Search for a measurement uncertainty PDF and you’ll get a hundred results: national standards, accreditation body guides, university lecture notes, and a great many re-uploaded copies of documents whose edition nobody has checked in a decade.
Most of them are describing the same method. A few of them are excellent. One or two are teaching a framework that was superseded in the 1990s.
This page is a guide to the literature — which documents exist, what each one is actually for, which are free, and which to read first depending on why you’re asking. There’s also a one-page summary near the end you can turn into your own reference sheet.
What People Actually Want When They Search This
Three different needs hide behind the same search:
The authoritative reference. Someone needs the official method, usually because an auditor asked or a procedure has to cite something.
A readable introduction. Someone is learning and the standard is too dense to start with.
A working reference sheet. Someone already knows the method and wants the formulas and divisors in one place, printable, on the bench.
These need completely different documents. The rest of this page sorts out which is which.
The Essential Documents
These are the ones that matter. Everything else is derivative.
JCGM 100:2008 – the GUM
Evaluation of measurement data Guide to the expression of uncertainty in measurement.
The foundational document. Published jointly by BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML which is why it carries the authority it does. Around 130 pages, and also published as ISO/IEC Guide 98-3.
This is where Type A and Type B come from, where the law of propagation of uncertainty is stated, and where the coverage factor and Welch–Satterthwaite formula are defined. If a procedure needs to cite something, it cites this.
It is not a tutorial. The maths is presented formally and the worked examples are at the back. Excellent as a reference, difficult as a first read.
Free from BIPM.
JCGM 200:2012 the VIM
International vocabulary of metrology basic and general concepts and associated terms.
The dictionary. This is where measurement uncertainty, measurand, trueness, accuracy, precision and error are formally defined. When two people are arguing about what “accuracy” means, this settles it.
Worth having open alongside the GUM. Free from BIPM.
NPL Good Practice Guide No. 11
A Beginner’s Guide to Uncertainty of Measurement.
Around 30 pages, written in plain English, and the best starting point in the entire literature for someone new to the subject. It covers the same eight-step method as the GUM without the formalism.
If you read one document before any other, read this one. Free from NPL.
UKAS M3003
The Expression of Uncertainty and Confidence in Measurement.
The practical guide. Substantially longer than the NPL guide and considerably more useful than the GUM if you’re actually building budgets. Extensive worked examples across different measurement disciplines, and a thorough treatment of distributions, degrees of freedom and coverage factors.
For a working calibration engineer, this is often the most valuable single document on the list. Free from UKAS.
EA-4/02
Evaluation of the Uncertainty of Measurement in Calibration.
The European Accreditation equivalent, and similarly practical. Worked examples for gauge blocks, temperature, mass, electrical and other common calibrations. If your budget resembles one of its examples, it will save you hours.
Free from European Accreditation.
The GUM Supplements
Four further JCGM documents, all free, and all more useful than their obscurity suggests.
JCGM 101:2008 — Supplement 1. Propagation of distributions using a Monte Carlo method. What to do when your model is strongly non-linear or your output distribution isn’t normal. Increasingly relevant as people run simulations rather than derivatives.
JCGM 102:2011 — Supplement 2. Extension to any number of output quantities. For measurements producing several correlated results at once.
JCGM 104:2009. An introduction to the GUM and related documents. Short, and genuinely introductory. A good bridge between the NPL guide and the GUM proper.
JCGM 106:2012. The role of measurement uncertainty in conformity assessment. Decision rules, guard banding, and the risk of accepting bad parts or rejecting good ones. If you make pass/fail calls, this is the one that changes how you work.
That last document is the most under-read on this entire list relative to its practical value.
Sector-Specific Guides
Eurachem/CITAC Guide CG4 — Quantifying Uncertainty in Analytical Measurement. The chemistry guide. Covers volumetric glassware, recovery, calibration curves and reference materials in a way the general documents don’t. Essential for analytical laboratories. Free.
NIST Technical Note 1297 — Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. A concise American treatment, closely aligned with the GUM. Free.
ILAC-G17 — Introducing the Concept of Uncertainty of Measurement in Testing. Aimed at testing rather than calibration laboratories, where uncertainty evaluation is often less straightforward.
ISO 5725 series — Accuracy (trueness and precision) of measurement methods and results. The source of the formal trueness/precision definitions. Purchased, not free.
ISO 21748 — guidance on using repeatability, reproducibility and trueness estimates in uncertainty evaluation. Useful where interlaboratory data exists. Purchased.
Which One Should You Read First?
Depends entirely on why you’re asking.
| If you are… | Start with |
|---|---|
| A student meeting this for the first time | NPL Good Practice Guide No. 11 |
| An engineer who has to build a budget next week | UKAS M3003 or EA-4/02 |
| An analytical chemist | Eurachem/CITAC CG4 |
| Writing a procedure that needs a citation | JCGM 100 (the GUM) |
| Preparing for an ISO/IEC 17025 assessment | UKAS M3003, then JCGM 106 |
| Making conformity decisions | JCGM 106 |
| Dealing with a non-linear model | JCGM 101 |
| Just trying to settle a definition | JCGM 200 (the VIM) |
The common mistake is starting with the GUM. It’s the authoritative document, so it feels like the right place to begin — and it’s the wrong one. It assumes you already understand the framework it’s formalising. Read the NPL guide first, then M3003, and the GUM becomes readable.
What’s Free and What Isn’t
Worth knowing, because people buy documents that were free all along.
Free, from the issuing organisation:
- All JCGM documents the GUM, the VIM, and all four supplements, from BIPM
- NPL Good Practice Guides
- UKAS M3003
- EA-4/02
- Eurachem/CITAC guides
- NIST Technical Notes
- ILAC guidance documents
Purchased:
- ISO/IEC 17025 — the standard itself
- The ISO 5725 series
- ISO 21748
- National adoptions of ISO standards
The pattern: guidance is free, standards cost money. Everything explaining how to evaluate uncertainty is available at no charge. What you pay for is the standard that says you must.
Get Documents From the Original Source
A practical warning, because it causes real problems.
Many of these PDFs circulate on document-sharing sites, uploaded years ago and never updated. Three risks:
Outdated editions. The GUM was substantially reissued as JCGM 100:2008; copies of the 1995 ISO edition are still widely circulated. The VIM is in its third edition; the second is still floating around with different definitions. M3003 and EA-4/02 have both been revised more than once.
Altered copies. Re-uploads are sometimes partial, sometimes re-scanned badly, occasionally annotated by whoever uploaded them.
Uncertain provenance. If you’re citing a document in a procedure that an assessor will read, “downloaded from a file-sharing site” is not a good answer.
All of the free documents above are available directly from BIPM, NPL, UKAS, European Accreditation, Eurachem or NIST. Go there. It takes an extra minute and you get the current edition.
What Lecture Notes and Course PDFs Get Wrong
University and training PDFs are often the most readable material available, and they carry two recurring problems.
Some still teach the pre-GUM framework. If a document’s central classification is random versus systematic error rather than Type A versus Type B, it’s teaching the framework that was superseded in the 1990s. Random and systematic remain useful for diagnosing what’s going wrong they are no longer how uncertainty is classified, because the distinction shifts depending on how you define the measurement.
Many stop before the useful part. A great many treatments cover the standard deviation and stop there. No Type B evaluation, no distributions and divisors, no combined uncertainty, no coverage factor. That’s the first quarter of the method presented as though it were the whole thing.
Neither makes a document useless a physics course teaching error analysis is doing something legitimate and different. Just know which framework you’re reading before you build a professional budget from it.
What a Good Uncertainty PDF Should Contain
If you’re evaluating a document you’ve found, check it covers these. A document missing more than two or three of them is an introduction, not a reference.
- A clear definition of the measurand and why it matters
- Type A evaluation, including the ÷√n step
- Type B evaluation with a table of distributions and divisors
- Where the divisors come from, or at least a complete table
- Sensitivity coefficients
- Combination in quadrature, stated explicitly
- Coverage factors, and what k = 2 means
- Degrees of freedom and Welch–Satterthwaite, at least mentioned
- At least one complete worked example, arithmetic shown
- Rounding and reporting conventions
That last one is more important than it looks. A document that calculates an uncertainty and never explains how to round and present it has left out the step people get wrong most often.
Make Your Own One-Page Reference
The most useful uncertainty PDF most people own is one they made themselves. Everything below fits on a single side of A4, and it’s most of what you need at the bench.
Type A
u = s ÷ √n (use the sample standard deviation, n − 1) Degrees of freedom: ν = n − 1
Type B — divisors
Certificate at k = 2 → ÷ 2 Certificate at k = 3 → ÷ 3 Rectangular (resolution, tolerance limits) → ÷ √3 Triangular → ÷ √6 U-shaped → ÷ √2 Resolution shortcut: u ≈ resolution × 0.289
Combining
u꜀ = √[ Σ (cᵢ · uᵢ)² ] Sensitivity coefficient c = ∂f/∂xᵢ; c = 1 for direct measurements; c = L × α for thermal
Expanding
U = k × u꜀, k = 2 for ≈95% ν_eff = u꜀⁴ ÷ Σ(uᵢ⁴ ÷ νᵢ) k from t-table: ν = 2 → 4.53; 4 → 2.87; 10 → 2.28; 20 → 2.13; ∞ → 2.00
Propagation
Add or subtract → add absolute uncertainties Multiply or divide → add relative uncertainties Power n → multiply relative uncertainty by n
Reporting
Round U to one significant figure (two if it starts with 1) Round the value to match U’s decimal place Always state k and the confidence level
Print it, laminate it, put it next to the instrument. It will get more use than any 130-page standard.
Common Problems With Uncertainty PDFs Found Online
No worked example. A document that states the formulas without applying them is a reference, not a guide. You need at least one complete calculation to check your understanding against.
Notation that shifts halfway through. Some documents use u for both standard and expanded uncertainty in different sections. Upper-case U should always mean expanded.
Divisors given without context. A table saying “rectangular: √3” is fine until you have to decide whether your source is rectangular.
No rounding guidance. Extremely common, and it’s the step that costs marks and audit points.
Sector assumptions left unstated. A chemistry guide assumes volumetric glassware; a dimensional guide assumes thermal expansion matters. Applying one to the other’s problems produces odd results.
Excessive length for the purpose. If you need the divisor for a triangular distribution, a 130-page standard is the wrong tool. Hence the one-page sheet above.
Frequently Asked Questions
What is the GUM?
JCGM 100:2008, Evaluation of measurement data Guide to the expression of uncertainty in measurement. The internationally agreed method for evaluating and expressing measurement uncertainty, published jointly by eight international organisations. It’s free from BIPM.
Is the GUM free to download?
Yes. All JCGM documents the GUM, the VIM and the four supplements are available at no charge from BIPM. The ISO standards that reference them are not free.
What’s the difference between the GUM and the VIM?
The GUM is the method. The VIM is the vocabulary the formal definitions of terms like measurand, uncertainty, trueness and accuracy.
Which measurement uncertainty PDF is best for beginners?
NPL Good Practice Guide No. 11, A Beginner’s Guide to Uncertainty of Measurement. About 30 pages, plain English, and it covers the whole method without the formalism.
What is UKAS M3003?
The Expression of Uncertainty and Confidence in Measurement a practical guide with extensive worked examples across measurement disciplines. Often more useful than the GUM for anyone actually building budgets. Free from UKAS.
What is EA-4/02?
European Accreditation’s guide to evaluating uncertainty in calibration, with worked examples for common calibration types. Free.
Do I need to buy ISO/IEC 17025 to evaluate uncertainty?
No. The standard states the requirement; the free guidance documents explain the method. You’d buy the standard if you need to read the requirements themselves, typically for accreditation.
Which document covers Monte Carlo uncertainty?
JCGM 101:2008, GUM Supplement 1, Propagation of distributions using a Monte Carlo method. Free.
Which document covers pass/fail decisions?
JCGM 106:2012, The role of measurement uncertainty in conformity assessment. It covers decision rules and guard banding, and it’s the most under-read document on the list relative to its usefulness.
Are lecture notes on uncertainty reliable?
Often readable and sometimes excellent, but check the framework. If the central classification is random versus systematic rather than Type A versus Type B, it’s teaching a pre-GUM approach fine for a physics course, not a basis for a professional budget.
Conclusion
There are a lot of measurement uncertainty PDFs. There are about six that matter.
Starting out? NPL Good Practice Guide No. 11.
Building budgets? UKAS M3003 or EA-4/02, for the worked examples.
Need to cite something? JCGM 100, the GUM.
Arguing about a definition? JCGM 200, the VIM.
Making pass/fail decisions? JCGM 106.
Working in analytical chemistry? Eurachem/CITAC CG4.
