Metrology

Uncertainties, propagation, budgets, and reporting. Designed around the GUM idea: report a measured value with a defensible uncertainty.

SI Base Units

Quantity	Unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Thermodynamic temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

Prefixes

Factor	Name	Sym

Reporting Helper (value ± uncertainty)

Value

Uncertainty

Is this expanded (U)?

k (if expanded)

Digits rule

—

Uncertainty Budget

Enter sources as half-width limits (±a). Choose distribution to convert to standard uncertainty. Sensitivity coefficient applies scaling (c·u). Add degrees of freedom to compute k via Welch–Satterthwaite.

Source	Type	±a (unit)	Distribution	Divisor	c (sens.)	u_i (std)	c·u_i	ν_i (DOF)

u_c (combined): — v_eff: — k (95%): — U (expanded): —

Divisors: normal = 1, rectangular = √3, triangular = √6. u_i = a / divisor. Combined u_c = √Σ(c·u_i)². v_eff by Welch–Satterthwaite; k from Student’s t (95%).

Propagation — Common Cases

Sum / Difference

u_Q = —

Q = ±A ± B ± … ; u_Q = √Σ u²

Product / Quotient / Powers

Symbol	Value	u (std)	Exponent (α)

Q = Π A^α → u_Q = —

Relative: (u_Q/|Q|) = √Σ(α·u/A)² ; for Q = A·B/C use α = {+1,+1,−1}

Propagation — General (Monte Carlo)

Expression f(a,b,c,...) (use a,b,c… as variables)

N samples k (expanded)

Var	Mean	Std u	Distrib.

mean: — std (u): — U (k·u): —

Distributions: normal (μ,σ), rectangular (μ, half-width a → std = a/√3), triangular (μ, a → std = a/√6).

Metrology Terms (quick reference)

Accuracy vs. Precision — closeness to true vs. spread of repeated results.
Resolution — smallest increment that can be discerned.
Repeatability — same operator/conditions; Reproducibility — different conditions/operators.
Linearity — deviation from best-fit line across range.
Bias — systematic error (often part of Type B).
Traceability — unbroken chain of calibrations to a standard.
Type A — evaluated by statistics of repeats; Type B — other info (specs, calibration, experience).
Expanded uncertainty U — k·u_c for chosen coverage (often k≈2 ≈95%).