Measurement Uncertainty Explained for Quality Managers

Every measurement is an estimate. When a calibration certificate says a pressure gauge has an error of +0.03 bar, the true error is not exactly 0.03 bar. It lies somewhere close to it. Measurement uncertainty states how close: the range within which the true value is expected to lie, at a stated level of confidence.

Measurement uncertainty is a quantified parameter that characterizes the spread of values that can reasonably be attributed to a measurement result. In practice, it defines the range around the result within which the true value is expected to lie, at a stated level of confidence (typically 95 % for k = 2).

A result without uncertainty is incomplete. “Error +0.03 bar ± 0.04 bar (k = 2)” tells you something you can act on, so you can expect the error lies in the range between -0.01 bar to 0.07bar.

“Error +0.03 bar” on its own does not.

Error and uncertainty are not the same thing

Error Uncertainty

What it is | The difference between the instrument | A quantified range around the result within which the true | reading and the reference value | value is expected to lie, at a stated level of confidence

Can it be corrected? | Yes, by applying a correction or | No. It can be reduced through better standards, methods or | adjusting the instrument | conditions, but never eliminated

Sign Positive or negative Always stated as ±

Where it appears | Results table on the certificate | The uncertainty results on the certificate, with its coverage | factor k

A workplace way to see it: think of two operators weighing the same 500 g standard weight ten times.

• Operator A gets 500.20 g every time. The error is large (+0.20 g), but the results are tight. Once you know the offset, you can correct it. That is large error, small uncertainty.
• Operator B gets results scattered between 499.80 g and 500.25 g. The average is close to 500 g, but no single reading can be trusted. That is small error, large uncertainty.

Operator A’s problem is solved with a correction. Operator B’s problem needs a better instrument, a better environment or a better method.

Where uncertainty comes from

Every calibration carries contributions from several sources. The main ones are Reference standard, Resolution, Repeatability, Environment, Instrument behavior for example Hysteresis in pressure gauges, eccentricity on balances, non-uniformity in ovens and baths, Drift of the standard, Method and operator for example Reading parallax on analogue dials, or reaction time.

A good uncertainty budget lists every significant source, even the ones that turn out to be small. Leaving a source out is the most common mistake auditors find.

How uncertainty is calculated, step by step

The internationally accepted method is the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100). Accredited laboratories follow it, with ILAC P14 setting how uncertainty is stated on calibration certificates.

Step 1: Define the measurement. Write down what is being measured and the equation behind it. For example: error = gauge reading − reference pressure.

Step 2: List every source. Use the uncertainty budget table as a starting checklist.

Step 3: Evaluate each source as a standard uncertainty. There are two types:

• Type A (statistical). Take repeated readings and calculate the standard deviation s. If the reported result is the mean of n readings, its standard uncertainty is s / √n.
• Type B (other information). Use certificates, specifications, resolution, handbook data or experience. Convert each to a standard uncertainty by dividing by the factor for its distribution: Divide the half-width (or Distribution When to use it (Examples) U) by

Normal A value from a calibration certificate stated with k k (usually 2)

Rectangular Resolution, or a limit where any value inside is equally likely √3 ≈ 1.73

Triangular Values are more likely near the center than the limits √6 ≈ 2.45

U-shaped Quantities that cycle between limits, such as temperature controlled √2 ≈ 1.41

Divide the half-width (or
Distribution | When to use it (Examples) | U) by

by a thermostat

Step 4: Apply sensitivity coefficients. Some sources are in different units from the result, so each is multiplied by a sensitivity coefficient cᵢ, which converts it into the unit of the result. Example: a 100 mm steel gauge block expands by about 11.5 µm per meter per °C. A ±1 °C temperature uncertainty therefore contributes about ±1.15 µm to the length.

Step 5: Combined Uncertainty. For independent sources, the combined standard uncertainty is the root-sum-of- squares: uc = √( (c₁·u₁)² + (c₂·u₂)² + … )

Step 6: Expanded Uncertainty Multiply by a coverage factor k to give the expanded uncertainty U = k · uc. With k = 2, the interval covers about 95 % of the likely values. If the Type A part is based on only a few readings, k may need to be larger, calculated from the effective degrees of freedom.

Step 7: Report Expanded Uncertainty State U, k and the confidence level. Round U to no more than two significant digits, and round the result to the applicable decimal place.

Worked example: a 0–10 bar pressure gauge checked at 6 bar

Reported result: error at 6 bar = +0.03 bar, U = 0.04 bar (k = 2, approximately 95 % confidence).

What the budget tells you: the reference standard is the smallest contribution. The gauge’s own resolution and repeatability dominate. This is typical, and it explains why a more expensive reference would hardly change the result.

CMC and the uncertainty on your certificate

The accredited scope of a calibration laboratory lists its calibration and measurement capability (CMC). This is the smallest uncertainty the laboratory can achieve, using its best standards on a nearly ideal instrument.

The uncertainty on your certificate also includes your instrument’s resolution, repeatability and behaviours. It will therefore normally be larger than the CMC, and under ILAC P14 it can never be smaller. When comparing laboratories, the CMC shows what each laboratory is capable of. The certificate shows what was achieved on your instrument.

How to use uncertainty in daily decisions

1. Check fitness for use. Add the absolute error and the uncertainty, then compare it with your tolerance: |error| + U ≤ tolerance → the instrument is fit for use with confidence

In the example: 0.03 + 0.04 = 0.07 bar , 0.03 – 0.04 = -0.1 bar. If the tolerance process is 0.1 bar, the gauge is fit. If it is ±0.05 bar, it is not, even though the error alone (0.03) looks acceptable.

2. Apply a decision rule near the limits. When a result sits close to the tolerance limit, the uncertainty decides whether conformity can be claimed. Agree with the decision rule with your calibration laboratory before the work. Decision rules and statements of conformity →

3. Use your own uncertainty when you measure. The uncertainty on a calibration certificate is one input to the uncertainty of every measurement you then make with that instrument. ISO/IEC 17025 requires testing and calibration laboratories to evaluate the uncertainty of their own results (clause 7.6).

Common mistakes

• Reporting a result with no uncertainty, or with no K value.
• Using the CMC from the laboratory’s scope as if it were the uncertainty of your instrument.
• Forgetting significant contributions such as resolution, hysteresis or temperature in the budget.
• Dividing a certificate’s expanded uncertainty by √3 instead of by k.
• Treating uncertainty as a “tolerance” the instrument is allowed to have.