Measurement science · GUM method

Measurement uncertainty
budget calculator

From individual uncertainty sources to a defensible combined and expanded uncertainty.

JCGM 100:2008First-order modelUncorrelated inputs

Build a budget from Type A evaluations of repeated readings and Type B evaluations of certificates and limits. Convert a normal expanded uncertainty using its coverage factor; divide rectangular, triangular, and U-shaped half-widths by √3, √6, and √2. For Type A, divide the standard deviation of individual readings by √n.

Each sensitivity coefficient maps an input uncertainty into the result unit. The calculator combines these contributions and uses the Welch–Satterthwaite effective degrees of freedom to find a two-sided Student t coverage factor. With few degrees of freedom, the factor needed for a given coverage probability can be greater than 2.

Your uncertainty budget

Define the sources. Review their contribution. Set your coverage.

Awaiting calculation

Result unit (unit symbol)

Used for combined and expanded uncertainty.

Coverage probability p (%)

0 < p ≤ 99.9999. At infinite ν, 68.27%, 95.45%, and 99.73% give approximately k = 1, 2, and 3.

Nine sources · nm · 99% coverage.
Expected: u_c = 31.66 nm, U = 92.47 nm.

Uncertainty sources

Use each source's own input unit; c converts it to the result unit.

SourceName · text	Distribution / typeDimensionless	Input valueU, a, or s · input unit	Divisor datak or n	Sensitivity cnm / input unit	Degrees of freedomν · dimensionless	Standard u(x)Input unit	Contribution u_inm	Variance share%	RemoveSource

Blank ν means infinity. Type A starts at n − 1; edit ν for a pooled s. Sensitivities may be negative or zero.

Outputs: 4 significant figures

Budget results

Two-sided coverage · k shown to 3 decimals

COMBINED UNCERTAINTY · u_c

—nm

Root-sum-square of contributions

EFFECTIVE DEGREES OF FREEDOM

—

Integer used for coverage: m = —

COVERAGE FACTOR · k

—

Student t / normal limit

EXPANDED UNCERTAINTY · U

—nm

Combined uncertainty × coverage factor

Enter a valid budget to calculate U = k × u_c.

Calculations use unrounded values. u(x) is in the source's input unit; u_i, u_c, and U are in the result unit. Degrees of freedom, k, and variance shares are dimensionless.

How to build the budget

Set the result unit and coverage.Enter the unit of the measurement result and the desired two-sided coverage probability in percent.
Add each uncertainty source.Choose its distribution. Enter U and k for Normal, half-width a for Rectangular, Triangular, or U-shaped, or standard deviation s and reading count n for Type A.
Apply the sensitivity and degrees of freedom.Enter c in result units per input unit. Supply ν when finite, or leave it blank for infinity. For a pooled Type A standard deviation, edit the automatically filled n − 1.
Review the combined result.Inspect the standard uncertainties and variance shares. The calculator reports u_c, ν_eff, its truncated integer m, the coverage factor k, and expanded uncertainty U.

Calculation method

Normal · expanded inputu(x) = U / k

Rectangular · half-widthu(x) = a / √3

Triangular · half-widthu(x) = a / √6

U-shaped · half-widthu(x) = a / √2

Type A · individual-reading standard deviationu(x) = s / √n

Contribution · result unitu_i = |c_i| × u(x_i)

u_c = √(Σ u_i²)

Share_i = 100 × u_i² / u_c²

ν_eff = u_c⁴ / Σ (u_i⁴ / ν_i)

m = floor(ν_eff), P(|T_m| ≤ k) = p / 100

U = k × u_c

Only nonzero contributions with finite ν enter the degrees-of-freedom denominator. A zero denominator gives ν_eff = ∞. GUM G.4.1 uses the next lower integer m; m = 0 cannot provide a Student t coverage factor.

Coverage-factor equations

For finite m, set θ = atan(t / √m) and c² = cos²θ. When m = 1, P = 2θ / π.

For odd m above 1: P = (2 / π)(θ + sinθ × S), where S sums T_j from j = 0 to (m − 3) / 2, T₀ = cosθ, and T_j = T_j−1 × c² × 2j / (2j + 1).

For even m: P = sinθ × S, where S sums T_j from j = 0 to (m − 2) / 2, T₀ = 1, and T_j = T_j−1 × c² × (2j − 1) / (2j).

These Abramowitz–Stegun 26.7.3–4 expressions are inverted by 80 bisection halvings on θ ∈ [0, π/2], giving k = √m × tanθ.

For infinite m or m > 100000, solve erf(k / √2) = p / 100 by bisection on k ∈ [0, 6]. The series is erf(x) = (2 / √π) Σ a_j / (2j + 1), with a₀ = x and a_j = −a_j−1 × x² / j, stopping when a term is below 10⁻¹⁷.

Frequently asked questions

How are the different uncertainty inputs converted?

Normal inputs use the certificate's expanded uncertainty divided by its coverage factor, U/k. Rectangular, triangular, and U-shaped inputs use the half-width divided by √3, √6, and √2, respectively. Type A inputs use the standard deviation of individual readings divided by √n.

Why can the coverage factor be greater than 2?

Few effective degrees of freedom require a larger Student t coverage factor for the same coverage probability. This calculator truncates the effective degrees of freedom to the next lower integer before finding the two-sided coverage factor. At infinite degrees of freedom, the normal distribution is used.

What should I enter for degrees of freedom?

Enter a positive degrees-of-freedom value, or leave it blank to represent infinity. Type A initially uses n − 1; you can edit this for a pooled standard deviation. Rows with zero uncertainty contribution do not affect the effective degrees of freedom.

Which measurement models does this calculator cover?

The calculator covers uncorrelated inputs in a first-order model using the supplied sensitivity coefficients. It does not include GUM 5.2 correlations or Monte Carlo propagation from GUM Supplement 1.