Uncertainty And Units

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 2026-07-26:

• pint 0.25.3, released 2026-03-19; requires Python 3.11+.
• uncertainties 3.2.3, released 2025-04-21; requires Python 3.8+.
• NumPy 2.5.1 and SciPy 1.18.0; both require Python 3.12+.
• scipy.constants in SciPy 1.18.0 serves CODATA 2022. SciPy 1.11 and earlier

served CODATA 2018, and several recommended values differ between them.

uv venv --python 3.13
source .venv/bin/activate
uv pip install "pint==0.25.3" "uncertainties==3.2.3" "numpy==2.5.1" "scipy==1.18.0"

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

installs.

Non-negotiable workflow

• Attach units at input and strip them only at output. Convert at function

boundaries with ureg.wraps or m_as("unit"), never mid-calculation.

• 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.

• Give every input four things: an estimate, a standard uncertainty, the

distribution the uncertainty came from, and its degrees of freedom.

• Convert Type B statements with the right divisor. A certificate's expanded

uncertainty divides by its stated k; rectangular limits divide by sqrt(3).

• Identify correlations before combining. Inputs calibrated against the same

standard, measured on the same instrument, or drawn from the same fit are correlated.

• Compute sensitivity coefficients, and read the budget from c_i * u(x_i) rather

than from the raw uncertainties.

• Check the linearization. Run Monte Carlo alongside the GUM framework and apply

the JCGM 101 clause 8 comparison. Report the Monte Carlo result when it fails.

• Choose k from the effective degrees of freedom, not by habit.
• 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, so the parameter uncertainties

absorb the goodness of fit and match what you would get by passing no sigma at all. 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 model and applies the JCGM 101 clause 8

validation test.

python skills/uncertainty-and-units/scripts/propagate_uncertainty.py \
  --expression "m / (pi * (d / 2) ** 2 * h)" \
  --variable "m=250.0,0.05" \
  --variable "d=20.0,0.02,rectangular" \
  --variable "h=40.0,0.05,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. Correlations go in as --correlation "a,b=0.9". A JSON

--spec file holds the same model for anything long-lived.

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

verdict on whether the linearized result may be reported.

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 computes u_c, the Welch-Satterthwaite 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, which matters because

context conversions are reciprocal rather than proportional. 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

| Rule | Severity | Detects |

| --- | --- | --- |

| UNIT001 | medium | a second UnitRegistry in one module — cross-registry ValueError |

| UNIT002 | medium | offset temperature units with no delta_ unit anywhere |

| UNIT003 | high | .magnitude without a preceding .to(...) or .m_as(...) |

| UNIT004 | medium | logarithmic units, whose + multiplies |

| UNC001 | high | curve_fit without absolute_sigma |

| UNC002 | medium | np.std / np.var without ddof |

| UNC003 | medium | math or numpy functions in a module that uses uncertainties |

| UNC004 | high | a ufloat rebuilt from .nominal_value and .std_dev |

| CONST001 | low | a literal within 0.1% of a CODATA constant |

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 curated magnitude

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, and are read from scipy.constants at run time rather

than written as literals, so they track the CODATA release SciPy ships.

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

| Situation | Method |

| --- | --- |

| Linear or near-linear model, normal-ish inputs, large dof | GUM framework alone |

| Any nonlinearity across ±2u of an input | run both, apply the clause 8 test |

| Relative uncertainty above ~20% on any input | Monte Carlo |

| Dominant rectangular or otherwise non-normal component | Monte Carlo |

| Output bounded below (variance, concentration, squared quantity) | Monte Carlo |

| Asymmetric output distribution | Monte Carlo, shortest coverage interval |

| Correlated inputs | either, but supply the covariance matrix, not the standard uncertainties alone |

A model dominated by rectangular contributions fails 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 2019 SI redefinition fixed c, h, e, k,

and N_A exactly, so their relative standard uncertainty is zero; everything else is a

measured value that moves between CODATA releases.

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 value 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 2026-07-26:

• [JCGM 100:2008, Evaluation of measurement data — Guide to the expression of

uncertainty in measurement](https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf)

• [JCGM 101:2008, Supplement 1 — Propagation of distributions using a Monte Carlo

method](https://www.bipm.org/documents/20126/2071204/JCGM_101_2008_E.pdf)

• NIST Technical Note 1297
• CODATA internationally recommended values
• Pint on PyPI — 0.25.3, released 2026-03-19.
• Pint documentation, including

non-multiplicative units

and contexts.

• uncertainties on PyPI — 3.2.3, released

2025-04-21.

• uncertainties documentation
• scipy.constants reference