What is the difference between Type A and Type B uncertainty?

Type A is worked out statistically from repeated readings: s ÷ √n with m − 1 degrees of freedom. Type B comes from any other information, such as a calibration certificate, a specification, the resolution or a handbook value (GUM 4.2 and 4.3). Once both are standard uncertainties they are combined the same way.

Why divide by √3 for resolution?

When all you know is that the value lies within ±a, the GUM models it as a rectangular distribution, whose standard deviation is a ÷ √3 (GUM 4.3.7). For a 0.001 mm digit, a = 0.0005 mm and u = 0.29 µm.

Is the coverage factor k always 2?

No. k = 2 gives about 95% only when the effective degrees of freedom are large. The template takes k from the t-distribution at νeff: 2.06 at 26 degrees of freedom and 95%, 3.18 at 3. Pick 95.45% if your customer expects k = 2.00 at large νeff, as UKAS M3003 Appendix B does.

What is a good test uncertainty ratio?

The GUM sets no value. Ratios such as 4:1 are industry practice and often loosely defined; JCGM 106 defines Cm = T ÷ 4u, which equals T ÷ 2U at k = 2. Agree a target with your customer, and use a guard band when the ratio is low.

How is this different from a gage R&R study?

A gage R&R splits observed variation into gauge, appraiser and part variation to show whether the gauge can tell parts apart. An uncertainty budget asks how far one result could be from the true value, and adds what an R&R cannot see: the reference standard's calibration, temperature and the instrument's specification. The R&R repeatability can be the Type A row.

Free template

Updated October 9, 2026

Measurement uncertainty budget template

A measurement uncertainty budget lists every source of doubt in one measurement, turns each into a standard uncertainty and combines them into the expanded uncertainty U that you report as y ± U. This template follows the GUM (JCGM 100:2008): Type A rows from repeated readings (s ÷ √n), Type B rows from certificates, specifications and resolution with the divisor for each distribution (k, √3, √6 or √2), sensitivity coefficients, the combined standard uncertainty by root sum of squares, effective degrees of freedom by the Welch-Satterthwaite formula and a coverage factor from the t-distribution. Page 1 of the PDF is a blank budget for 12 sources. Page 2 is a filled example for an outside micrometer on a 25 mm shaft. Page 3 has the steps, the formulas with their GUM clauses, a t-table, the decision rules and the common mistakes. The Excel version takes up to 30 repeated readings, looks up each divisor, works out u(xi), ui(y), each source's share of uc², νeff, k at 95% or 95.45% from an embedded t-table, U to two significant digits and the result statement. It then compares U with the tolerance: test uncertainty ratio, guard band, and a pass, conditional pass, conditional fail or fail statement for each part you measure.

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Page 1 of the measurement uncertainty budget — Type A and Type B rowsDivisor by distributionSensitivity coefficientsWelch-Satterthwaite νeffk from the t-tableExpanded uncertainty UTUR and guard bandWorked example

When to use it

When to use a measurement uncertainty budget

Your lab works to ISO/IEC 17025, which asks you to evaluate the uncertainty of your measurements (clause 7.6, Evaluation of measurement uncertainty).
A customer wants results as y ± U, or the calibration certificates you issue must state U and k.
Before you trust a gauge on a tight tolerance: is U small enough against the tolerance, and how much of the tolerance does a guard band cost?
When parts measured near a limit start an argument between supplier and customer. The budget shows how far a reading could be from the true value.
Not instead of a gage R&R study. The R&R shows whether the gauge can tell parts apart; its repeatability can become the Type A row here.

How to fill it in

1
Define the measurand and the model
Write what is measured, where and under which conditions: the diameter of journal A, two-point, at 20 °C. Then write the model: the reading plus every correction, even those estimated as zero. Each term becomes a row.
2
Take repeated readings
Measure the same item 10 or more times the way production does. The Excel works out s. The divisor is √n, where n is the number of readings averaged in each reported result, and the degrees of freedom are m − 1.
3
Add the Type B rows
From a certificate, enter U and its k. From limits only, enter the half-width a and pick rectangular (÷ √3), triangular (÷ √6) or U-shaped (÷ √2). Resolution is half a digit, rectangular.
4
Set the sensitivity coefficients
ci turns each standard uncertainty into the unit of the result. Length terms are 1. A temperature limit in °C needs α × L, which the sheet's helper works out: 11.5 × 10⁻⁶ per °C × 25 mm = 0.2875 µm per °C.
5
Combine and expand
uc is the square root of the sum of the squared ui(y). νeff = uc⁴ ÷ Σ(ui⁴ ÷ νi). k is the t value at νeff, rounded down to a table row, and U = k × uc, reported to two significant digits.
6
Apply your decision rule
Enter the tolerance. The sheet gives TUR = T ÷ 2U, moves each limit in by w = r × U and labels each measured part. Agree the rule with your customer before you measure.

Divisors

From what you know to a standard uncertainty

Every input is turned into a standard uncertainty u(xi) by dividing by the divisor of its distribution. Sources: JCGM 100:2008 clauses 4.2 and 4.3, UKAS M3003 Edition 6 Table 1.

Distribution, divisor	What you know	Example	Source
Normal (Type A), ÷ √n	s of single readings, with n readings averaged in each result	s = 0.843 µm, n = 1: u = 0.843 µm	GUM 4.2.3
Normal (Type B), ÷ k	U and its coverage factor k, from a certificate or spec	U = 0.5 µm, k = 2: u = 0.250 µm	GUM 4.3.3
Rectangular, ÷ √3 = 1.732	Only limits ±a: resolution, or a spec with no level stated	1 µm digit, a = 0.5 µm: u = 0.289 µm	GUM 4.3.7
Triangular, ÷ √6 = 2.449	Limits ±a, values near the middle more likely	a = 2 µm: u = 0.816 µm	GUM 4.3.9
U-shaped, ÷ √2 = 1.414	Limits ±a, values near the limits more likely	a = 1 °C: u = 0.707 °C	M3003 Table 1

Normal (Type A), ÷ √n
What you know
s of single readings, with n readings averaged in each result
Example
s = 0.843 µm, n = 1: u = 0.843 µm
Source
GUM 4.2.3
Normal (Type B), ÷ k
What you know
U and its coverage factor k, from a certificate or spec
Example
U = 0.5 µm, k = 2: u = 0.250 µm
Source
GUM 4.3.3
Rectangular, ÷ √3 = 1.732
What you know
Only limits ±a: resolution, or a spec with no level stated
Example
1 µm digit, a = 0.5 µm: u = 0.289 µm
Source
GUM 4.3.7
Triangular, ÷ √6 = 2.449
What you know
Limits ±a, values near the middle more likely
Example
a = 2 µm: u = 0.816 µm
Source
GUM 4.3.9
U-shaped, ÷ √2 = 1.414
What you know
Limits ±a, values near the limits more likely
Example
a = 1 °C: u = 0.707 °C
Source
M3003 Table 1

A filled-in example

Illustrative, not a benchmark

An example: an outside micrometer reading a 25 mm shaft journal with a tolerance of 24.990 to 25.010 mm (illustrative numbers, the same as the Excel Worked example sheet and page 2 of the PDF).

Ten readings on the shaft give s = 0.843 µm. Production reads each shaft once, so n = 1 and the Type A row is 0.843 µm with 9 degrees of freedom.
Type B rows: setting standard U = 0.5 µm at k = 2 (0.250 µm), the 1 µm digit (0.289 µm), anvil flatness and parallelism ±1 µm (0.577 µm), shaft and micrometer within 1 °C of each other (0.166 µm), room within 20 ± 2 °C with expansion coefficients differing by up to 2 × 10⁻⁶ per °C (0.058 µm).
uc = 1.105 µm. Repeatability is 58.2% of uc² and the anvils 27.3%; resolution is 6.8%.
νeff = 26.5, so k is t at 26 degrees of freedom: 2.056 at 95%. U = 2.272 µm, reported as 2.3 µm.
TUR = 20 µm ÷ (2 × 2.3 µm) = 4.3. With a guard band w = U, parts are accepted from 24.9923 to 25.0077 mm.

Result: 25.0040 mm ± 0.0023 mm (k = 2.06, νeff = 26.5, about 95%). Of five shafts, 25.004 mm passes; 24.991 and 25.008 mm pass simple acceptance but are conditional passes under the guard band; 25.011 mm is a conditional fail and 25.013 mm fails.

Common mistakes

Double counting
If several operators took the repeated readings, operator effects are already in s. An operator row on top counts them twice (GUM 4.3.10).
Using resolution as the whole story
Half a digit ÷ √3 is one row. In the example it is 0.289 µm of a 1.105 µm uc, 6.8% of the variance. U from resolution alone would be about 0.6 µm instead of 2.3 µm.
Dividing by √n you never use
s ÷ √10 is right only if every reported result is the mean of 10 readings. In the example that shortcut would report U = 1.5 µm for a shop that reads each shaft once.
Using k = 2 with few readings
When a Type A row from 4 readings dominates, νeff falls toward 3, where k at 95% is 3.18 (GUM Table G.2). Let the t-table set k.
Reporting too many digits
Give U to at most two significant digits and round y to the same decimal place (GUM 7.2.6). 2.2720 µm is reported as 2.3 µm, and the reading as 25.0040 mm with U = 0.0023 mm.
Adding a safety factor
Inflating inputs to be safe makes the stated coverage false (UKAS M3003 3.48). Keep U realistic and handle risk with the guard band.

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