How to Build an Uncertainty Budget

An uncertainty budget is a table. That is the useful thing to know before starting one: you are not deriving anything from first principles, you are filling in rows, standardising each one, and combining them. The awkward part is knowing which rows belong in the table and what to divide each by.

This guide is the worked version — the divisor table you need, a complete budget filled in line by line with real arithmetic, and the reporting rules that apply once you have a number. If you want the concepts first (what uncertainty is, why root-sum-square, what a coverage factor does), the measurement uncertainty guide covers that ground and this post assumes it.

Everything here follows UKAS M3003 Edition 6, March 2024 — the UK application of the GUM. The Measurement Uncertainty Calculator will do the combining if you would rather not do it by hand.

Type A and Type B: A Distinction About Method, Not Importance

Worth clearing up first, because the labels mislead people into thinking one kind matters more.

M3003 section 4.1: "If an uncertainty is evaluated by statistical analysis of a series of observations, it is known as a Type A evaluation."

M3003 section 5.1: "If an uncertainty is evaluated by non-statistical analysis, it is known as a Type B evaluation."

That is the entire distinction. It describes where the number came from, not how significant it is. In most calibration budgets the largest contribution is Type B — usually the reference standard's own uncertainty — and the Type A repeatability term is comparatively small. Both become standard uncertainties and both go into the same combination.

Step 1: List the Contributions

This is where budgets go wrong, by omission. M3003 section 5.2 gives the minimum set of Type B sources you must "consider and include":

"(a) For measuring instruments — the imported uncertainties associated with their calibration and any drift or instability in their values or readings. (b) The reported uncertainty assigned to reference materials and any drift or instability in their values. (c) Effects arising from the use of ancillary equipment, including items such as connecting leads, pipework, heaters etc., and any drift or instability in their values or readings. (d) The equipment or item being measured, for example its resolution and any instability during the measurement… (e) The operational procedure. (f) The effects of environmental conditions on any or all of the above."

Read (c) again. Connecting leads, pipework, heaters — the ancillary equipment nobody lists. And (f) is a multiplier across everything above it, not a single row.

There is also a discipline in section 5.4: "Whenever possible, corrections should be made for known errors revealed by calibration or other sources." A known error is corrected, not carried as uncertainty. Only where the correction is inconvenient and the error is small relative to other terms does M3003 permit treating it "as if it were a systematic uncertainty of the uncorrected error magnitude."

What not to include

Section 5.2(d) contains a caveat that surprises most people:

"It should be noted that the anticipated long-term performance of an item being calibrated is not normally included in the uncertainty evaluation for that calibration."

A calibration describes the instrument at the moment it was calibrated. What it might do over the next twelve months is not part of this uncertainty — it is the reason calibration intervals exist. Padding the budget with the unit's expected drift inflates your reported uncertainty and confuses two separate questions.

Drift in your reference since its own calibration is different. That is section 5.2(a) and it does belong.

Step 2: Standardise Each Contribution

Contributions arrive in different shapes. Some are limits ("no more than ±0.02 °C"), some are standard deviations, some are expanded uncertainties from a certificate. Each needs converting to a standard uncertainty before anything can be combined.

M3003 section 3.26 puts it as: "Whenever input uncertainties are expressed in terms of limit values (e.g., limits of a rectangular distribution) rather than standard deviations, some processing is needed to 'standardise' them."

M3003 Table 1 gives the divisors. The worked figures below are M3003's own examples:

Choosing between them:

• Rectangular is the default when you know only the bounds and have no reason to think the middle is more likely. Resolution and digital rounding are always rectangular. Section 3.27 is explicit that this is the assumption for "the case of digital rounding".
• Triangular applies when values near the centre are genuinely more likely and the limits are hard. M3003's example earns it through physics — a large thermal mass tracks the mean air temperature rather than the extremes.
• U-shaped is a specialist case, mainly RF mismatch, where values cluster at the extremes rather than the middle.
• Normal covers Type A repeatability and any figure already given as a standard deviation.

An expanded uncertainty from a certificate is a fifth case: divide by its stated coverage factor. If a certificate says U = 0.06 °C at k = 2, the standard uncertainty is 0.030 °C. Which is why section 1.4.2 of the UKAS traceability policy insists every certificate in the chain carries an uncertainty statement — without k you cannot standardise it.

M3003 also warns against over-reaching on distribution choice. Around section 3.20 it notes you need grounds to "justify choosing a more 'informative' distribution than a uniform, or rectangular, probability" — so where you are unsure, rectangular is the honest choice, not the lazy one.

Step 3: A Worked Budget, Line by Line

Calibrating a digital thermometer that reads to 0.1 °C against a reference thermometer in a stirred bath, near 20 °C. All contributions are already in °C, so every sensitivity coefficient is 1 — which keeps the arithmetic visible. Where a contribution is in different units from the result, its sensitivity coefficient converts it.

Combine in root-sum-square. Square each standard uncertainty, add, take the square root:

0.0300² + 0.0115² + 0.0289² + 0.0126² + 0.0173² = 0.000900 + 0.000133 + 0.000833 + 0.000160 + 0.000300 = 0.002327

Combined standard uncertainty uc = √0.002327 = 0.0482 °C

Expand. Multiply by the coverage factor k = 2 for approximately 95% coverage:

U = 2 × 0.0482 = 0.0964 °C

Round, once, at the end. Two significant figures gives U = 0.096 °C.

Two things to notice about that table. Line 1 — the reference standard's own uncertainty — dominates: it contributes 0.000900 of the total 0.002327, about 39%. Line 3, the resolution of the instrument being calibrated, is the second largest at 36%. Squaring is unforgiving, so the two largest terms decide the answer and the small ones barely register. If you want a smaller reported uncertainty, a better reference or a finer-resolution instrument is the only route — taking more repeat readings would shrink line 4, which is already contributing under 7%.

That is the real value of writing the budget out. It tells you which contribution to attack.

Step 4: Report It Properly

M3003 section 6.1 requires the result be reported as a coverage interval, y ± U, with a specific accompanying statement. Section 6.2 gives the wording for the standard k = 2 case:

"The reported expanded uncertainty is based on a standard uncertainty multiplied by a coverage factor k = 2, providing a coverage probability of approximately 95 %. The uncertainty evaluation has been carried out in accordance with UKAS requirements".

Where the coverage factor came from a t-distribution via M3003 Appendix B, section 6.4 substitutes a form stating the actual factor and the effective degrees of freedom, at a coverage probability of 95.45%.

The presentation rules matter too:

• Two significant figures on the uncertainty (§6.10) — "it is seldom justified to report more".
• Match the decimal places — the measured value is "reported with the same number of decimal places as the measurement uncertainty" (§6.10).
• Round at the end only (§6.11), "to avoid the effects of cumulative rounding errors".

So for the budget above, if the thermometer read 20.1 °C where the reference indicated 20.043 °C, the error is +0.057 °C and the result is reported as error = +0.057 °C ± 0.096 °C, with the section 6.2 statement. Three decimal places on both, because the uncertainty has three.

Uncertainties may be reported in units of the measurand or as relative values — percent, ppm, parts in 10ˣ (§6.9). Edition 6 added Appendix Q specifically for input uncertainties expressed as relative quantities.

Mistakes Are Not Uncertainties

A short section in M3003 worth internalising, because it draws a line people blur. Section 5.5:

"Measurement errors should not be confused with mistakes. Common examples of mistakes are incorrectly applied corrections, transcription errors, and faults in software designed to control or report on a measurement process. The effects of such mistakes cannot readily be included in the evaluation of uncertainty and care is needed to avoid them."

You cannot budget for a transcription error. There is no divisor for "someone typed it in wrong". This is why the uncertainty budget is not a substitute for checking — and why a spreadsheet formula error in a budget is particularly nasty: it produces a confident number with no symptom.

When the Standard Approach Does Not Fit

Two limits M3003 states on its own method, worth knowing before you force a budget into a shape that does not suit it.

Section 6.12: where the input distributions are asymmetric, or the measurement model is non-linear, the result's distribution may also be asymmetric — and in those cases Monte Carlo simulation "offers a more suitable approach to evaluation of measurement uncertainty and coverage intervals". That points to JCGM 101 rather than M3003.

And where a single contribution dominates, the flat k = 2 assumption weakens. M3003 provides Appendix B for a dominant Type A input and Appendix C for a dominant Type B input, each giving a different coverage factor. If one line in your budget is much larger than everything else, check those appendices before defaulting to k = 2.

The Short Version

• Type A = evaluated statistically; Type B = evaluated non-statistically (§4.1, §5.1). Nothing to do with importance.
• List contributions against M3003 §5.2 (a)–(f) — including ancillary equipment and environment, the two most often missed.
• Do not include the calibrated item's anticipated future drift (§5.2(d)). Do include your reference's drift since its own calibration.
• Standardise with Table 1: rectangular /√3, U-shaped /√2, triangular /√6, normal s/√n. Certificate values: divide by the stated k.
• Combine in root-sum-square, then multiply by k = 2 for ~95%.
• Report to two significant figures, match the value's decimal places, and round only at the end (§6.10, §6.11).
• Squaring means the largest one or two contributions decide the answer — the budget tells you what to improve.
• Budgets do not cover mistakes (§5.5).

How CalProof Fits

An uncertainty budget is only worth building once per calibration type, but it has to stay attached to the calibrations it applies to. The common failure is a folder of budget spreadsheets that nobody can map to the instruments they were built for — so at audit the question "how was this uncertainty arrived at?" takes an afternoon to answer.

Each calibration record in CalProof has an optional uncertainty field where you record the uncertainty stated on the certificate (for example the expanded uncertainty and coverage factor), with the certificate and the instrument's full history in the same place — it stores what you enter, it does not compute or propagate the budget itself. Your reference standards are instruments in the same register, each with their own intervals and due dates — so a reference approaching or passing its due date is visible in the same recall view rather than tracked separately.

GBP pricing from £29 a month. No per-user fees on Pro and above. UK data hosting. No long-term contract.

To see the shape of the output, the sample audit pack is ungated and downloads as a single PDF.

Sources

• UKAS M3003 Edition 6, March 2024 — The expression of uncertainty and confidence in measurement
• JCGM 100:2008 — Evaluation of measurement data: Guide to the expression of uncertainty in measurement (the GUM)
• UKAS TPS 41 Edition 6, December 2022 — UKAS policy on metrological traceability

This guide follows UKAS M3003 Edition 6 (March 2024) as published; section, table and appendix references are to Edition 6. The worked budget is an illustrative example constructed to demonstrate the method — the contributions, values and distributions that apply to your measurements are yours to establish, and the figures here should not be copied into a real budget. Verify against the current published edition and your UKAS assessor. This is not legal or compliance advice.