Measurement
Uncertainty: When Your Result Isn’t a Number — But an Interval That Tells
the Truth About Your Measurement System
When you last reported a measurement result, how did you write it?
“The average is 12.47 mm.” How would you react if someone
asked you: “And how certain are you about that number?” Most
people in quality would shrug and say: “Well, we measured it three
times and it came out the same.”
The First Day a Number Lied
to Me
I was at an audit at a supplier of precision components for the
automotive industry. Everything looked perfect — control plans filled
out, SPC charts on the wall, calibration certificates in place. Then I
asked the operator on the line: “Can you tell me what the uncertainty
of this measurement is?”
He stopped. He looked at me as if I had asked about the meaning of
life. “Uncertainty? I measure what’s on the drawing. The
instrument is calibrated.”
I picked up his drawing. The tolerance on a critical dimension was
±0.02 mm. His measuring instrument had a calibration certificate with
a resolution of 0.001 mm. Everything looked fine — until I asked the
next question: “And what is the uncertainty of your measurement
system including the operator, environment, and method?”
Silence. He didn’t know the answer. And when we figured it out
together, it turned out that the uncertainty of his measurement system
was ±0.015 mm. Against a tolerance of ±0.02 mm. Suddenly his “we
meet the specification” didn’t sound so confident anymore.
That was the day I understood that the most dangerous thing in
quality is not poor inspection. It is false certainty.
What Is Measurement
Uncertainty, Really?
Measurement uncertainty is not error. That is the first and most
important distinction you need to make.
Measurement error is the difference between the
measured value and the true value. The problem is, you never know the
true value — so you never know the error. It is a paradox that has
troubled metrologists for centuries.
Measurement uncertainty is a parameter associated
with the result of a measurement that characterizes the dispersion of
values that could reasonably be attributed to the measured quantity. Put
simply: it is the interval within which the true value lies with a
certain probability.
The GUM (Guide to the Expression of Uncertainty in Measurement)
defines two types of evaluation:
- Type A — statistical analysis of a series of
observations. You repeat the measurement, calculate the standard
deviation, and derive the uncertainty from it. - Type B — everything else. Experience, calibration
certificates, manufacturer specifications, knowledge of material
behavior.
The real uncertainty is a combination of both. And ignoring either
one means lying to yourself.
Why
Should You Care? Because Your Decisions Are Built on Sand
Imagine the following scenario. You manufacture a shaft with a
diameter of 25.000 mm and a tolerance of ±0.010 mm. You measure
25.008 mm. Within specification. You ship it to the customer.
But your measurement uncertainty is ±0.006 mm. That means the true
value could be anywhere from 25.002 to 25.014 mm. The upper limit is
25.010 mm. And you may have just shipped a part that is out of
specification.
This is not a theoretical problem. This happens
every day in factories around the world. Decisions to accept or reject
parts are made based on numbers that are not as certain as we think.
In the automotive industry, this problem is addressed through the
concept of guard banding — you create a safety margin
around the specification limits. If the uncertainty is U, your
acceptance zone narrows by 2U. Suddenly, parts you previously accepted
are now on the borderline. And parts you rejected may actually be
fine.
A Real Story:
When Uncertainty Saved a Million
One of my clients manufactured precision pistons for hydraulic
cylinders. Their customer was reporting complaints — pistons too tight
or too loose. All control plans were in order. All measurements met
specification.
After two weeks of intensive investigation, we found the problem.
Their measurement system had an uncertainty of ±0.004 mm. The piston
tolerance was ±0.005 mm. The Test Uncertainty Ratio (TUR) was 1.25:1.
According to AIAG recommendations, it should be at least 4:1, ideally
10:1.
Their measurement system simply could not distinguish whether a part
was good or bad. Quality decisions were essentially guesswork.
The solution? Investment in a more precise measuring device with an
uncertainty of ±0.0005 mm. The TUR improved to 10:1. Complaints dropped
to zero. The return on investment was 14 months.
A million euros in complaints saved because we learned to ask:
“And what is the uncertainty?”
How
to Calculate Measurement Uncertainty — Without Getting Lost in the
Math
The process of calculating uncertainty according to the GUM is
systematic and understandable if you break it down into steps:
Step 1: Define the Measurand
What exactly are you measuring? What is the mathematical model? For
example, when measuring the diameter of a cylinder: D = M + δcal +
δtemp + δmethod, where D is the diameter, M is the measured value,
δcal is the calibration correction, δtemp is the temperature effect,
δmethod is the method effect.
Step 2: Identify
All Sources of Uncertainty
This is where most people stumble. Uncertainty is not just about
instrument repeatability. Sources include:
- Repeatability — same operator, same instrument,
same conditions - Reproducibility — different operators, different
instruments - Calibration — uncertainty of the calibration
certificate - Resolution — the smallest step of the
instrument - Temperature — effect of thermal expansion
- Pressure — effect on measurement points
- Method — how the measurement is performed
- Sample — representativeness of the measured
piece - Environment — humidity, vibrations,
electromagnetic interference
Step 3: Quantify Each
Contribution
For each source, calculate the standard uncertainty u_i. For Type A,
you use the standard deviation of repeated measurements. For Type B, you
use information from certificates, specifications, and expert
estimates.
Step 4: Calculate
Combined Uncertainty
You use the law of propagation of uncertainty: u_c = √(Σ u_i²). Yes,
it is essentially the Pythagorean theorem — you are summing variances,
not uncertainties directly.
Step 5: Expanded Uncertainty
U = k × u_c, where k is the coverage factor. For a normal
distribution and 95% confidence level, k = 2.
Your result then looks like this: 25.008 ± 0.006 mm (k = 2, 95%).
And that is a number you can trust.
Uncertainty Budget
— Where Your Problem Hides
One of the most useful tools is the uncertainty
budget — a table that shows each uncertainty contribution and
its share of the total uncertainty.
| Source | Standard Uncertainty (mm) | Share (%) |
|---|---|---|
| Repeatability | 0.0020 | 25 |
| Reproducibility | 0.0018 | 20 |
| Calibration | 0.0025 | 39 |
| Temperature | 0.0010 | 6 |
| Resolution | 0.0003 | 1 |
| Method | 0.0015 | 9 |
When you see this breakdown, you immediately know where the greatest
room for improvement is. In this case, calibration accounts for 39% of
the total uncertainty. Improve calibration by half and your overall
uncertainty drops significantly.
Without an uncertainty budget, you’re guessing. With one, you’re
targeting.
Uncertainty and
SPC — Two Worlds That Must Meet
SPC (Statistical Process Control) and measurement uncertainty are
related but often live in separate worlds. SPC assumes that measurement
is accurate. Measurement uncertainty says: “No, it isn’t.”
When you implement SPC without knowledge of measurement uncertainty,
you can make two types of errors:
- False alarm — the control chart signals outside
control limits, but in reality it is measurement uncertainty, not a
process change. - Undetected change — the process has changed, but
the measurement uncertainty is so large that you cannot detect the
change.
This is precisely why Gage R&R (Repeatability &
Reproducibility) is only the beginning. Gage R&R tells you what
portion of total variation is caused by the measurement system. But
measurement uncertainty tells you the quantifiable limit to which you
can trust your numbers.
ISO 17025 — Why It Is
the Gold Standard
The ISO 17025 standard (currently the 2017 version) is the only
standard that explicitly requires the evaluation of measurement
uncertainty. Every accredited calibration laboratory must state the
uncertainty on its certificates.
Why is this important for you? Because your calibration certificate
without uncertainty is like a train ticket without a destination. You
know you were calibrated. But we don’t know with what certainty.
If your calibration service provider is not accredited to ISO 17025,
their certificates may have zero value in terms of uncertainty. And that
means your entire chain of measurement confidence is broken.
Practical Steps — How to
Start Tomorrow
You don’t need to be a metrology expert to start taking measurement
uncertainty seriously. Here is a plan for the first three months:
Month 1: Inventory. List all measurement systems,
their resolution, calibration certificates, and TUR against critical
tolerances. Identify where TUR is < 4:1.
Month 2: Uncertainty budget for critical measurements.
Select the 5 most critical dimensions and create an uncertainty budget.
You will see where the largest contributions are.
Month 3: Action plan. Based on the budgets, propose
improvements — better calibration, more stable environment, better
instrument, better method.
Three months. And suddenly you know whether your numbers are telling
the truth.
The
Philosophy of Uncertainty — Why It Is More Than Mathematics
Measurement uncertainty is not just a technical concept. It is a
philosophy. It is the acknowledgment that our knowledge of the world is
always limited. That every number has its uncertainty. And that the only
thing we can do is quantify that uncertainty and work with it.
In quality, this means shifting from the mindset of “I measured it,
so I know” to “I measured it, and I know with what certainty I know
it.” That difference is between a factory that relies on luck and a
factory that relies on science.
Conclusion: Your
Number Is Only Half the Story
Next time you write down a measurement result, don’t forget the
other half. Uncertainty. That little “±” symbol is not a formality. It
is a promise — a promise that you know the boundary of your
knowledge.
In the automotive industry, in pharmaceuticals, in aerospace — in
every field where quality decides lives — measurement uncertainty is
something you cannot afford to ignore.
Because when your customer asks “are you sure?”, your answer will no
longer be “yes.” Your answer will be: “With an uncertainty of ±0.003 mm
at k = 2. And here is the budget that proves it.”
And that is an answer that buys trust.
Peter Stasko is a Quality Architect with 25+ years of experience
in automotive, manufacturing, and continuous improvement. He helps
companies turn data into decisions and uncertainty into competitive
advantage.
