Uncertainty and units

Scope

Use this skill whenever a calculation carries physical units or a reported number needs an uncertainty. Concretely:

• converting between units, including conversions that need a physical context (wavelength to photon energy, mass to amount of substance, energy to temperature);
• propagating uncertainty through a measurement model, with or without correlated inputs;
• building a GUM uncertainty budget from calibration certificates, specifications, and repeatability data;
• choosing a coverage factor and deciding whether k = 2 is defensible;
• rounding and writing a result so a reader knows what the ± means;
• extracting parameter uncertainties from a curve fit without discarding correlations;
• reviewing existing analysis code for silent unit and uncertainty defects;
• checking that a dimensionally consistent answer is also physically possible — the order of magnitude, the dimensionless group, and the regime it implies.

This skill covers the metrology and the two libraries that implement it. It does not cover statistical inference, model selection, or study design — see statistical-analysis, statistical-power, and experimental-design.

Current release and installation

Verified 2026-10-01:

• pint 0.26.1, released 2026-09-10; requires Python 3.12+.
• uncertainties 3.2.3, released 2025-04-21; requires Python 3.8+.
• NumPy 2.5.3 and SciPy 1.18.1; both require Python 3.12+.
• scipy.constants in SciPy 1.18.1 serves CODATA 2022. SciPy 1.11 and earlier served CODATA 2018; the switch to CODATA 2022 occurred in SciPy 1.15.

uv venv --python 3.13
source .venv/bin/activate
uv pip install "pint==0.26.1" "uncertainties==3.2.3" "numpy==2.5.3" "scipy==1.18.1"

pint-pandas and pint-xarray add unit-aware columns and arrays and are separate installs.

Non-negotiable workflow

1. Attach units at input and strip them only at output. Convert at function boundaries with ureg.wraps or m_as("unit"), never mid-calculation.
2. Write the measurement model explicitly before computing anything, including corrections whose estimated value is zero. A correction left out of the model leaves its uncertainty out of the budget.
3. Give every input four things: an estimate, a standard uncertainty, the distribution the uncertainty came from, and its degrees of freedom.
4. Convert Type B statements with the right divisor. A certificate's expanded uncertainty divides by its stated k; rectangular limits divide by sqrt(3).
5. Identify correlations before combining. A shared calibration, instrument correction, or fit can induce correlation; quantify shared components instead of assuming every pair is correlated.
6. Compute sensitivity coefficients, and read the budget from c_i * u(x_i) rather than from the raw uncertainties.
7. Check the linearization. Compare propagated distributions with the GUM interval when in doubt. Establish Monte Carlo numerical stability before a JCGM 101 clause 8 claim; neither method validates the measurement model or the assigned input PDFs.
8. Justify coverage. A Student-t factor from effective degrees of freedom is an approximation requiring suitable distributional assumptions.
9. Round the uncertainty first, then the value to the same decimal place.
10. **State what the ± is** — standard or expanded, with k, the coverage probability, and the method.
11. Sanity-check the magnitude before reporting. A dimensionally consistent result can still be impossible. Compare it against a known scale or a dimensionless group, and confirm every assumption you relied on still holds in that regime.

The failures this skill exists to prevent

Each of the following runs without error and produces a plausible number.

A unit stripped at an unknown scale

length = (12.7 * ureg.mm).magnitude          # 12.7 -- of what?
length = (12.7 * ureg.mm).m_as("m")          # 0.0127 metres, stated

.magnitude returns whatever the quantity happened to be carrying. Name the unit at the point of extraction, every time.

Offset temperature arithmetic

Q(20, "degC") + Q(5, "degC")     # OffsetUnitCalculusError -- correctly refused
Q(20, "degC") + Q(5, "delta_degC")   # 25 degree_Celsius
Q(25, "degC") - Q(20, "degC")        # 5 delta_degree_Celsius

Celsius and Fahrenheit are interval scales. An uncertainty on a temperature is always a difference and belongs in a delta_ unit: converting 20 ± 0.5 degC to Fahrenheit gives 68 degF ± 0.9 delta_degF, two different conversions on one line.

Logarithmic units that add by multiplying

Q(10, "dBm") + Q(10, "dBm")   # 0.0001 kilogram**2 * meter**4 / second**6

That is 10 mW × 10 mW, not 20 mW and not 13 dBm. Nothing raises. Convert to a linear unit before any arithmetic.

A correlation destroyed by a round trip

x = ufloat(1.0, 0.1)
x - x                                     # 0.0+/-0
x - ufloat(x.nominal_value, x.std_dev)    # 0.00+/-0.14

Rebuilding a variable from its nominal value and standard deviation creates an independent variable. So does any serialization that passes through a pair of floats. Use correlated_values(values, covariance_matrix) to rebuild a correlated set.

A covariance matrix silently rescaled

popt, pcov = curve_fit(f, x, y, sigma=sigma)                        # default
popt, pcov = curve_fit(f, x, y, sigma=sigma, absolute_sigma=True)

The default rescales pcov by the reduced chi-square. Relative weights from sigma still affect the fit and covariance: equality with an unweighted fit holds for constant sigma, not generally for heteroscedastic inputs. On one synthetic straight-line fit the two give [0.0364, 0.2154] and [0.0477, 0.2820] — a 31% difference. Pass absolute_sigma=True whenever sigma holds real standard uncertainties.

A linearization that was never checked

For y = x² with x = 1.0 ± 0.5, the GUM framework gives y = 1.0, u_c = 1.0, and a 95% interval of [-0.96, 2.96] — mostly negative, for a squared quantity. Monte Carlo gives a mean of 1.25, u_c = 1.06, and a shortest 95% interval of [0, 3.32]. Nothing in a linear-propagation library will tell you this happened.

Bundled local CLIs

All helpers run offline, reject URLs and symlinks, bound their inputs, write output atomically with private permissions, and refuse to overwrite without --force.

python skills/uncertainty-and-units/scripts/propagate_uncertainty.py --help
python skills/uncertainty-and-units/scripts/uncertainty_budget.py --help
python skills/uncertainty-and-units/scripts/format_result.py --help
python skills/uncertainty-and-units/scripts/convert_units.py --help
python skills/uncertainty-and-units/scripts/audit_units.py --help
python skills/uncertainty-and-units/scripts/check_plausibility.py --help

propagate_uncertainty.py

Runs both propagation methods on the same numerical model and compares interval endpoints. This fixed-trial diagnostic does not implement the adaptive stabilization in JCGM 101 7.9/8.2 and never labels the GUM result fully validated.

python skills/uncertainty-and-units/scripts/propagate_uncertainty.py \
  --expression "m / (pi * (d / 2) ** 2 * h)" \
  --variable "m=250.0,0.05" \
  --variable "d=2.0,0.002,rectangular" \
  --variable "h=4.0,0.005,rectangular" \
  --measurand density --unit "g/cm3" --format markdown

Each --variable is name=value,standard_uncertainty[,distribution[,dof]], where the distribution is normal, rectangular, triangular, arcsine, or exact and controls Monte Carlo sampling only. Here mass is in g and lengths in cm: --unit is a report label, not dimensional validation or conversion. JSON variable units are also labels; convert inputs to a consistent numerical model first. Correlations go in as --correlation "a,b=0.9". A JSON --spec file holds the same model for anything long-lived. The joint normal sampler supports positive semidefinite correlation matrices, including perfect correlations. Other joint distributions need a separate sampler. Correlated finite-dof inputs are refused because independent-input Welch-Satterthwaite is invalid. For independent finite-dof inputs the CLI's MC PDFs remain fixed: dof affects the GUM factor only, not MC sampling. A small-sample t input needs a separate justified model.

The expression is parsed into an abstract syntax tree and reduced by an explicit walk over + - * / ** and a fixed list of functions. It is never compiled or executed.

The report gives the estimate, u_c, sensitivity coefficients, the budget in percent, effective degrees of freedom, k, U, both Monte Carlo coverage intervals, and the endpoint-agreement diagnostic. Repeat/increase sampling and verify model/PDF assumptions before deciding what to report.

uncertainty_budget.py

Combines components stated the way certificates and data sheets state them.

python skills/uncertainty-and-units/scripts/uncertainty_budget.py --template > budget.json
python skills/uncertainty-and-units/scripts/uncertainty_budget.py --spec budget.json --format markdown

Each component names a distribution that fixes its divisor — expanded divides by its coverage_factor, rectangular by sqrt(3), triangular by sqrt(6), arcsine by sqrt(2), normal by 1 — with an optional sensitivity, dof, and relative: true. The tool assumes independent components, requires an explicit factor for expanded, and rejects nonzero exact components. It computes u_c, effective degrees of freedom, k from the t-distribution, and U, and warns when a Type A component has no degrees of freedom, when nu_eff is small enough that k = 2 is wrong, when one component dominates, and when a Type B component declared normal is probably an undivided expanded uncertainty.

format_result.py

python skills/uncertainty-and-units/scripts/format_result.py \
  --value 12.34567 --uncertainty 0.02345 --unit mm \
  --coverage-factor 2.26 --coverage-probability 0.95

Returns 12.346 ± 0.023 mm, 12.346(23) mm, the scientific and LaTeX forms, and the sentence that has to accompany the number. Warns when one significant digit is requested for an uncertainty beginning in 1 or 2, and when the uncertainty exceeds the estimate.

convert_units.py

python skills/uncertainty-and-units/scripts/convert_units.py \
  --value 532 --unit nm --to eV --context spectroscopy --uncertainty 0.5

python skills/uncertainty-and-units/scripts/convert_units.py \
  --value 1.0 --unit g --to mol --context chemistry --context-parameter "mw=180.156 g/mol"

Carries the uncertainty through the conversion's local derivative. Some contexts (such as wavelength to energy) are reciprocal, whereas others are proportional. Context parameters (mw, n) are treated as exact; their uncertainties need an explicit measurement model. Offset-temperature uncertainty units are differences. Names the context in the error message when a conversion needs one, and flags offset and logarithmic units. --list-contexts shows what the registry defines.

audit_units.py

Static review of existing analysis code. Parses, never imports or runs.

python skills/uncertainty-and-units/scripts/audit_units.py \
  --input analysis.py --format markdown --fail-on medium

Exit status is 1 when a finding meets --fail-on (default high), which makes it usable as a pre-commit or CI check.

The rules are heuristics, so a false positive is suppressed with a directive comment — trailing to cover its own line, or alone on a line to cover the next one:

value = quantity.magnitude  # audit-units: ignore UNIT003 -- already converted upstream

# audit-units: ignore UNC003 -- the argument here is a plain float array
scaled = np.log10(counts)

# audit-units: ignore-file CONST001 covers a whole module, and naming no rule suppresses all of them. Suppressions are counted in the report rather than hidden, so a file that silences everything still says so.

check_plausibility.py

Dimensional consistency is not physical possibility. A cell 2 m across and a Reynolds number of 4e7 in a capillary both pass every unit check. This tool tests a set of quantities against dimensionless groups, characteristic scales, and illustrative typical-value bands, and verifies each formula's dimensionality before reporting a number.

python skills/uncertainty-and-units/scripts/check_plausibility.py \
  --quantity "density=1060 kg/m**3" --quantity "velocity=0.5 mm/s" \
  --quantity "length=8 um" --quantity "viscosity=3.5 mPa*s" \
  --group reynolds --format markdown
# Re = 0.001211 -- laminar (circular pipe, length = diameter)

python skills/uncertainty-and-units/scripts/check_plausibility.py \
  --quantity "diameter=2 m" --band "eukaryotic_cell_diameter=diameter"
# implausible: 4.3 decades outside the 5-100 um range

--group evaluates one of 14 dimensionless groups and names the regime it places the system in; --scale computes a characteristic scale such as a diffusion time, Debye length, or Stokes settling velocity; --band compares a supplied quantity against an observed range. --list prints the whole catalogue with the inputs each formula needs.

Physical constants (k_B, N_A, R_gas, g_earth, and the rest) are available to every formula without being supplied. Their nominal values come from scipy.constants; this plausibility helper does not propagate their uncertainties. g_earth is standard gravity, not a local measurement. Bands are screening heuristics, not physical limits.

The dimensionality check is the point. Passing a kinematic viscosity where the formula needs a dynamic one — both called "viscosity", both tabulated for water, differing by a factor of ρ — is refused before any number is computed:

error: viscosity must have dimensionality [mass] / ([length] * [time]),
       but m²/s is [length] ** 2 / [time]

Exit status is 1 when the verdict meets --fail-on (default implausible; a value within one decade of a band is questionable). The thresholds are conventions with soft edges and assume the geometry their correlation was fitted for — see references/plausibility-scales.md for the characteristic length to use in each case.

Choosing a propagation method

A model dominated by rectangular contributions can fail the clause 8 test even when it is perfectly linear: the framework's k = 1.96 over-covers a nearly trapezoidal output. The estimate and u_c are still right; only the interval is too wide.

Constants

Never type a constant from memory. The SI has seven exact defining constants, including c, h, e, k, and N_A; quantities derived solely from exact constants can also be exact (for example, R = N_A * k). Other constants may be measured and change between CODATA releases. Check each constant's stated uncertainty rather than assuming every other value is measured.

import scipy.constants as constants

constants.value("electron mass")        # 9.1093837139e-31
constants.unit("electron mass")         # kg
constants.precision("electron mass")    # 3.07e-10, relative standard uncertainty
constants.precision("Planck constant")  # 0.0, exact by definition

precision returns a relative standard uncertainty; multiply by the absolute value of the constant for the absolute one.

Reference files

• references/gum-methodology.md — Type A and Type B evaluation, distribution divisors, the law of propagation, Welch-Satterthwaite, when the framework fails, the Monte Carlo procedure, and the clause 8 validation test.
• references/pint-recipes.md — registries, offset and logarithmic units, contexts, boundary enforcement with wraps and check, NumPy interoperability, custom units, formatting.
• references/uncertainties-recipes.md — variable identity and correlation, correlated_values, umath and unumpy, format specs, fit covariance matrices, and the package's limits.
• references/domain-conversions.md — the energy ladder, spectroscopy, concentration, pressure, radiation and magnetism, mass spectrometry, logarithmic quantities, and the pairs that share dimensions without sharing meaning.
• references/reporting-rules.md — rounding, notations, the sentence that must accompany a result, SD versus SEM versus CI in figures, non-detects, and conformity decision rules.
• references/plausibility-scales.md — choosing the characteristic length, the dimensionless groups and the modelling assumption each one gates, characteristic scales, the observed magnitude bands and their sources, and the caveats on every threshold.

Dated sources

Checked 2026-10-01:

• JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement
• JCGM 101:2008, Supplement 1 — Propagation of distributions using a Monte Carlo method
• NIST Technical Note 1297
• CODATA internationally recommended values
• Pint on PyPI — 0.26.1, released 2026-09-10.
• Pint documentation, including non-multiplicative units and contexts.
• uncertainties on PyPI — 3.2.3, released 2025-04-21.
• uncertainties documentation
• See references/review.md for current API/source verification and execution limits.
• scipy.constants reference