consequently

All four angle measurements have the same standard uncertainty, so ordinary least squares minimizesDifferentiation givesThus the unique minimum isThe overbars denote sample means over the four observations. Numerically,soThe fixed $x_i$ have negligible uncertainty. Assuming independent errors in the four $y_i$, propagation of uncertainty in quadrature givesHenceIf the period measurement is independent of the angle measurements, including its calibration uncertainty givesThis last uncertainty includes the stated random measurement errors only. Whether the model describes the data must be checked before treating it as a reliable measurement of the gravitational constant.

Write $X=1/(\omega C)=200\Omega$. The magnitude of the series RC impedance isFor independent small errors interpreted as standard uncertainties, propagation of uncertainty givesThe input uncertainties are $u_R=2\Omega$, $u_{\omega }/\omega =0.01$, and $u_C/C=0.01$. Thus $u(Z_m{)}^2=0.80+3.20+3.20=7.20{\Omega }^2$, andThe fractional uncertainty is $1.20\%$. If the stated errors are instead worst-case bounds, their first-order contributions add in magnitude, giving approximately $4.47\Omega$, or $2.0\%$, rather than the quadrature result. Both conventions require the error interpretation to be specified.

Rearrange the supplied form of Kepler's third law to make the gravitational constant the subject:Convert the semi-major axis and orbital period to SI units before substitution:The central value is thereforeFor monomial uncertainty propagation, logarithmic differentiation givesIf the quoted errors are independent standard uncertainties, the variances of these contributions add:Thus ${\sigma }_G=1.61335\times 10^{-12}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, and the measured result under that statistical convention isThe semi-major axis measurement supplies the largest individual contribution to the relative uncertainty.

The paper does not specify whether its errors are statistical uncertainties or bounds. If they are maximum input errors, the first-order worst-case error propagation estimate adds magnitudes:Correlation between input errors would instead require the covariance terms in the general propagation of uncertainty formula; they are not specified.

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.

For a fixed magnitude zero point, the derivative of $m=-2.5{\log }_{10}(f/f_0)$ gives the small-error propagation of uncertainty formula ${\sigma }_m=(2.5/\ln 10){\sigma }_f/f$. A fractional observed radiative flux uncertainty of $0.1$ gives about $0.109$ mag of absolute uncertainty. A fractional magnitude uncertainty additionally needs the value of $m$, is undefined at $m=0$, and depends on the adopted magnitude zero point.

Propagation of uncertainty estimates how measurement errors in inputs affect a calculated result. For small input errors with covariance matrix $\Sigma$, linearizing a differentiable function $g$ gives the displayed approximation for its variance. For independent errors in a product $g=2fd$, the squared relative standard deviations add. Correlated errors need the covariance terms; systematic offsets require separate assessment.

At fixed measured count slope, $N_A^{\mathrm{sed}}\propto a^{-3}$, whereas at fixed measured diffusion coefficient, $N_A^{\mathrm{diff}}\propto a^{-1}$. First-order propagation of uncertainty follows by taking logarithmic derivatives:For independent small errors, the radius contributes $9({\sigma }_a/a{)}^2$ to the relative variance of $N_A^{\mathrm{sed}}$, compared with $({\sigma }_a/a{)}^2$ for $N_A^{\mathrm{diff}}$. Shared radius and temperature calibrations make the two estimates correlated; their agreement is therefore not two wholly independent determinations. As a finite-error example, overestimating $a$ by $5\%$ multiplies the respective answers by $1.05^{-3}\simeq 0.8638$ and $1.05^{-1}\simeq 0.9524$.

This explains the importance of the radius cross-checks in the 1909 memoir: testing the Stokes drag law through an independent radius estimate helps prevent circular inference.^[ref] Convection must also be excluded: a residual horizontal speed $u$ adds $u^2{\tau }^2$ to an uncentred mean squared displacement, making a linear fit misleading.

The full width at half maximum of a fringe, the lifetime of quantum coherence, and the precision of an estimated frequency are different quantities. In the ideal short-pulse limit, write the transfer probability with contrast $0<V\le 1$ and second-pulse oscillation phase $\phi$ asAt quadrature, $P=1/2$ and $|\partial P/\partial \nu |=\pi VT$. For $N$ independent, equally prepared particles with ideal binary readout, the binomial distribution gives ${\sigma }_P=\sqrt{P(1-P)/N}=1/(2\sqrt{N})$. Local propagation of uncertainty therefore yields the displayed ${\sigma }_{\nu }$. For chosen $T=5\mathrm{ms}$, $V=0.8$ and $N=10^4$, it is approximately $0.398\mathrm{Hz}$, although the ideal fringe width is $100\mathrm{Hz}$. This is an original counting-noise estimate at a known quadrature, not the accuracy of a historical clock or a guarantee of finding the correct fringe.

For the illustrative dephasing model $V(T)=e^{-T/T_2}$ with fixed $N$, minimizing ${\sigma }_{\nu }$ maximizes $T{e}^{-T/T_2}$ and gives $T=T_2$: its derivative is $e^{-T/T_2}(1-T/T_2)$. This conclusion changes if longer interrogation reduces the number of accepted particles or the repetition rate. Quantum decoherence, a distribution of flight times, reference oscillator noise and readout errors can reduce contrast; a fitted decay time need not identify one microscopic mechanism. More counting can improve precision while a reproducible cavity quantum phase offset still biases the frequency, as in atomic clock frequency shifts.

For simple pendulum oscillation period $T$ in the small-angle approximation, $g=4{\pi }^{2}L/T^2$. If the length measurement and $N$ identically precise timing measurements have independent errors, the timing sample mean has standard error ${\sigma }_T/\sqrt{N}$. Propagation of uncertainty gives the title formula. A target fractional standard uncertainty $ϵ$ requires $N\ge 4({\sigma }_T/T{)}^2/[{ϵ}^2-({\sigma }_L/L{)}^2]$, rounded upward to an integer. No amount of repeated timing can beat the floor ${\sigma }_L/L$ without improving the length measurement; correlated errors or a timing systematic error require a different model.

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.

Using images from different viewpoints to convey or recover depth. In scientific photography, comparing corresponding features in calibrated views can help reconstruct their three-dimensional positions; a single image generally leaves depth ambiguous.