Calibration is often performed without technical justification regarding the attributes that get calibrated and the resulting calibration uncertainty. Worse yet, calibrations are sometimes based solely on whatever auditors demand. This approach leaves companies with the cost of excessive calibration in some areas, while leaving them exposed in other areas where they "undercalibrate". The paper presents an uncertainty budget driven technique, which determines what attributes of each standard and instrument should be calibrated and to what uncertainty. The technique provides clear, data based documentation of the real calibration needs to management and auditors. The underlying concept is that the purpose of calibration is to limit the maximum error of the metrological characteristics of calibrated standards or instruments, that enters into the uncertainty budgets of subsequent measurements.
This paper is written assuming the reader is familiar with the ISO "Guide to the Expression of Uncertainty in Measurement (GUM) (1), which is the internationally recognized way of expressing measuring uncertainty. It also treats the determination of the individual uncertainty contributors and their combination without much detail. For more details on the concepts used in that respect, see ISO 14253-2(2), currently in draft stage from ISO Technical Committee 213.
The paper concentrates on exploring the conclusions that can be drawn from an uncertainty budget regarding the "Metrological Confirmation System", or calibration system used to maintain the capabilities assumed in various parts of uncertainty budgets. This system may have global elements controlled by NIST or equivalent agencies in other countries as well as elements maintained within an individual company.
The concepts presented in the paper can therefore be used both at a tactical company level to determine what capabilities are important and of value to the company as well as at a strategic National or International level determining what capabilities are important to us as a nation or as a global society.
The first example is a very simple one, regarding ordinary workshop micrometers 0-25 mm, as they are used in every machine shop all over the world.
If we look at the ISO standard for micrometers, ISO 3611 (3), we see a number of requirements to these micrometers (national standards of various countries have essentially the same requirements as the ISO standards). If we combine these requirements with the environmental conditions in a normal workshop, airconditioned for creature comfort, if at all, we get the following contributors to the uncertainty budget for the measurements performed with the micrometers:
| Contributor | Variation Limit | Equivalent influence at 1 s level |
| Scale error of the micrometer | 3 µm | 1.73 µm |
| Zeropoint error | 2 µm | 1.15 µm |
| Average temperature | +/- 5 oC | 0.09 µm |
| Temperature difference | 3 oC | 0.61 µm |
| Parallelism of anvils | 2 µm | 0.58 µm |
| Repeatability | 6 = 2 µm | 0.33 µm |
| Combined Standard Uncertainty, u | 2.27 µm | |
| Expanded Uncertainty, U | 5.54 µm | |
Table 1: Uncertainty budget for measurement using 0-25 mm micrometer in a workshop environment. Assumptions: Micrometer calibrated according to ISO 3611, workshop temperature 20 oC +/- 5 oC, maximum difference in temperature between micrometer and workpiece 3oC.
If we look closer at the contributions, we can derive several very useful pieces of information, that can help us decide where to invest (or not to invest) to improve the uncertainty of our measurement.
The first observation is that temperature is not the main contributor. This tells us that if we had planned to move the measurement to a temperature controlled environment, we would have been wasting our money, as there are much larger contributors to the uncertainty than the temperature influence. Developing an uncertainty budget up front just saved us the cost of a temperature controlled enclosure for our micrometer measurements!
The next observation is that the major contributor is the scale error of the micrometer. Assuming that we calibrate our micrometers in-house, that leads us to focus on the procedure we use to calibrate the scale of the micrometers.
The normal procedure for this calibration is to use a special set of gage blocks, that are dimensioned to check different points along the scale, as well as pitch errors within each revolution of the spindle of the micrometer. Let us assume that we are planning our capability to calibrate the spindle error of our own micrometers in-house.
Our initial assumption is that we need a good set of steel gage blocks,
ISO 3650 grade 0, as well as a temperature controlled environment running at 20 oC
+/- 1 oC in order to accomplish this task. To verify these assumptions, we look
at the uncertainty budget for the calibration of the scale error:
Contributor |
Variation Limit | Equivalent influence at 1 s level |
| Gage Block | 0.14 µm | 0.08 µm |
| Digital Step/Repeatability | 1 µm | 0.29 µm |
| Average Temperature | 1 oC | 0.017 µm |
| Temperature Difference | 0.5 oC | 0.1 µm |
| Combined Standard Uncertainty, u | 0.31 µm | |
| Expanded Uncertainty, U | 0.62 µm | |
Table 2: Uncertainty budget for the calibration of the scale error of a 0-25 mm micrometer using grade 0 gage blocks in a laboratory environment. Assumptions: Laboratory temperature: 20 oC +/- 1 oC, maximum difference in temperature between micrometer and gage block: 0.5 oC.
We find from this initial analysis, that the major contributor to the uncertainty in the calibration is the resolution of the micrometers themselves. That means we can redesign our measurement process for calibrating the scale error of our micrometers.
First of all we can use less expensive grade 1 gage blocks. We can also extend the calibration interval for the gage blocks, since the tolerance for grade 1 blocks is more than double that of grade 0 blocks. Additionally it is also generally less costly to have grade 1 blocks calibrated than grade 0, since the tolerances are larger, so each calibration is cheaper.
Secondly, we see that the average temperature the calibration is
performed at has negligible influence, as long as we are using steel gage blocks to
calibrate a steel gage. The temperature difference between the gage block and the
micrometer is the second largest contributor, but even if it is doubled, u will only
change by 13% and still be small compared to the tolerance for the micrometer scale error.
Contributor |
Variation Limit |
Equivalent influence at 1 s level |
| Gage Block | 0.3 µm | 0.17 µm |
| Digital Step/Repeatability | 1 µm | 0.29 µm |
| Average Temperature | 5 oC | 0.09 µm |
| Temperature Difference | 1 oC | 0.2 µm |
| Combined Standard Uncertainty, u | 0.4 µm | |
| Expanded Uncertainty, U | 0.8 µm | |
Table 3: Uncertainty budget for the calibration of the scale error of a 0-25 mm micrometer using grade 1 gage blocks in a laboratory environment. Assumptions: Laboratory temperature: 20 oC +/- 5 oC, maximum difference in temperature between micrometer and gage block: 1 oC.
The example shows us that we can use uncertainty budgeting, not only to evaluate the actual measurement and the environment we use to perform it in, we can also use it to determine the calibration requirements for our gages, both in terms of the standards and the environment necessary.
We found that by changing our measurement procedure in areas, where the influence on the uncertainty is small, we can achieve dramatic savings on our measurement cost.
Air gages are often used to measure tight tolerances on the order of a few microns. They are measuring either the variation in flow or pressure as a function of the difference in diameter between the air gage plug or ring and the corresponding hole or shaft being measured.
Since the air gage is a relative gage, it relies on a pair of master
rings or pins to set its upper and lower tolerance limit (or zero point and gain). If we
look at the uncertainty budget for the measurement of a nominal ø 10 mm hole in a steel
workpiece using a steel air gage plug, we get the following uncertainty budget:
Contributor |
Variation Limit |
Equivalent influence at 1 s level |
| Setting ring certificate: | 1 µm | 0.58 µm |
| Average temperature: | +/- 1oC | 0.007 µm |
| Temperature difference: | 0.5 oC | 0.04 µm |
| Reproducibility (drift): | 0.5 µm | 0.29 µm |
| Combined Standard Uncertainty, u | 0.65 µm | |
| Expanded Uncertainty, U | 1.30 µm | |
Table 4: Uncertainty budget for the measurement of the diameter of a 10 mm hole using an air gage calibrated with a gage ring in a laboratory environment. Assumptions: Laboratory temperature: 20 oC +/- 1 oC, maximum difference in temperature between air gage and work piece: 0.5 oC.
The uncertainty budget is based on using 2 rings, each calibrated to 1 µm, for setting upper and lower tolerance limit, an average temperature of 20 oC +/- 1oC during the measurement and a temperature difference between the air gage and the workpiece of less than 0.5 oC. These conditions are used both for the calibration of the air gage and the subsequent measurements.
If we assume that our target (desired) uncertainty is 1 µm, then we will have to improve our measurement process to achieve that. The uncertainty budget shows, that the main contributor is the uncertainty of the setting rings. Therefore nothing can be gained by shortening up the calibration intervals, as that will only improve the reproducibility and drift contributor.
Similarly the temperature conditions of the environment are so good relative to the uncertainty of the setting rings, that the conditions can be relaxed by a factor of 5 without affecting the uncertainty.
Making an uncertainty budget has shown us that the controlling factor for this measurement is the uncertainty of the setting rings. This provides us with the knowledge necessary to aggressively seek to improve in this area. Unfortunately, we find that very few commercial calibration laboratories are able to calibrate rings much better than 1 µm. The best are performing on the order of 0.5 µm. If we use such a lab to provide us our traceability for this measurement, we can amend our uncertainty budget to the following:
Contributor |
Variation Limit |
Equivalent influence at 1 s level |
| Setting ring certificate: | 0.5 µm | 0.29 µm |
| Average temperature: | +/- 1 oC | 0.007 µm |
| Temperature difference: | 0.5 oC | 0.04 µm |
| Reproducibility (drift): | 0.5 µm | 0.29 µm |
| Combined Standard Uncertainty, u | 0.41 µm | |
| Expanded Uncertainty, U | 0.82 µm | |
Table 5: Uncertainty budget for the measurement of the diameter of a 10 mm hole using an air gage calibrated with a gage ring in a laboratory environment. Assumptions: Laboratory temperature: 20 oC +/- 1 oC, maximum difference in temperature between air gage and work piece: 0.5 oC.
At this point, the uncertainty originating from the reproducibility and drift is on the same order of magnitude as that originating from the setting ring uncertainty. Therefore it is now possible to further improve the uncertainty by shortening the interval between calibrations of the air gage.
Using uncertainty budgeting in this example has shown us the right sequence of improvements to our measuring process to achieve maximum results. Without it we might have changed calibration intervals or tried to improve the thermal environment before working on the uncertainty of our setting rings. Again uncertainty budgeting has provided us clear guidance on how to improve our measurement process.
This paper has discussed two very simple examples of how uncertainty budgeting can be used as a tool to improve measurement processes.
The first example showed how we can use the uncertainty budget to reduce the cost of measurements, while suffering only a minimal increase in uncertainty, by targeting the areas of high cost and low influence on the uncertainty for a relaxation of requirements.
The second example showed how we can use the uncertainty budget to identify which area of a measurement process we need to improve, if we want to improve the overall uncertainty of the measurement process.
1 Guide to the Expression of Uncertainty in Measurement
2 ISO/DTR 14253-2:1997 "Geometrical Product Specifications (GPS) - Inspection by measurement of workpieces and measuring instruments - Part 2: Guide to the estimation of uncertainty in GPS measurement, in calibration of measuring instruments and in product verification.
3 ISO 3611:1978 Micrometer callipers for external measurement
4 ISO 3650:1978 Gauge blocks