Introduction
The uncertainty of measurement is the range of values within which the true value of a measured quantity is likely to lie. It is not a mistake and it is not an error. It is a number that tells you how much doubt exists in a measurement so that anyone reading your result knows how far to trust it.
Measure a shaft as 25.000 mm and you have given someone a number. Report it as 25.000 ± 0.007 mm and you have given them a result they can actually use.
This guide explains what measurement uncertainty means, where it comes from, the different types, and how engineers reduce it in plain language, with examples you can picture.
The Official Definition
The internationally agreed definition comes from the International Vocabulary of Metrology (VIM, JCGM 200) and is repeated in the GUM, ISO/IEC 17025 and ISO 15189:
Measurement uncertainty is a non-negative parameter that characterises the dispersion of the values that could reasonably be attributed to the measurand.
Three phrases are doing all the work there, so let’s unpack them.
“Non-negative” – uncertainty is never below zero and never exactly zero. Every real measurement has some.
“Dispersion”– spread. It describes a band of plausible values, not a single number.
“The measurand” – the specific quantity you set out to measure. Not the part, not the instrument: the quantity. The diameter of this shaft, at this position, at this temperature.
Put together in everyday language: uncertainty is an honest statement of how much your measurement could plausibly wobble.
Why Measurement Uncertainty Matters
It is tempting to treat uncertainty as academic paperwork. It isn’t. It changes decisions.
It decides whether a part passes or fails. Suppose a shaft must be 25.00 mm with a tolerance of ±0.01 mm, and you measure 25.009 mm. Pass? Your measurement uncertainty is ±0.007 mm, so the true diameter could be anywhere from 25.002 to 25.016 mm. Part of that range is outside tolerance. You cannot honestly declare a pass. ISO 14253-1 handles exactly this situation by shrinking the acceptance zone a practice called guard banding.
It makes results comparable. Two labs measuring the same part will get slightly different numbers. Without stated uncertainties, you cannot tell whether they agree. With them, you can check whether the bands overlap.
It is a legal and accreditation requirement. ISO/IEC 17025 requires accredited testing and calibration laboratories to evaluate and report measurement uncertainty. No uncertainty statement, no accreditation.
It shows where to spend money. An uncertainty budget ranks your sources of doubt. Usually one or two dominate. Fixing those is cheap; buying a better instrument to fix the small ones is not.
Uncertainty vs. Error vs. Accuracy vs. Precision vs. Tolerance
These five words get used interchangeably in workshops, and they mean five different things. This table is worth memorising.
| Term | What it describes | Key point |
|---|---|---|
| Uncertainty | The range the true value probably lies in | You can calculate it from your own data |
| Error | The difference between your value and the true value | You can only know it if the true value is known |
| Accuracy | How close your result is to the true value | Compared against a reference standard |
| Precision | How close repeated readings are to each other | Says nothing about correctness |
| Tolerance | The variation the design permits in the part | A property of the part, not the measurement |
Two confusions cause most of the trouble:
Uncertainty is not error. Error needs the true value. Uncertainty doesn’t that’s the whole point of it. You can state a defensible uncertainty for a part nobody has ever measured before.
Uncertainty is not tolerance. Tolerance is what the drawing allows the part to be. Uncertainty is how well you know what the part actually is. A ±0.05 mm tolerance and a ±0.05 mm measurement uncertainty is a serious problem: your doubt is as wide as the entire permitted range, and you cannot make a reliable pass/fail call. A common rule of thumb is that measurement uncertainty should be no more than a quarter of the tolerance the 4:1 test uncertainty ratio.
You can be precise and wrong. Five readings clustered within a micron of each other, from a micrometer with a 0.02 mm zero error, are precise and badly inaccurate. Repeating a measurement can never reveal that.
What Causes Measurement Uncertainty?
Uncertainty is not one thing. It is many small contributions stacking up. Engineers usually group them into six families:
1. The instrument. Its resolution, its own calibration uncertainty, drift since it was last calibrated, wear on the anvils or jaws, and hysteresis.
2. The reference standard. Your slip gauge or setting ring has its own uncertainty from its calibration. That uncertainty passes straight into your measurement.
3. The environment. Temperature is usually the biggest offender. Steel expands roughly 11.5 µm per metre per °C, so a 1 metre steel part measured at 25 °C instead of the standard 20 °C is about 58 µm longer than its drawing value. Vibration, humidity and air pressure matter too.
4. The operator. Measuring force on a micrometer thimble, parallax when reading a scale, alignment errors, judgement about where the surface “really” is. Two skilled inspectors will not get identical numbers.
5. The workpiece. A “round” shaft is slightly oval. A “flat” surface has roughness. Where exactly on the part did you measure? Form error is a genuine uncertainty contributor, not a rounding issue.
6. The method. Clamping, fixturing, probe strategy on a CMM, how many points you take and where. Change the method and you change the answer.
Identifying and quantifying these sources is the first real step of any uncertainty evaluation and it’s covered in detail in GaugeHow’s Uncertainty Measurement course.
Types of Uncertainty in Measurement
The GUM sorts uncertainty by how you evaluate it, not by what causes it. There are exactly two types.
Type A Uncertainty Evaluated From Statistics
Type A uncertainty is worked out from repeated measurements, using statistics.
You measure the same feature several times, calculate the standard deviation of your readings, and divide by the square root of the number of readings:
u = s ÷ √n
Five readings on a shaft: 25.002, 25.004, 25.001, 25.003, 25.002 mm Standard deviation s = 0.0011 mm Type A uncertainty = 0.0011 ÷ √5 = 0.0005 mm
This is the type most students meet first, because it comes straight out of your own data.
Type B Uncertainty Evaluated By Other Means
Type B uncertainty is everything you cannot get from repeating the measurement. You take it from documents, specifications and knowledge:
- Calibration certificates
- Manufacturer’s accuracy specifications
- Published data, such as thermal expansion coefficients
- The instrument’s resolution
- Professional judgement and experience
Because these figures arrive in different forms, each one is converted into a standard uncertainty by assuming a probability distribution and dividing by the right factor:
| Source | Distribution | Divide by |
|---|---|---|
| Calibration certificate (k = 2) | Normal | 2 |
| Instrument resolution | Rectangular | √3 |
| Manufacturer tolerance limits | Rectangular | √3 |
| Temperature variation between limits | Rectangular | √3 |
| Value most likely near the centre | Triangular | √6 |
A micrometer with 0.01 mm resolution: half-resolution is 0.005 mm, rectangular distribution Standard uncertainty = 0.005 ÷ √3 = 0.0029 mm
Why Not “Random” and “Systematic”?
Older textbooks split uncertainty into random and systematic errors. The GUM replaced this because the split describes the cause, and the same cause can be evaluated either way depending on your method. Type A and Type B describe how you got the number, which is what actually matters when you combine them. Random and systematic error are still useful ideas for understanding what’s going wrong they’re just no longer how uncertainty is classified.
Standard, Combined and Expanded Uncertainty
Once every contribution is expressed as a standard uncertainty, three more terms appear and they show up on every calibration certificate you will ever read.
Standard uncertainty (u) – a single contribution, expressed as one standard deviation. Roughly 68% confidence.
Combined standard uncertainty (u꜀) – all contributions merged by adding them in quadrature (square them, add, take the square root). Not simple addition, because the contributions are independent and unlikely all to conspire in the same direction.
u꜀ = √(u₁² + u₂² + u₃² + …)
Expanded uncertainty (U) – the combined value multiplied by a coverage factor, k.
U = k × u꜀
k = 2 is the near-universal choice, giving about 95% confidence. k = 3 gives about 99.7% and is used where the consequences of being wrong are severe.
So a certificate reading “±0.007 mm (k = 2)” means the combined standard uncertainty was about 0.0035 mm, and the laboratory is roughly 95% confident the true value lies inside that band.
What Is an Uncertainty Budget?
An uncertainty budget is simply a table listing every source of uncertainty and what each one contributes. Here is a realistic one for a micrometer measuring a 25 mm feature:
| Source | Value | Type | Distribution | Divisor | Std. uncertainty |
|---|---|---|---|---|---|
| Repeatability | 0.0011 mm | A | Normal | 1 | 0.0011 mm |
| Micrometer calibration | 0.0020 mm (k=2) | B | Normal | 2 | 0.0010 mm |
| Resolution (0.01 mm) | 0.0050 mm | B | Rectangular | √3 | 0.0029 mm |
| Temperature effect | 0.0015 mm | B | Rectangular | √3 | 0.0009 mm |
| Measuring force / operator | 0.0020 mm | B | Rectangular | √3 | 0.0012 mm |
Combined standard uncertainty = √(0.0011² + 0.0010² + 0.0029² + 0.0009² + 0.0012²) = 0.0036 mm
Expanded uncertainty (k = 2) = 2 × 0.0036 = 0.007 mm
Final result: 25.000 ± 0.007 mm (k = 2)
Look at which row dominates. Resolution contributes 0.0029 mm more than everything else combined. Buying a more repeatable operator or a warmer room would barely move the total. Switching to a digital micrometer with 0.001 mm resolution would roughly halve it. That is what an uncertainty budget is for: it tells you where the doubt actually lives.
Measurement Uncertainty and ISO/IEC 17025
For anyone working in or towards an accredited laboratory, uncertainty is not optional.
ISO/IEC 17025:2017 requires laboratories to identify the contributions to measurement uncertainty, evaluate it using appropriate methods, and report it on calibration certificates. Calibration laboratories must always report it. Testing laboratories must evaluate it, and report it when it is relevant to the validity of the result or when a customer asks.
It also underpins two related requirements:
- Traceability – an unbroken chain of calibrations back to the SI, where every link carries its own stated uncertainty.
- Decision rules – how you use uncertainty to make a conformity statement, and who carries the risk of a wrong call.
The laboratory requirements behind all of this are covered in the ISO/IEC 17025:2017 awareness programme, and the practical calibration workflow in the Calibration Process course.
How Is Measurement Uncertainty Calculated?
In brief, it is a seven-step process: define the measurand, identify every source of uncertainty, quantify each one, convert them all to standard uncertainties, combine them in quadrature, multiply by a coverage factor, and report the result as value ± U.
For the full formulas, rounding rules and worked examples half the least count, the range method, standard deviation ÷ √n, and the rules for propagating uncertainty through calculations see our step-by-step guide: How to Find the Uncertainty of a Measurement.
How to Reduce Measurement Uncertainty
You can never eliminate uncertainty, but you can shrink it usually a lot, and usually cheaply. Work down this list in order.
1. Read your uncertainty budget first. Attack the biggest contributor. Improving a source that contributes 5% of the total is wasted effort.
2. Take more readings. Uncertainty falls as √n, so four times as many readings halves the Type A contribution. Beyond about ten readings, the returns get thin.
3. Use an instrument with finer resolution. As the budget above showed, resolution is very often the dominant term. A 0.001 mm digital micrometer instead of a 0.01 mm analogue one is frequently the single biggest available improvement.
4. Control temperature. Measure at the standard reference temperature of 20 °C, and let parts and instruments soak until they reach it. Handle precision parts with gloves or tongs body heat alone can expand a gauge block measurably.
5. Calibrate regularly, and use traceable standards. A calibration certificate converts an unknown instrument error into a known correction plus a small, quantified uncertainty.
6. Remove systematic errors rather than absorbing them. Zero your micrometer before use. Apply the correction stated on your setting ring’s certificate. A known bias should be corrected, not carried around as uncertainty.
7. Standardise the method. Use fixtures instead of hand-holding. Use the ratchet or friction thimble so measuring force is repeatable. Write the procedure down so everyone follows it.
8. Follow the Abbe principle. Keep the measurement axis in line with the scale axis. Offsetting them as a vernier caliper’s jaws do — introduces error that grows with the offset. This is exactly why a micrometer outperforms a caliper.
9. Train the operators. A Gauge R&R study will tell you how much of your variation is the people rather than the equipment. Often it’s more than anyone expects.
10. Measure closer to the source. Fewer links in the traceability chain means fewer accumulated uncertainties.
Instrument selection, method planning and 3D measurement strategy are covered practically in Engineering Metrology & 3D Measurement.
Common Misconceptions About Measurement Uncertainty
“Uncertainty means I made a mistake.” No. A mistake is a blunder misreading a scale, writing down the wrong number. Uncertainty is present in a perfectly executed measurement. A result without a stated uncertainty is the incomplete one.
“Uncertainty and error are the same thing.” Error is your value minus the true value, and needs the true value to exist. Uncertainty is a range you calculate from your own data. They are different quantities that usually have different magnitudes.
“A more expensive instrument has no uncertainty.” Every instrument has uncertainty, including the primary standards at national metrology institutes. Better equipment makes uncertainty smaller, never zero.
“Repeating the measurement removes uncertainty.” Repeating reduces the random part. A zero error, a miscalibrated instrument or a room at the wrong temperature shifts every reading identically, and no amount of averaging will reveal it.
“Uncertainty is the same as tolerance.” Tolerance is what the design permits the part to be. Uncertainty is how well you know what the part is. Confusing them leads directly to shipping bad parts.
“Smaller uncertainty is always better.” Uncertainty needs to be fit for purpose, not minimal. Chasing nanometre uncertainty on a part with a ±0.1 mm tolerance wastes time and money. Aim for roughly a quarter of the tolerance and stop.
“You can only find uncertainty by repeating measurements.” That’s Type A only. Type B certificates, specifications, resolution, published data — is often the larger share, and sometimes it’s all you have.
“Digital instruments have no uncertainty because they show exact numbers.” A crisp display is not the same as a correct one. Digital instruments have resolution limits, calibration uncertainty and drift like everything else. The clean readout just hides it better.
“Calibration eliminates uncertainty.” Calibration quantifies it and lets you correct known bias. The certificate itself states a residual uncertainty, which then feeds into your budget.
Frequently Asked Questions
What is meant by uncertainty of measurement?
It is the range of values within which the true value of a measured quantity is likely to lie. It quantifies the doubt in a measurement result, and is written as value ± uncertainty.
What is uncertainty of measurement with an example?
If a shaft is reported as 25.000 ± 0.007 mm, the measured value is 25.000 mm and the uncertainty is 0.007 mm. The true diameter probably lies between 24.993 and 25.007 mm.
What are the two types of uncertainty in measurement?
Type A, evaluated statistically from repeated measurements, and Type B, evaluated by other means such as calibration certificates, manufacturer specifications and instrument resolution.
What is the difference between error and uncertainty?
Error is the difference between a measured value and the true value, and can only be calculated if the true value is known. Uncertainty is the range the true value probably falls within, and can be calculated from your own measurements alone.
What causes uncertainty in measurement?
Six main families: the instrument, the reference standard, the environment (especially temperature), the operator, the workpiece itself, and the measurement method.
What does k = 2 mean on a calibration certificate?
k is the coverage factor. k = 2 means the combined standard uncertainty was multiplied by two, giving a confidence level of approximately 95%.
Can measurement uncertainty ever be zero?
No. By definition it is non-negative and never zero. Counted quantities such as “12 parts” are exact, but counting is not measuring. An uncertainty of zero means the evaluation is wrong.
Is measurement uncertainty required by ISO 17025?
Yes. Accredited laboratories must identify contributions to uncertainty, evaluate it appropriately, and report it always for calibration, and for testing where it affects the validity of results or the customer requests it.
How much uncertainty is acceptable?
It depends on what the measurement is for. A widely used guideline is that measurement uncertainty should be no more than a quarter of the part’s tolerance the 4:1 ratio.
Conclusion
Measurement uncertainty is not a confession of sloppiness. It is what turns a number into a result.
The essentials:
- Uncertainty is the range within which the true value probably lies never zero, never negative.
- It is different from error, accuracy, precision and tolerance, and confusing it with tolerance is the costliest mistake of the five.
- It comes from six families of sources: instrument, reference standard, environment, operator, workpiece and method.
- There are two types by evaluation method: Type A from statistics, Type B from certificates, specifications and published data.
- Contributions combine in quadrature into a combined standard uncertainty, then multiply by a coverage factor usually k = 2 for 95% confidence to give the expanded uncertainty you report.
- Build an uncertainty budget, find the dominant term, and fix that one first.
