When uncertainty sources have different units of measurement

This article follows on from my introduction to uncertainty budgets. In that article, I explained how different sources of uncertainty are evaluated and combined to give the uncertainty of a measurement result. The example I used was a simple one in which uncertainty sources mapped directly to the measurement result. This meant that no understanding of sensitivity coefficients was required.

In this article, I explain the meaning of sensitivity coefficients within uncertainty evaluation. I will also use examples to show exactly how sensitivity coefficients can be calculated and used within an uncertainty budget.

### Uncertainty Sources Which Map Directly to the Measurement Result

In the example from my previous article, each source of uncertainty resulted in an equal uncertainty in the measurement result. This meant that the sensitivity coefficients were all equal to one and I provided no further explanation of these. In order to introduce sensitivity coefficients, let’s start by considering the simple case in more detail. This is most easily understood by considering the errors in an individual measurement resulting from the sources of uncertainty.

It is important to remember that it is impossible to know the true value of the thing we are measuring (the ‘measurand’)—we can only know the result of a measurement. This measurement will have some error, which is the difference between the true value and the measurement result. Since we cannot know the true value, we also cannot know the error. The uncertainty of the result of a measurement and the uncertainty of the error are mathematically equivalent. We will consider these unknowable quantities, the true value and the error, in a theoretical analysis.

Let’s now define the simple case in which each source of uncertainty results in an equivalent uncertainty in the measurement result. Imagine there are three sources of uncertainty which result in three errors: x_{1}, x_{2} and x_{3}. The measurement result *Y* is then the sum of the true value (*y*) and the three errors:

Y = y + x_{1} + x_{2} + x_{3}

Hence, if the error x_{1} increases by 5 µm then the measurement result *Y* will also increase by 5 µm. Therefore the sensitivity coefficient for each term is one. The *law of propagation of uncertainty* states this as:

This complicated looking equation simply says that the combined uncertainty in the measurement result (*U _{C}*) is the square root of the sum of each individual uncertainty multiplied by its sensitivity coefficient

and squared. In the simple case above, all of the sensitivity coefficients are one. If you’re still confused, don’t worry, with a few more examples this will all become clear.

### An Example Where Sensitivity Coefficients are Not One

Let’s imagine that we are measuring the height of a building using a tape to measure the horizontal distance along the ground and a clinometer to measure the angle. Our final measurement result *H* is the result of three initial measurements: *h _{1}* the height of the clinometer;

*L*the horizontal distance along the ground; and

*theta*, the angle. The measurement result is given by the equation:

In this case, an error of 1 mm in *h _{1}* will result in an error of 1 mm in H. Therefore we can say that

*h*has a sensitivity coefficient of one.

_{1}However, in general, an error of 1 mm in *L* will not result in an error of 1 mm in *H* and therefore the sensitivity coefficient is not equal to one. The exception to this, of course, is when the angle *theta* is 45 degrees. The sensitivity coefficient for *L* is given by tan (*theta*) and, if you remember your calculus, the sensitivity coefficient for *theta* is *L*sec^{2} (*theta*).

There is, however, a more intuitive way to consider these sensitivities by considering the actual measured values.Let’s say we have carried out the measurements and found that *h _{1}*=1.65 m,

*L=10*m and

*theta*=58°, giving a height

*H*of 17.653 m. If we increase L by 10 mm (ΔL=10mm) then this results in a change of height of 16 mm (ΔH = 16 mm). The sensitivity coefficient for the length is therefore approximately ΔH/ΔL = 1.6. Similarly if we increase the angle by 0.5° we see an increase in the height of 316 mm so ΔH/Δθ = 632 mm / deg.

Let’s see how this fits into an uncertainty budget. The uncertainty for the tape measurements is stated as 0.5 + 5 mm/m. Since no further information is given, we assume this is normally distributed with a 95 percent confidence, as is standard practice for calibration certificates. Over the lengths measured, this gives uncertainties of 50.5 mm in *L* and 8.75 mm in *h1*. The clinometer measurement has a standard uncertainty of 1 degree. These uncertainties can now be combined, considering the sensitives calculated above, using an uncertainty budget.

Working along the rows, the standard uncertainties for each source are calculated first by dividing the value by the divisor and multiplying the result by the sensitivity coefficient. Note that where the source and the standard uncertainty have the same units, the sensitivity coefficient should be dimensionless. Where they have different units, the units of the sensitivity coefficient should make the necessary correction. Ensuring that this is the case is a useful check. The combined uncertainty is then calculated as the root sum square of the final column containing the standard uncertainties, as explained in my previous article.

### A More Detailed Example for the Clinometer Measurement

This simple example has assumed that we know the uncertainties for each individual measurement. In practice, each of these would have a number of components which must be combined. We will continue with the same example of measuring the height of a building, but considered in a more realistic way.

Let’s assume that the values previously used for the uncertainty of the tape and the clinometer measurements were actually the calibration uncertainties. We will now also consider repeatability, resolution and thermal expansion of the steel tape. A complete uncertainty budget considering all of these sources is shown below.

In this example, the calibration, repeatability and resolution for the horizontal length are calculated in the same way as the simple uncertainty for the horizontal length in the previous example. The repeatability would be determined by making repeated measurements, ideally at least 30, and then calculating the standard deviation. The resolution is the uncertainty due to rounding errors and therefore has an equal probability of resulting in errors of plus or minus half of the smallest increment for the instrument reading. Therefore the resolution has an uncertainty value which is half of the smallest increment and follows a uniform distribution. Remember that the divisor for a uniform distribution is the square root of three, which is approximately 1.73.

The uncertainty in the temperature for the horizontal length requires some more explanation. When the tape was calibrated, it was at 20 C, since this is the standard reference temperature for all measurements (ISO 1). We are assuming that our measurements were made at close to 20 C, but we have some uncertainty about the actual temperature which we have estimated has a standard uncertainty of two degrees Celsius.

The tape is made from steel, which has a rate of thermal expansion of 0.012 mm per meter for every degree of temperature increase. This would normally be stated at the coefficient of thermal expansion (CTE) is 12 x 10^{-6} C^{-1}. The sensitivity of the length *L* to a change in temperature is therefore

Since we already know the sensitivity of the measurement result *H* to a change in the length *L* we can now find the sensitivity of interest, the sensitivity of the measurement result *H* to a change in the temperature over the length *L*:

We can also find the sensitivity of the measurement result *H* to a change in the temperature over the view height *h _{1}*:

The remaining calculations to determine the combined standard uncertainty and expanded uncertainty are the same as for the previous example.

### Cases Where You Need to Use Sensitivity Coefficients

For many simple uncertainty evaluations it will be possible to assume that all of the sensitivity coefficients are simply equal to one. This will also be generally true where sources of uncertainty are evaluated using ‘Type A’ methods involving a repeatability or reproducibility study for the full measurement process. In these cases, the effect of variations in the influence quantity will be directly observed as changes in the measurement result. Therefore, the sensitivity coefficient is exactly one.

For many other types of influence quantity, however, it will be necessary to consider the sensitivity coefficient more carefully. Some typical examples include:

- Measurements involving a number of intermediate measurements which are mathematically combined to give the measurement result. If the intermediate measurements are simply summed then the sensitivity is one, in all other cases it must be calculated. The Examples above are of this type.
- Uncertainty due to temperature variation causing thermal expansion or other influences on the measurand
- Uncertainty due to variation in alignment, where the uncertainty in the angle is known
- Uncertainty due to environmental effects, such as temperature, pressure, humidity and carbon dioxide level influencing the refractive index. This effects both laser range measurements and any optical measurement which depends on the angle of a line-of-sight. Examples of measurements where angle of line-of-sight is important include theodolites, laser scanners and photogrammetry. Changes in refractive index also affect time of flight measurements and interferometric measurements, since the laser wavelength is influenced.
- Consideration of environmental effects such as temperature or humidity acting on individual components of an instrument, either mechanical or electronic.

### Sensitivities Without a Constant Value

Some sensitivities do not have a constant value. For example, it is common for an alignment to have a nominal value of zero but for there to be some uncertainty about the actual angle, which may be positive or negative. An example of this is a cosine error. The error in alignment is nominally zero but may take a positive or negative value, in either case it produces a positive error in the measured length. This error does not increase linearly with increased error in angular alignment, since it is determined by the cosine function. This means that the sensitivity of the length to a change in the angle is not a constant, as shown below:

In cases where the sensitivity coefficient is not a constant, it is normal practice to evaluate it at the value of its uncertainty. This is acceptable for cases such as cosine error, since the sensitivity is zero when the angle is zero and the sensitivity increases as the angle increases. Therefore, this may be considered a worst case value for the sensitivity coefficient.

There are two things to be careful of here. Firstly, for some functions the sensitivity may not increase as the error magnitude increases and therefore using the value of the uncertainty may not result in the largest expected sensitivity. Secondly, if a standard uncertainty is used to determine the sensitivity then this may result in a significant underestimation of the sensitivity when the expanded uncertainty is calculated. Therefore it is good practice to determine the sensitivity for the expanded uncertainty which will ultimately be used.

### Conclusion

I’ve introduced sensitivity coefficients in this article. Together with my previous introduction to uncertainty budgets, you should now have enough understanding to calculate uncertainty budgets for real measurements. In my next article on this subject, I will look at some of the limitations of these analytical methods and the advantages of a numerical simulation approach to uncertainty.

*Dr. Jody Muelaner’s 20-year engineering career began in machine design, working on everything from medical devices to saw mills. Since 2007 he has been developing novel metrology at the University of Bath, working closely with leading aerospace companies. This research is currently focused on uncertainty modelling of production systems, bringing together elements of SPC, MSA and metrology with novel numerical methods. He also has an interest in bicycle design. Visit his website for more information.*