Calibration uncertainty can sometimes feel complicated. There are formulas, distributions, coverage factors, sensitivity coefficients, and many other terms that can make uncertainty seem like something mainly for mathematicians.

But uncertainty should also be understood by the people who actually perform calibrations. After all, knowing the calibration result is not enough. We also need to know how much we can trust that result.

A few years ago, we published a more generic article explaining calibration uncertainty and some of the theory behind it. In this article, I want to take a more practical approach. The idea is to go through a real-world calibration example step by step and see where the different uncertainty components come from, how significant they are, and how they can be combined.

For the example, we will calibrate a pressure transmitter. Pressure calibration is a common task in the process industry, and it provides a relatively straightforward example for learning the basics of uncertainty analysis.

We will start with a short reminder of the basic uncertainty concepts and then move on to the practical example. If you are already familiar with uncertainty theory, feel free to jump directly to the pressure calibration example.

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Table of contents

• What is calibration uncertainty

• Where do the uncertainties come from? 

• How do we know the uncertainty?

• How do we combine the uncertainties?

• Example: pressure calibration

• Uncertainty of the reference standard (calibrator)
• Converting the mA uncertainty to pressure
  • Resolution of the reference
  • Combined uncertainty of the reference standard
  • Repeatability and hysteresis
  • Our repeatability test
  • Normalized output
  • Summary of the results

• Seeing calibrator uncertainty during calibration

• Other possible uncertainty components

• Summary

What is calibration uncertainty

A simple way to think about calibration uncertainty is that it describes the doubt associated with a calibration result.

For example, suppose you calibrate a digital pressure indicator at 100 kPa and it indicates exactly 100.00 kPa. That sounds like a perfect result.

But how certain are we that the pressure really was 100.00 kPa?

When all the relevant uncertainty components are taken into account, the result might be expressed as:

100.00 kPa ± 0.10 kPa

But with a different calibration setup or method, the uncertainty might instead be:

100.00 kPa ± 1 kPa

or perhaps even:

100.00 kPa ± 10 kPa

The indicated value is the same in all three cases, but the amount of confidence we can have in that result is very different. This is why a measurement or calibration result without information about its uncertainty does not tell the whole story.

Where do the uncertainties come from?

There can be many sources of uncertainty, and the important ones depend on what is being calibrated and how the calibration is performed. Some common sources are:

• The reference instrument or calibrator.
  No reference instrument is perfect. It has its own measurement uncertainty. In some calibrations this may be the largest uncertainty component, but that should not automatically be assumed. In other applications, other uncertainty sources can be considerably larger.
• The calibration process.
  The way the calibration is performed can introduce uncertainty. Perhaps insufficient time is allowed for the measurement to stabilize, the reference is not properly zeroed or warmed up, or different technicians perform the calibration in slightly different ways. A good calibration procedure and proper training help reduce these effects.
• The instrument being calibrated.
  The device under test (DUT) can itself contribute to the uncertainty. It may have limited resolution, poor repeatability, hysteresis, a slow response, or sensitivity to environmental conditions.
• Environmental conditions.
  Temperature and other environmental conditions may change between calibrations. If the reference or the DUT is sensitive to these changes, they can contribute to the uncertainty.

And there can be many others. The important point is that uncertainty is not necessarily just the specification of the calibrator. We need to consider the whole calibration process.

How do we know the uncertainty?

Some uncertainty components are relatively easy to identify. The uncertainty of the reference calibrator, for example, may be available from its specifications or calibration certificate.

Other components need to be determined experimentally. Repeatability is a good example: we can repeat the same calibration several times and see how much the results vary.

Some components need to be estimated based on available information, experience, specifications, environmental limits, or knowledge of the calibration process.

The objective is not to find every imaginable source of uncertainty. It is to identify the components that can have a meaningful effect on the calibration result.

How do we combine the uncertainties?

Before different uncertainty components can be combined, they need to be expressed as standard uncertainties and in the same measurement quantity.

Calibration equipment specifications are often stated as expanded uncertainty, for example with a coverage factor of k=2, corresponding approximately to a 95% coverage probability for a normal distribution. To use such a value in the uncertainty calculation, we first divide it by the appropriate coverage factor to convert it into a standard uncertainty.

Other uncertainty components may have a different probability distribution. A common example is a uniform, or rectangular, distribution. In that case, the half-width of the possible range is divided by √3 to obtain the standard uncertainty.

The uncertainty components also need to be expressed in the same unit. For example, in our pressure transmitter calibration we will have uncertainty components expressed in both pressure and current. Before combining them, we will convert the mA uncertainty into an equivalent pressure uncertainty.

Once all the components are expressed as standard uncertainties in the same unit, they can be combined using the root sum of squares (RSS) method:

Combined uncertainty = √(a² + b² + c² + …)

Why not simply add them together? Adding them directly would assume that all uncertainty components reach their maximum value at the same time and in the same direction. For independent uncertainty components, this is generally an unnecessarily conservative assumption.

RSS also means that the largest uncertainty components tend to dominate the result. For example, if you have two same size uncertainties of both 10 units, then the RSS sum is:

√(10² + 10²) = 14.14

But if you have uncertainties of 10 and 1 unit, then:

√(10² + 1²) = 10.05

So, when one uncertainty component is much smaller than the dominant component, it has very little effect on the combined uncertainty. This is an important practical lesson: uncertainty analysis is not about desperately trying to find every tiny possible contributor. It is about identifying and properly evaluating the components that actually matter.

Finally, once we have calculated the combined standard uncertainty, we can calculate an expanded uncertainty. In this example we will use a coverage factor of k=2, so the combined standard uncertainty is multiplied by two.

With that short theory reminder behind us, let's move on to the practical pressure calibration example.

Example: pressure calibration

Let's take a practical example of pressure calibration. We will calibrate a pressure transmitter using a Beamex MC6 calibrator. The transmitter has a pressure range of 0 to 20 bar (gauge) and an output of 4 to 20 mA.

In the calibrator, we use the 20 bar pressure module for measuring the applied pressure and the mA measurement module for measuring the transmitter output.

This type of pressure calibration is in many ways simpler than, for example, a temperature calibration, where we may have an external reference sensor and a temperature dry block, both introducing additional uncertainty components. Temperature calibration could be a topic for another article.

Uncertainty of the reference standard (calibrator)

As mentioned earlier, the reference standard, or calibrator, is often one of the main sources of uncertainty, so it is a good place to start.

The simplest way to determine the uncertainty of the calibrator is to look at its specifications.

If suitable uncertainty specifications are not available, determining the uncertainty becomes more complicated. You may need to look at the calibrator's calibration history, how much it has drifted between calibrations, and the uncertainty stated on its calibration certificate. Based on this information, you can estimate the uncertainty of the calibrator during its calibration interval.

In our example, we have 1-year uncertainty specifications (k=2) for the calibrator, so let's start with those.

The P20 pressure module has the following 1-year uncertainty specification:

0.01% FS + 0.025% RDG

The full scale (FS) of the module is 20 bar. This gives us:

• At 0 bar (0%): (0.01% of 20 bar + 0.025% of 0 bar) = 0.002 bar
• At 10 bar (50%): 0.0045 bar
• At 20 bar (100%): 0.007 bar

The mA measurement has a 1-year uncertainty specification of:

1 µA + 0.01% RDG

This gives us:

• At 4 mA (0%): (1 µA + 0.01% of 4 mA) = 0.0014 mA
• At 12 mA (50%): 0.0022 mA
• At 20 mA (100%): 0.0030 mA

The uncertainties in these specifications are expanded uncertainties with a coverage factor of k=2. This means that the standard uncertainty has been multiplied by a factor of two. Before we combine these values with the other uncertainty components later, we therefore need to divide them by two to convert them back to standard uncertainties.

Converting the mA uncertainty to pressure

To combine different uncertainty components, they eventually need to be expressed in the same unit. In our example, we will express the final uncertainty in pressure, so the uncertainty of the mA measurement needs to be converted to bar.

Our transmitter has a linear transfer function with an input range of 0 to 20 bar and an output range of 4 to 20 mA. The mA span is therefore 16 mA and the pressure span is 20 bar:

20 bar / 16 mA = 1.25 bar/mA

We can now convert the mA measurement uncertainties into pressure:

• At 4 mA (0%): (1.25 bar/mA × 0.0014 mA) = 0.00175 bar
• At 12 mA (50%): 0.00275 bar
• At 20 mA (100%): 0.00375 bar

These are still expanded uncertainties (k=2). We will convert them to standard uncertainties when we put all the uncertainty components together later.

Resolution of the reference

The resolution of the calibrator also needs to be taken into account. We cannot read the calibrator more accurately than its resolution allows. There are always some "hidden" digits below the last displayed digit, and we don't know their exact value.

The resolutions of our calibrator's pressure and mA measurements are:

• Pressure measurement: 0.0001 bar
• mA measurement: 0.0001 mA

A common way to evaluate the uncertainty caused by digital resolution is to assume that the actual value can be anywhere within half of the least significant digit in either direction. This gives us a range of ±0.5 digit.

Assuming a uniform distribution, the half-width of this range is divided by √3 to convert it to standard uncertainty. Therefore:

Standard uncertainty due to resolution = resolution / (2 * √3)

For our example:

• Pressure: 0.0001 bar / (2 * √3) = 0.00002887 bar, or approximately 0.00003 bar
• mA measurement: 0.0001 mA / (2 * √3) = 0.00002887 mA, or approximately 0.00003 mA

Later, we will need the mA resolution uncertainty expressed in pressure, so we use the same conversion factor of 1.25 bar/mA:

0.00002887 mA × 1.25 bar/mA = 0.0000361 bar

As a simple rule of thumb, the standard uncertainty caused by digital resolution is approximately 0.3 times the resolution. So, if the resolution is 0.1, the standard uncertainty caused by resolution is approximately 0.03. If the resolution is 1, it is approximately 0.3.

If the device being calibrated has a digital display, its resolution should also be considered as an uncertainty component. In some cases, especially with instruments that have limited resolution, this can become significant. In our example, however, the DUT is a transmitter with an analog mA output, so there is no DUT display resolution to include.

Combined uncertainty of the reference standard

We have now identified four uncertainty components coming from our reference standard:

• the pressure measurement uncertainty,
• pressure resolution,
• mA measurement uncertainty,
• mA resolution.

We could combine these four components already at this point. However, to keep everything in one place, we will do that later in the final uncertainty budget. There we will also show the combined uncertainty of the calibrator separately, so we can compare it with the total calibration uncertainty.

So now that we know the uncertainty contributions from our calibrator, you may ask: isn't that enough?

Well, let's find out.

Repeatability and hysteresis

If you perform only one calibration cycle, you don't know how repeatable the measurement results are.

Depending on the instrument and calibration process, repeatability can be a significant contributor to the uncertainty, or it can be so small that it has practically no effect. But we don't know that until we investigate it.

One way to evaluate repeatability is to perform the same calibration several times under the same conditions. In our example, we will repeat the calibration 10 times and use the variation in the results to estimate the repeatability uncertainty.

Obviously, in an industrial environment, it is usually not practical to repeat every routine calibration 10 times. But it can be useful to perform this kind of study when establishing or validating a calibration procedure for a certain type of instrument.

Once we know the typical repeatability of the calibration process, we can use that information as an uncertainty component in future routine calibrations, where only one normal calibration cycle is performed. This assumes, of course, that the instrument type, calibration equipment, procedure, and conditions remain reasonably similar.

Some of our customers use an interesting variation of this approach. They ask three calibration technicians to calibrate the same instrument three times each. This gives nine calibration results performed by three different people.

The standard deviation of these results can provide useful information about the variation in the overall calibration process. It can also reveal differences between technicians. If the results vary significantly depending on who performs the calibration, the variation may not be caused only by the instrument. It may indicate differences in how the calibration procedure is being performed.

This can be a useful way to identify hidden issues in calibration procedures, instructions, or training.

If the repeatability contribution turns out to be very small compared with the other uncertainty components, it may not need much attention in routine uncertainty calculations.

Our repeatability test

Back to our example. We perform 10 repeated calibrations using a three-point up-and-down sequence:

0% → 50% → 100% → 50% → 0%

Using an up-and-down calibration also allows us to investigate hysteresis. At the 50% point, for example, we can compare the result when approaching the point from below with the result when approaching it from above.

For each calibration cycle, we calculate hysteresis at the points where we have both an increasing and decreasing measurement. The Max hysteresis column below shows the largest difference of these values.

Normalized output

Please note that the table below does not show the actual input pressure. In practice, the applied pressure will not be exactly at the nominal calibration point in every repeat. Therefore, the measured mA values have been normalized to the nominal pressure points using the actual reference pressure reading. This makes the results directly comparable when evaluating repeatability and hysteresis.
For example, suppose the nominal 50% point is 10.0000 bar, but the actual reference pressure is 10.0030 bar. For a linear 0–20 bar / 4–20 mA transmitter, that 0.0030 bar difference corresponds to:
0.0030" bar"×16" mA" /20" bar" =0.0024" mA"

If the measured output was 12.0040 mA, the normalized result at exactly 10.0000 bar would therefore be:

12.0040-0.0024=12.0016" mA"

Then 12.0016 mA is the value you use for the repeatability and hysteresis calculations.

The results of our 10 repeated calibrations, with normalized mA readings, are shown in the table below.

The standard deviation of the repeated measurements gives us an estimate of repeatability. At the 50% point, we have standard deviations of 0.00040 mA in the increasing direction and 0.00049 mA in the decreasing direction.

To be conservative, we will use the larger value:

Repeatability at 50% = 0.00049 mA

Converting this to pressure:

0.00049 mA × 1.25 bar/mA = 0.0006125 bar

For hysteresis, the largest value observed at the 50% point is 0.0041 mA. Converted to pressure:

0.0041 mA × 1.25 bar/mA = 0.005125 bar

Later, when we add hysteresis to the uncertainty budget, we will treat this as a range with a uniform distribution. Therefore, the value will still need to be converted to standard uncertainty before it is combined with the other uncertainty components.

Putting it all together

We have now identified the uncertainty components that we want to include in our example. It is time to put them together and calculate the total calibration uncertainty.

Let's start with the 50% calibration point. The uncertainty components we have identified are:

• Pressure measurement uncertainty of the calibrator
• Pressure measurement resolution
• mA measurement uncertainty of the calibrator
• mA measurement resolution
• Repeatability
• Hysteresis

The first four components come from the calibrator specifications and resolution. Repeatability and hysteresis were determined experimentally from our repeated calibrations.

Before combining the components, they all need to be expressed as standard uncertainties in the same unit, which in our example is bar.

The table below shows the complete uncertainty budget for the 50% point.

The first four rows are the uncertainty components coming from the calibrator. Combining their standard uncertainties using the root sum of squares (RSS) method gives a combined standard uncertainty for the calibrator of 0.002637 bar. Note that this is a standard uncertainty, not an expanded uncertainty. If expressed as expanded uncertainty with k=2, it would be 0.005274 bar.

We show this subtotal separately because it allows us to compare the uncertainty coming from the calibrator with the total uncertainty of the complete calibration. The subtotal is not added again when calculating the final RSS, as that would count the same components twice.

We then include the repeatability and hysteresis contributions.

For repeatability, the standard deviation is already a standard uncertainty, so no further divisor is needed.

For hysteresis, the measured value represents the total difference between the increasing and decreasing results. We first take half of this range and, assuming a uniform distribution, divide it by √3. This is shown in the table as a divisor of 2 × √3.

Combining all the individual standard uncertainty components using RSS gives:

Combined standard uncertainty, u = 0.003085 bar

Finally, we calculate the expanded uncertainty using a coverage factor of k=2:

Expanded uncertainty, U = 0.006170 bar

At the 0% and 100% points

The same calculation can be performed at the other calibration points.

At 0%, we get:

• Combined standard uncertainty of the calibrator, u: 0.001330 bar
• Combined expanded uncertainty of the calibrator, U: 0.002660 bar
• Total expanded uncertainty, U (k=2): 0.003330 bar

At 100%, we get:

• Combined standard uncertainty of the calibrator: 0.003971 bar
• Combined expanded uncertainty of the calibrator, U: 0.007942 bar
• Total expanded uncertainty, U (k=2): 0.007990 bar

The results for all three calibration points are summarized below.

One thing to note about the 100% point is that we cannot determine hysteresis directly from our measurement sequence. We increased the pressure to 100% and then started decreasing it, so there is only one measurement at 100%. To measure hysteresis at that point, we would need to go above 100% and then approach the 100% point again from the decreasing direction.

For this example, we have therefore left the hysteresis contribution out of the 100% result. Another option could be to use a representative hysteresis value determined at the other calibration points, but we have not done that here.

Do I need to do 10 repeats?

So, does this mean that you need to perform 10 repeated cycles every time you calibrate an instrument? Fortunately, no.

We performed 10 repeats in this example to learn about the typical repeatability and hysteresis of this type of instrument and calibration process. In normal routine calibration, you can perform just one normal calibration cycle and use the repeatability and other process-related uncertainty components determined in this kind of more extensive study.

Of course, this is only valid as long as the instrument type, calibration equipment, procedure, and conditions remain reasonably comparable. It is also good practice to repeat this kind of study from time to time to make sure that the assumptions are still valid.

This is one of the practical benefits of doing a more thorough uncertainty evaluation: once you understand your calibration process and know its typical uncertainty contributions, you don't need to repeat the full investigation for every routine calibration.

Summary of the results

Looking at the results, we can see that the uncertainty of the calibrator is the largest contributor to the total calibration uncertainty. In this particular example, we did not find any other large uncertainty components. Repeatability is relatively small, while hysteresis has a more noticeable effect, but the calibrator uncertainty still dominates the final result.

This is also partly due to the way uncertainty components are combined. With the root sum of squares (RSS) method, the largest uncertainty components tend to dominate the final result. Adding an uncertainty component that is much smaller than the largest component has very little effect on the combined uncertainty. Therefore, it is especially important to identify the largest contributors rather than simply trying to find as many uncertainty components as possible.

The calibrator uncertainty increases with pressure, mainly because both the pressure and mA measurement specifications include a percentage-of-reading component. This is why the expanded uncertainty is smaller at the 0% point and larger toward the upper end of the range.

We can also see that the resolution of the pressure and mA measurements has a very small effect in this example. If the calibrator has sufficiently good resolution compared with its measurement uncertainty and the other uncertainty components, its effect may be negligible. This does not mean that resolution can always be ignored, but with the equipment and ranges used here its contribution is insignificant compared with the larger uncertainty components.

It is important not to generalize the result of this example too far. In this pressure calibration, the calibrator happens to be the dominant source of uncertainty. In another calibration application, other uncertainty sources may be much larger and the uncertainty of the calibrator may represent only a small part of the total uncertainty.

This is one of the main reasons for doing an uncertainty evaluation like this. Instead of assuming which uncertainty components are important, we evaluate them and find out which ones actually matter in the calibration process.

This example also illustrates a limitation of using a simple Test Accuracy Ratio (TAR) to evaluate whether a calibrator is suitable for a calibration. TAR typically focuses on the relationship between the DUT tolerance and the accuracy or uncertainty of the reference calibrator. This can be a useful quick indication, but it may hide other uncertainty sources in the actual calibration process.

In our example, the calibrator happens to be the dominant uncertainty contributor, so a TAR based mainly on the calibrator may give a reasonably good indication. But this should not be assumed to be true for every calibration. If repeatability, environmental conditions, the calibration method, or other uncertainty sources are significant, a good TAR alone does not necessarily mean that the total calibration uncertainty is sufficiently small.

Seeing calibrator uncertainty during calibration

As we have seen, calculating calibration uncertainty requires some work. In our example, we also found that the uncertainty of the calibrator was the largest contributor to the total uncertainty.

To make this part easier, a recent free firmware update for the Beamex MC6 family added the ability to calculate and display the calibrator's own uncertainty during calibration. The uncertainty is calculated from the measurement modules used for the input and output measurements and can be shown visually during the calibration.

This means that you can see how the combined uncertainty of the calibrator changes at different calibration points without having to calculate it manually. The calibrator uncertainty can also be taken into account in the pass/fail decision. 

The screenshot above shows how the Beamex MC6 family calibrator displays its own combined uncertainty during calibration. The uncertainty is shown graphically as the small segments above and below each saved calibration point.

See how it works in practice

Watch the video to see how the Beamex MC6 family calculates and displays expanded uncertainty during calibration when configured in CMX.

It is important to remember that this is the uncertainty of the calibrator, not necessarily the total uncertainty of the calibration. As we have seen in our example, there may be additional uncertainty components coming from the DUT, calibration process, environmental conditions, and other sources.

Additional uncertainty components, such as the ones we determined in our example, can be configured in the Beamex calibration management software and included in the overall uncertainty evaluation.

If your MC6 family calibrator does not yet have this functionality, you can download and install the latest free firmware update from the Beamex website’s Download Center or contact Beamex. The uncertainty functionality also needs to be enabled in the calibrator’s settings.

Other possible uncertainty components

The number and size of uncertainty components depend on the type of calibration and how it is performed. In addition to the components included in our example, the following may be relevant in industrial pressure calibration.

Height difference

A height difference between the reference and the device under test can cause a hydrostatic pressure difference and therefore an error in the calibration.

With air or another gas as the pressure medium, the effect is usually relatively small. With a liquid pressure medium, however, it can become significant.

Ideally, this effect should be corrected by taking the height difference into account. In routine field calibration, another practical approach may be to specify in the calibration procedure that the reference and DUT should be positioned at approximately the same height. You can then estimate the maximum remaining height difference and evaluate its effect as an uncertainty component.

For example, with water as the pressure medium (density approximately 998 kg/m³), local gravity of 9.8 m/s², and a height difference of 1 meter (3.3 feet), the resulting pressure difference is approximately 9.8 kPa (98 mbar or 1.42 psi). This can obviously be significant.

Environmental conditions

If the calibrator or DUT has a significant temperature coefficient and the environmental conditions vary between calibrations, the effect should be considered.

A practical approach is to estimate the temperature range that can occur during routine calibrations and use the temperature coefficients of the calibrator and DUT to determine how much this could affect the measurement. The resulting effect can then be included as an uncertainty component.

DUT resolution

If the device being calibrated has a display, its resolution can introduce additional uncertainty, as discussed earlier.

This can be particularly important when calibrating an analog pressure gauge, where the limited readability of the scale may become one of the largest uncertainty components.

Leakage and pressure stability

Any leakage in the pressure system can cause the pressure to change during calibration. Pressure may also change after adjustment due to thermal or adiabatic effects, even when there is no actual leak. It is therefore important to make sure that the applied pressure is sufficiently stable before recording the results.

Settling time and response time

Even when the applied pressure is stable, the DUT and reference may have different response times or filtering. If a measurement is recorded too quickly after changing the pressure, one instrument may have reached the new value while the other is still settling.

A consistent procedure with sufficient settling time helps reduce this uncertainty.

Reference standard drift

If a suitable 1-year uncertainty specification is available for the calibrator, as in our example, expected drift during the calibration interval may already be included in that specification.

If we instead base the reference uncertainty mainly on its latest calibration certificate, we also need to consider how much the reference may have drifted since that calibration.

If several calibration results are available from previous years, the calibration history can help us estimate the typical drift between calibrations and include an appropriate contribution in the uncertainty budget.

Other uncertainty sources

The examples above are not a complete list. Depending on the type of instrument, pressure range, pressure medium, calibration equipment, procedure, environmental conditions, and required accuracy, there may be other uncertainty sources that need to be considered.

The important thing is to look at the complete calibration process and ask what could have a meaningful effect on the result. Not every possible uncertainty source needs to be included, especially if its effect is insignificant compared with the larger contributors. But it should first be considered before deciding that it can safely be ignored.

Summary

Calibration uncertainty can seem complicated, but the basic idea is quite practical: identify the things that can cause doubt in your calibration result, estimate how large their effects can be, and combine the significant uncertainty components.

In our pressure transmitter example, we started with the uncertainty specifications of the reference calibrator. We then looked beyond the calibrator itself and considered other possible contributors, such as resolution, repeatability, hysteresis, environmental conditions, height difference, leakage, settling time, and reference drift.

After doing the calculations, we may find that the uncertainty of the reference calibrator is by far the largest contributor and that the other components have very little effect on the final combined uncertainty. If that is the case, the uncertainty calculation for routine calibrations can potentially be simplified.

However, we should not simply assume that the calibrator uncertainty is always enough. The other relevant uncertainty sources should first be evaluated. Once we have demonstrated that certain components are insignificant, there is little benefit in making every routine uncertainty calculation unnecessarily complicated.

It is also important to remember that this conclusion is specific to the calibration application. In another type of calibration, the reference instrument may represent only a small part of the total uncertainty.

Temperature calibration is a good example. In addition to the uncertainty of the reference measurement, the total uncertainty may be affected by the reference sensor, stability and uniformity of the temperature source, immersion depth, stem conduction, loading effects, environmental conditions, repeatability, and the characteristics of the DUT. Insufficient stabilization time can also be important: if the reference and DUT have not reached thermal equilibrium before the readings are taken, this can introduce a significant additional uncertainty.

So, the purpose of an uncertainty analysis is not to find as many uncertainty components as possible. It is to identify the components that actually matter in your calibration process, understand their magnitude, and build an uncertainty estimate that realistically describes how much doubt there is in the calibration result.

There is also another practical benefit in doing this kind of uncertainty evaluation. It does not mean that every routine calibration needs to be repeated ten times or that all these investigations need to be performed every time.

By performing a more thorough study for a certain type of instrument and calibration process, you can learn about the typical repeatability and other process-related uncertainty components. If the calibration method, equipment, conditions, and type of instrument remain comparable, these experimentally determined values can then be used as typical uncertainty components in routine calibrations.

This allows the routine calibration to be performed with just one normal calibration cycle, while the uncertainty estimate still takes into account what has been learned from the more extensive study.

Of course, these assumptions should be reviewed from time to time, especially if the equipment, calibration procedure, environmental conditions, personnel, or type of instrument changes significantly. But this is also one of the benefits of uncertainty analysis: once you understand your calibration process and know which components really matter, routine calibration can be both efficient and technically justified. 

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