Measurement uncertainty is a critical calculation in the calibration process. Many calibration laboratories must report it to customers to maintain compliance with quality standards such as ISO 17025. But more importantly, without an accurate, reliable sense of a measurement tool’s uncertainties, users risk signing off on systems that might be unsafe to operate — something that can quickly become dangerous, especially in industrial operations, like oil refineries.
That said, calculating measurement uncertainty — known as uncertainty analysis — can be complicated. Many variables and factors at play impact measurement uncertainty: environmental conditions like local gravity and humidity levels, limitations of the specific measurement instrument, and human error, for example.
In this guide, we’ll walk through those complications in more detail and provide a helpful step-by-step guide to help you better calculate measurement uncertainty.
What Is Measurement Uncertainty?
Measurement uncertainty is a range of values that indicates doubt about the reading of whatever physical quantity a tool is measuring; in other words, it’s a quantified expression of the accuracy of the reading. The true, accurate value of the physical quantity lies within the range of uncertainty.
For example, a digital multimeter that can measure DC voltage might have a listed measurement uncertainty of ±0.2%. In this case, an accurate DC voltage reading would lie within plus or minus 0.2% of whatever reading appears on the screen. If the multimeter displayed a reading of 5.000 V, the true voltage would be anywhere between 5.010 V-4.990 V.
Measurement Uncertainty Versus Measurement Error
Note that uncertainty is not the same as error. Uncertainty expresses the range within which the true measurement value falls. However, error indicates the amount by which the reading differs from the true value.
Key Factors Affecting Measurement Uncertainty
As we mentioned earlier, there are many different variables and factors to consider during uncertainty analysis. This isn’t an exhaustive list, but these specific considerations are most likely to cause incorrect calculations:
- Environment: The physical laboratory environment you’re in can impact measurement results. Take into account factors such as pressure, relative humidity, dirt and dust, temperature, local gravity due to elevation — even other machinery or devices in the room that might produce electrical noise that could alter readings. Factors such as these can impact the ability of the device you’re measuring, also called a device under test (DUT), and of your calibration equipment to take accurate readings.
- Equipment: The equipment involved in your calculation can also have an effect on results. Both your calibration equipment and the DUT might be old and worn out, or they could both could suffer from bias errors. Pieces of test or measurement equipment might also have a limited resolution: a factor that can be included in uncertainty analysis.
- Operator: Human error is another variable to consider. Eventually, you or another metrologist or engineer on your team will make errors when performing calibrations and tests; those errors will contribute to measurement uncertainty. One’s previous training, skills, and judgement can all impact the final results. Even physical attributes, such as an operator’s eyesight and reaction times, may play a role.
- Reference Standards: As we mentioned above, the equipment involved in calculating measurement uncertainty should be part of your uncertainty analysis; that equipment includes your reference standards, i.e., your calibration devices and systems. Ensure calibration devices have good traceability by checking their calibration certificates to see when they were last calibrated. Also, ensure other elements of the calibration system are in good shape. For instance, check for leaks in any pressure systems and confirm any solvents or other media are free of contaminants.
How to Calculate Measurement Uncertainty in a Calibration Lab
1. Identify and Eliminate External Sources of Uncertainty
Use the uncertainty analysis considerations we discussed above as a checklist to help you identify and eliminate any external sources of uncertainty:
- Environmental factors (local gravity, temperature, signal noise, etc.)
- Equipment limitations (age, resolution, etc.)
- Operator limitations (human error, reaction times, etc.)
- Reference standards (traceability, system leaks, media or solvent contamination, etc.)
Eliminating these extraneous sources of uncertainty will help ensure the data you use for calculation is as clean as possible.
2. Estimate Standard Uncertainties
Once you’ve eliminated as many external sources of uncertainty as possible, you’ll need to understand what other uncertainties remain and if you should categorize them as Type A uncertainties or Type B uncertainties — also known as standard uncertainties.
Type A uncertainties are those that 1) you can tie back to a quantitative measurement and, most importantly, 2) result from aseries of observations. In other words, you need to collect multiple data points per uncertainty in order to classify it as a Type A uncertainty. Analyses like the average, standard deviation, and degrees of freedom will help you evaluate any Type A standard uncertainties.
Type B uncertainties are any data points that aren’t Type A. In other words, Type B uncertainties are data points that you can’t collect through a repeated series of observations. Information gathered from sources such as calibration certificates, manufacturer’s specifications, scientific journals and interlaboratory comparison reports (like the BIPM’s KCDB) are all Type B uncertainty data.
3. Determine the Probability Distribution of Standard Uncertainties
Once you’ve categorized uncertainties as either Type A or Type B, choose an appropriate distribution to plot those data points on. In most cases, one of two options will work: a normal distribution (i.e., a Gaussian distribution or a bell curve) or a rectangular distribution (i.e., a uniform distribution).
For Type A uncertainties, a normal distribution will likely be appropriate; for Type B uncertainties, a rectangular distribution should suffice.
4. Convert Standard Uncertainties to Standard Deviations
With the Type A and Type B uncertainty data plotted onto an appropriate probability distribution, you’ll be able to convert the data for each uncertainty to a standard deviation (a requirement for the next step of our calculation: combining uncertainties).
In order to make this conversion to standard deviation, you must use the correct divisor for the data’s distribution:
- For a normal distribution, use the coverage factor, or k value, as the divisor.
Tip for finding the k value: If the device has an ISO/IEC 17025 accredited calibration report, you can find the k value in the report. You can also determine the k value by using the confidence levels reported in the manufacturer’s specifications sheet. The ISO Guide to the Expression of Uncertainty of Measurement (GUM)provides a helpful conversion table for common confidence levels and details for calculating k manually in Annex G, section 1.
- For a rectangular distribution, your divisor will be the square root of 3 (1.7321).
Note that you’ll need to do this conversion for each source of uncertainty you’ve collected. If, for example, you’ve identified four different Type A uncertainties and seven Type B uncertainties, you should have a matching standard deviation for each source — so 11 in total in this case.
5. Combine Uncertainties
With the standard deviation calculated for each source of uncertainty, you can then combine them using the root sum of squares (RSS) method:
- Square the standard deviation of each source of uncertainty.
- Add each squared standard deviation together.
- Find the square root of the sum.
The square root of the result represents the total combined standard uncertainty.
6. Calculate Expanded Uncertainty
Finally, if you want your uncertainty to fall within a specific confidence level, you’ll need to calculate the expanded uncertainty.
To do this, you’ll again need a k value, or coverage factor; the k value for this step should correspond to your preferred confidence level. For most calibration purposes, a confidence level of 95% (k=2) should suffice.
Once you’ve determined the k value that corresponds to your preferred confidence level, simply multiply the combined uncertainty you found in step five by the k value.
How to Report Uncertainty Results According to Industry Standards and Customer Requirements
Reporting ISO 17025 uncertainty gives customers a clear picture of a laboratory’s calibration accuracy. This aligns with the ISO/IEC 17025:2017’s standard of providing clear, accurate, and objective results with all the information needed to interpret them.
The ISO/IEC 17025 standard states that accredited testing laboratories may need to report uncertainty results in several situations. For instance, when it’s required by the client and/or when measurement uncertainty is needed to properly interpret the results. ISO 17025 calibration laboratories, on the other hand, always need to include measurement uncertainty on calibration certificates.
When reporting measurement uncertainty, both testing and calibration laboratories need to present it in the same unit as the measurement or in a term relative to the measurement (this could be a percentage).
Recap: Calculating Measurement Uncertainty
By now, you should have a clearer, general idea of how to determine the measurement uncertainty of your devices. However, there are special considerations for specific types of measurement devices. To learn more about calculating measurement uncertainty in more specific situations, check out the below resources.