The uncertainty budget

How Open Gauge builds a GUM-compliant uncertainty budget, combines it, and expands it into a reported ± value.

Open Gauge expresses measurement uncertainty as an itemized budget (GUM Annex H.1 table format), not a single opaque number. Every calibration stores an array of budget rows, one per contribution, each expressed as a standard uncertainty in the measurand's own units.

Type A vs. Type B

Per GUM §0.7/§3.3.3, "Type A" and "Type B" describe how an uncertainty was evaluated, not whether it's "random" or "systematic" (GUM explicitly deprecates that older distinction):

Type A — evaluated by statistical analysis of a series of observations (e.g. the spread of calibration-fit residuals).
Type B — evaluated by any other means: manufacturer specs, calibration certificates, handbook data, resolution, judgement.

Contributions

Source	Type	How it's derived
`fit_residuals`	A	Always present. Standard deviation of the calibration curve's fit residuals. Degrees of freedom = points − parameters, or unknown if that's ≤ 0.
`reference_standard`	B	Optional. The reference standard's own stated expanded uncertainty, converted to a standard uncertainty via $u = U/k$ (GUM §4.3.3). For internal calibrations, auto-fetched from the selected reference asset's most recent calibration; otherwise entered manually.
`resolution`	B	Included automatically whenever the channel has a resolution value. Rectangular distribution per GUM §4.3.7 — see below.
`sensor_nominal_accuracy`	B	The channel's manufacturer-stated uncertainty, converted via $u = U/k$ . Pre-fills from the channel's Uncertainty (±) field but is freely editable per calibration. Opt-in only — folding it in by default risks double-counting against `fit_residuals`, since both can reflect the same underlying instrument imprecision.
`external_certificate_stated`	B	Only appears on coefficients-only calibrations that have a lab-stated uncertainty. Recorded as a single row holding the certificate's expanded uncertainty and coverage factor as-is, not decomposed further.

Resolution as a rectangular distribution

A digital reading's resolution defines a range within which the true value could equally likely fall — a rectangular (uniform) distribution, not normal. Per GUM §4.3.7, its standard uncertainty is:

u_{\text{resolution}} = \frac{\text{resolution}}{\sqrt{12}}

Combination

All budget rows are combined via root-sum-square (GUM Eq. 10), assuming the contributions are uncorrelated:

u_c(y) = \sqrt{\sum_i u(x_i)^2}

This is the combined standard uncertainty, stored as combined_uncertainty ("Combined (RSS)" in the wizard).

Expansion

The expanded uncertainty is:

U = k \cdot u_c(y)

giving an interval expected to encompass a stated fraction (the confidence level) of values reasonably attributable to the measurand.

Coverage factor

The coverage factor $k$ is always derived from the requested confidence level and distribution — there's no separate "coverage factor" input anywhere in Open Gauge, because a user-picked $k$ independent of confidence level and distribution shape isn't statistically meaningful:

Normal distribution — the normal quantile for the requested confidence level (GUM §6.3.3's "simple case"):
$k = \Phi^{-1}\!\left(0.5 + \frac{\text{confidence \%}}{200}\right)$
e.g. 95% confidence → $k \approx 1.96$ , 99% → $k \approx 2.58$ .
t-distribution / χ² distribution — $k$ is derived from the effective degrees of freedom $\nu_{\text{eff}}$ , via the Welch-Satterthwaite formula (GUM Eq. G.2b):
$\nu_{\text{eff}} = \frac{u_c(y)^4}{\displaystyle\sum_i \frac{u(x_i)^4}{\nu_i}}$
Rows with unknown (infinite) degrees of freedom — exactly-known Type B contributions — drop out of this sum entirely. If no row has finite degrees of freedom, $\nu_{\text{eff}}$ is undefined and Open Gauge falls back to the normal-distribution quantile above (the correct limit as $\nu \to \infty$ ).

Reporting rounding

Per GUM §7.2.6, combined and expanded uncertainty are rounded to at most 2 significant figures for display and on the certificate — everywhere else (R², RMSE, max error, hysteresis, ...) uses a plain fixed-decimal display. See Rounding for exactly how this plays out in practice.