On every calibration certificate for a weight, next to the measured error there is a second figure: the measurement uncertainty. Many users overlook it and read only the conclusion, within or outside tolerance. Yet it is precisely that uncertainty that determines how much use the result is to you, and how well you can check your own scales with it.
In this article we explain what measurement uncertainty is, where the figures on the certificate come from and how to use them in practice. How a calibration of weights is carried out is explained on our page about weight calibration.
What is measurement uncertainty?
No measurement is perfect. Even in a climate-controlled laboratory with an accurate mass comparator, there remains a margin within which the true value lies. That margin is called measurement uncertainty. It is not an error and not carelessness, but an honest statement about how good the measurement is.
In the calibration of weights, the following sources contribute, among others:
- the uncertainty of the reference weight itself, as stated on its own certificate;
- the repeatability and readability of the mass comparator;
- the correction for air buoyancy, including the uncertainty in air density and volume;
- the positioning of the weight on the comparator and any magnetic effects;
- the possible change in the reference weight since its last calibration.
The laboratory combines all these contributions in an uncertainty budget. Read more about the influence of the air in conventional mass and air buoyancy.
Expanded uncertainty and coverage factor k=2
The certificate usually states the expanded uncertainty, denoted U. It is calculated by multiplying the combined standard uncertainty by a coverage factor k. Almost always this is k=2. That means the true value lies within the stated interval with a probability of approximately 95%.
An example: if the certificate states an error of +2.0 mg with U = 1.2 mg (k=2), the true error lies between +0.8 mg and +3.2 mg with approximately 95% confidence. If you want to use the uncertainty in your own calculation, you often need the standard uncertainty. You find it by dividing U by k, in this example 0.6 mg.
Why the uncertainty is at most one third of the mpe
OIML R111 sets a limit on the measurement uncertainty in the calibration of weights. The expanded uncertainty may be no more than one third of the maximum permissible error (mpe) of the class. This requirement ensures that an assessment as within or outside the class actually means something.
| Weight | Class | mpe (±) | Maximum U |
|---|---|---|---|
| 100 g | F2 | 1.6 mg | 0.53 mg |
| 1 kg | F1 | 5 mg | 1.67 mg |
| 1 kg | M1 | 50 mg | 16.7 mg |
| 20 kg | M1 | 1,000 mg | 333 mg |
| 1,000 kg | M3 | 500 g | 167 g |
For the fine classes, this means the laboratory needs a reference of a higher class, a suitable mass comparator and stable conditions. A laboratory that cannot achieve that uncertainty cannot assess a weight in that class, even if the error appears small.
How to read the in-tolerance assessment
This is where things often go wrong in practice. An error that is smaller than the mpe does not automatically mean that the weight conforms to the class. OIML R111 requires that the error plus the expanded uncertainty remains within the mpe. Three examples for a 1 kg F1 weight, mpe ±5 mg, with U = 1.2 mg:
| Measured error | Error (without sign) plus U | Assessment |
|---|---|---|
| +2.0 mg | 3.2 mg | Conforms to F1 |
| −4.1 mg | 5.3 mg | Does not meet the R111 condition, although the error itself is smaller than 5 mg |
| +5.6 mg | 6.8 mg | Outside F1 |
The middle case is the most interesting. The weight is probably still usable, but there is a real chance that the true error is greater than 5 mg. Check on the certificate which decision rule was used, so that you know how the conclusion was reached. A general explanation of the structure of a certificate can be found in the calibration certificate explained.
Working with correction values
Most users work with the nominal value: to them, a 1 kg weight is 1 kg. That is acceptable as long as the weight falls within its class. You can also work with the measured value. The conventional mass is then the nominal value plus the error. With an error of +3.2 mg, that is 1,000.0032 g. The correction value is the opposite of the error, here −3.2 mg.
Working with the measured value reduces the uncertainty but requires discipline. The value applies at the moment of calibration. If you use it until the next calibration, include an additional uncertainty for possible change. The history of previous certificates helps you estimate this. After an adjustment you start again; see adjusting weights.
Effect on your own scale calibrations
The uncertainty of your weights carries through into every calibration of a scale you perform with them. EURAMET cg-18 describes two ways of accounting for this:
- With the nominal value. The standard uncertainty of the weight is then the mpe divided by the square root of 3. For a 20 kg M1 weight, that is 1,000 mg / 1.73, approximately 0.58 g.
- With the conventional mass from the certificate. The standard uncertainty is then U/2, supplemented by a contribution for possible change since the calibration. With U = 300 mg you start at 0.15 g.
Whether that difference matters depends on the scale. For a floor scale with a readability of 20 g, 0.58 g is small. For a precision scale with a readability of 1 g, the weight can become the main source of uncertainty. In that case, choose a finer class or work with the measured values. How the uncertainty budget of a scale is built up is explained in EURAMET cg-18 explained.
How MASSLAB helps you
MASSLAB calibrates weights in classes F1 to M3 with reference weights of a higher class. Our own R111 LIMS software calculates a complete uncertainty budget for every weight, and the certificate states the error, the expanded uncertainty with coverage factor and the assessment within the class. Our calibrations are traceable to national standards and we are a Recognised Verifier (Erkend Keurder) under the Dutch Metrology Act. Calibrations are performed according to internally established procedures based on the relevant requirements of ISO/IEC 17025. Where applicable, measurement results are stated with measurement uncertainty and supported by traceable measurement references. If you need an accredited certificate, we can arrange this on request through an accredited partner. All certificates are available in the customer portal. Read more about our working method at weight calibration.