Measurements are affected by the environment, which means that the conditions in which measurements are made are critical. It is almost always necessary to control temperature and humidity. There may also be requirements related to dust, pressure and lighting. To properly understand appropriate tolerances for these parameters, it is necessary to consider tolerances of product specifications with the uncertainty of measurement, including consideration of its sensitivity to environmental variation.

Perhaps the most easily understood environmental effect is the way that thermal expansion impacts dimensional measurements. Consider that a nominally 1m long aluminum alloy part with the tolerance of +/-0.01mm is measured giving a result of 99.98mm. This is clearly out of specification, however, steel expands by approximately 0.01mm/m for every 1 C that it is heated. Therefore, if the part is heated by 2 C and remeasured, the measurement result would now show that it conforms with the specification. Whether or not the part conforms does not depend on how warm it is when it is measured. In reality, the specification implies that the part should be within tolerance when it is at a temperature of 20 C. This is specified in the easy-to-remember standard ISO 1, which states that any product specification that does not give a specific temperature should be interpreted as giving the product dimensions at this standard reference temperature.

It is not only temperature that affects product dimensions. Many polymer and composite parts are hygroscopic, meaning that they can absorb moisture leading to a change in their physical properties. In addition to the effect of the environment on the product itself, environmental parameters also affect measurement instruments. For traditional gauges and contact instruments, this is most likely due to thermal expansion of the components in the instrument. However, for optical instruments, which use lasers or CCD arrays, things get more complicated. The refractive index of air is strongly affected by temperature, pressure, humidity and carbon dioxide concentration. Changes in these parameters can, therefore, cause rays of light to bend and laser wavelengths to change.

### Uncertainty of Measurement

All measurements have uncertainty caused by sources such as repeatability, calibration and the environment. The combined uncertainty is evaluated by estimating the contribution from each source and then calculating the combined uncertainty, which is not as simple as summing the individual sources. The de facto standard for evaluating uncertainty is the Guide to the Expression of Uncertainty in Measurement (GUM), which defines uncertainty sources as Type A sources if they are estimated by statistical analysis of repeated measurements and Type B if they are estimated using any other information. Repeatability, which is normally evaluated by making a number of measurements and then calculating the standard deviation, would be a Type A source of uncertainty. The uncertainty of a calibration reference standard is normally taken from the calibration certificate, making it a Type B uncertainty.

When multiple sources of uncertainty are combined, this usually results in a normally distributed combined uncertainty, due to the central limit theorem. Making this assumption allows confidence levels to be estimated, giving a range of values about the measurement result within which we can be confident that the true value of the object being measured lies. Typically, 95% confidence is used, which means the true value is within two standard deviations of the measurement result.

Sources of uncertainty may be recorded and combined using an uncertainty budget. This is a table listing each source of uncertainty in its own row with columns for its value, distribution, sensitivity coefficient (something that will be explained in a moment) and normalized effect on the measurement result—known as a standard uncertainty.

Many common sources, such as repeatability and uncertainty of the calibration standard, can be combined in a relatively simple way. This is because there is a direct one-to-one relationship between an error arising from one of these sources and an error in the measurement result. If the calibration certificate is out by 1mm, then all the measurement results will have a 1mm error caused by this in addition to any other errors caused by other sources of uncertainty. If you’re getting confused about the difference between error and uncertainty, then read my introduction to metrology and quality.

When it comes to environmental parameters, the effect measurement result is not quite so simple. A deviation of one unit in temperature or humidity will probably not translate into one-unit error in the measurement result. This is most obvious when the quantities do not have the same units of measurement, but it may also be the case for sources of uncertainty that do have the same units as the measurement result. The way that change in an input quantity such as an environmental parameter results in a change in the measurement result is described by a sensitivity coefficient.

### Sensitivity Coefficients

Sensitivity coefficients are central to understanding environmental effects on measurements. Within the GUM, each source of uncertainty is considered to be an input quantity. The measurement result is a function of these input quantities. The sensitivity of the measurement result to a change in each input quantity can be calculated from the measurement function. This is explained in detail in my article on sensitivity coefficients where I use the example of measuring the height of the building using a tape measure and clinometer. The measurement result is the height of the building, *H,* which is given by the input quantities *h _{1}*, the height of the clinometer;

*L*, the horizontal distance along the ground; and

*theta*, the angle, according to the measurement function:

*h*will always result in an error of 1mm in H. This means that

_{1}*h*has a sensitivity coefficient of one. The sensitivity coefficients for the other two input quantities depend on the value that those quantities take. An error of 1mm in

_{1}*L*will not necessarily result in an error of 1mm in

*H*. Therefore, unless the angle

*theta*is 45 degrees, the sensitivity coefficient is not equal to one. Using calculus sensitivities can be calculated as tan (

*theta*) for

*L*and

*L*sec

^{2}(

*theta*) for

*theta*, but it is often easier to use finite differences to find the sensitivity coefficients for specific values of the input quantities. This approach is much more intuitive.

If the input quantities have been measured with values of *h _{1}*=1.65m,

*L=10*m and

*theta*=58° then the measurement result,

*H*is 17.653m. A finite difference means simply making a small change to one of the quantities and seeing what effect it has. If

*L*is increased by 10mm (ΔL=10mm) then this results in a change of height of 16mm (ΔH = 16mm). The sensitivity coefficient for the length is approximately ΔH/ΔL = 1.6. Note that in this case, both finite differences have the same units, so the sensitivity coefficient is dimensionless. Using finite differences to find the sensitivity coefficient for the angle, we could increase the angle by 0.5° giving a change in height of 316mm, which gives a sensitivity coefficient of ΔH/Δθ = 632mm / deg. Note that here the sensitivity coefficient has units that allow the conversion quantity’s units to the units of the measurement result.

The sensitivity coefficients allow the uncertainties in the input quantities to be combined to give the uncertainty of the measurement result as shown in the uncertainty budget below.

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Using an Uncertainty Budget to Establish Environmental Parameters

### Case Study with Rolls-Royce

I was involved with a Rolls-Royce study for the dimensional verification of a large aero gas turbine engine. The product verification measurements were taken at the main interface points where the engine is attached to the aircraft wing, which act as datums for all subsequent measurements on the engine. Special tooling was used to locate targets at these interfaces, and a laser tracker was used to make the measurements. The measurement process involved positioning the laser tracker at three locations in order to gain line of sight to all of the datum features on the engine. Each of the six datum features consisted of a pair of lugs with a coaxial bore running through them. A datum point was constructed for each datum feature by measuring the bore axis and finding the midpoint along it between the two inner faces of the lugs. The resulting six datum points were used to create a coordinate frame.

_{}

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Standard Laboratory Control

The above case study highlights the importance of establishing impact of environmental parameters on the measurement process. This is well-known, and ISO 17025 sets out general requirements for the competence of testing and calibration laboratories. As well as standard procedures for managing the laboratory and maintaining records, control of the environment is a key requirement. However, the standard does not specify tolerances for environmental parameters. Instead, it states that “Lighting and environmental conditions shall be such as to facilitate correct performance of the tests and/or calibrations. The laboratory shall ensure that the environmental conditions do not invalidate the results or adversely affect the required quality of any measurement.” This means that it is the responsibility of quality engineers to determine what environmental conditions are required to obtain measurements of sufficient quality to prove conformance nonconformance with product specifications. This is why an understanding of uncertainty principles, such as sensitivity coefficients, as detailed above, is so important.

ISO 1725 draws attention to biological sterility, dust, electromagnetic disturbances, radiation, humidity, electrical supply, temperature, and sound and vibration levels as being environmental conditions that might potentially affect the quality of measurements. This will depend on the type of measurements being carried out. The standard also mandates that environmental conditions are monitored and recorded, and that tests of your calibrations should not be carried out when the environmental conditions jeopardize the results of the tests.

Typical specifications for a metrology lab might include a temperature that is regulated to within less than 1° C of 20° C and relative humidity is less than 5 percent. However, blindly applying such rules of thumb cannot be considered good practice. What is important is that all significant influences on the measurements to be carried out have been considered and no influences are allowed to deviate in such a way that the uncertainty of measurement reaches an unacceptable level. Most importantly, decisions should not be taken without statistical validity taking into account the uncertainty of measurement.