Dr Jody Muelaner

Evaluating Uncertainty of Measurement

All measurements have uncertainty as explained in the introduction to metrology. Uncertainty of measurement arises from a number of sources. Alignment, repeatability, resolution, temperature and calibration are a few common sources of uncertainty. The evaluation of the combined uncertainty involves first estimating the contribution from each source and then determining how these will combine to give a combined standard uncertainty. The combined standard uncertainty can finally be expanded by the coverage factor for the required confidence level.

This page provides an overview each of these stages. More detail on each stage can be found in the detailed explanation of creating an uncertainty budget.

Sources of Uncertainty of Measurement

Uncertainty of measurement arises from a number of sources. Alignment, repeatability, resolution, temperature and calibration are a few common sources of uncertainty. Sources of uncertainty are classified as Type A if they are estimated by statistical analysis of repeated measurements or Type B if they are estimated using any other available information.

Alignment Errors

Alignment is a common source of uncertainty in dimensional measurement. For example when a measurement of the distance between two parallel surfaces is made it should be perpendicular to the surfaces. Any angular deviation from the perpendicular measurement path will result in a cosine error in which the actual distance is the measured distance multiplied by the cosine of the angular deviation.

Cosine Error, Common Source of Uncertainty of Measurement

Cosine Error is a Common Source of Uncertainty of Measurement

Another common alignment error is parallax error. This results from viewing a marker, which is separated by some distance from the scale or object being measured, at an incorrect angle. Parallax error commonly observed when a passenger in a car reads the speedometer. Another common example is when the markings on the upper surface of a ruler are used to measure between edges on a surface. If the viewing angle is not perpendicular to the ruler this will result in parallax error.

Parallax error is commonly observed when a passenger in a car reads the speedometer or when using a ruler with the markings on the upper surface

Parallax error is commonly observed when a passenger in a car reads the speedometer or when using a ruler with the markings on the upper surface

Abbe Error is similar to parallax error but rather than resulting from alignment of viewing angles it results from alignment of machine axes. The distance between the axis along which an object is being measured and the axis of the instruments measurement scale is known as the Abbe Offset. If the distance along the object is not transferred to the distance along the scale in a direction perpendicular to the scale then this will result in an error. The size of this error will be the tangent of the angular error multiplied by the Abbe Offset. Instruments such as Vernier callipers are susceptible to Abbe Error as the measurement scale is not co-axial with the object being measured. Micrometers are not susceptible to Abbe Error.

Calipers are susceptible to Abbe Error since the Measurement Scale is not Co-Axial with the Axis of Measurement

An Instrument is Susceptible to Abbe Error if the Measurement Scale is not Co-Axial with the Axis of Measurement

Repeatability and Reproducibility

Repeatability is estimated by making a series of measurements, generally by the same person and under the same conditions, and then finding the standard deviation of these measurements. Reproducibility is estimated by making a series of measurements, each by a different person.

One challenge in evaluating uncertainty of measurement is determining which sources of uncertainty contribute to the observed repeatability. For example it may be that alignment errors vary randomly, contributing to repeatability, and therefore do not need to be evaluated as a separate component of uncertainty. Experience and judgement often play a role in such evaluations.

One method of establishing both repeatability and reproducibility in a single test is a ‘Gage Repeatability and Reproducibility (Gage R&R) Analysis of Variance (ANOVA)’. Using this method several different parts are measured by a number of different people but each part is usually only measured 2 or 3 times by each person. The order of the parts is randomized. The ANOVA statistical analysis is then used to separate out the variation in the results which is due to three sources; the actual component variation; the repeatability of the measurement system; and the reproducibility of results between different people. Gage R&R is a very good way to establish the repeatability and reproducibility components of measurement uncertainty but effort spent on this should not be seen as a substitute for evaluating other sources of uncertainty such as calibration and environment.

Resolution

Uncertainty of measurement due to resolution is a result of rounding errors. For many digital instruments the readout resolution is many times smaller than the actual instrument uncertainty. In such cases rounding errors due the instrument resolution are insignificant.

For more traditional instruments resolution is often a significant source of uncertainty. The maximum possible error due to rounding is half of the resolution. For example when measuring with a ruler which has a resolution of 1 mm the rounding error will be +/- 0.5 mm which has a rectangular distribution. Converting a tolerance with a rectangular distribution into a standard uncertainty is covered later.

Rounding Errors Can be up to Half of the Resolution of an Instrument

Rounding Errors Can be up to Half of the Resolution of an Instrument

Temperature

Temperature variations effect measurements in a number of ways:-

  • Thermal expansion of the object being measured and of the instrument used to measure it
  • For interferometric measurements changes in the refractive index
  • For optical measurements which depend on light following a straight line path temperature gradients will cause refraction leading to bending of the light and therefore distortions

Calibration

Any errors in the reference standard used to calibrate a measurement instrument are transferred during calibration. Instruments therefore inherit uncertainty from their calibration standard. The actual process of calibration is also not perfectly repeatable; therefore additional uncertainty is introduced through the calibration process.

If calibration has been carried out by an accredited calibration lab then an uncertainty will be given on the calibration certificate. This is not the uncertainty for measurements made using the instrument; it is simply the component of uncertainty due to calibration. This point is often overlooked.

When carrying out a calibration a complete uncertainty evaluation must be carried out for the calibration process. The combined uncertainty for the calibration then becomes a component of uncertainty for measurements taken using the instrument.

Combining individual sources of Uncertainty of Measurement

Once the individual sources for the uncertainty of a measurement have been identified and quantified a combined uncertainty should be calculated. This is the actual uncertainty of measurement for the process being considered.

Combining uncertainties requires some basic statistics, for example an understanding of standard deviations, normal distributions and the central limit theorem. The central limit theorem states that a number of different independent distributions, each of which is not normally distributed, combine to give a normal distribution. This can be demonstrated by rolling dice. The outcome for rolls of a single die is a rectangular distribution between 1 and 6. For two dice the results vary between 2 and 12 but the distribution is triangular and as the number of dice increases the distribution tends towards a normal distribution. Combined uncertainty is therefore generally assumed to be normally distributed even when many of the components of uncertainty are not.

The Central Limit Theorem – One Die Gives and Equal Chance of a Number Between 1 and 6 but the Sum of 2 Dice is a Triangular Distribution between 2 and 12. More Dice tend towards a Normal Distribution

The Central Limit Theorem – One Die Gives and Equal Chance of a Number Between 1 and 6 but the Sum of 2 Dice is a Triangular Distribution between 2 and 12. More Dice tend towards a Normal Distribution

As an introduction to the process of combining uncertainty we will first assume that all sources are normally distributed. Considering the measurement of a bolt, let’s assume that there are just two sources of uncertainty, the thermal expansion of the bolt and the uncertainty of the calliper measurement itself. The measurement result (y) will therefore be given by

y=X+ΔxT+ΔxC
where x is the true length, ΔxT is the error due to thermal expansion and ΔxC is the error due to the caliper measurement itself.

A Simple Example of Combining Uncertainty for the Measurement of a Bolt using Callipers

A Simple Example of Combining Uncertainty for the Measurement of a Bolt using Callipers

The error due to each source is not known; each error source has an uncertainty which defines the range of values which we might expect it to take. The probability that both errors will be maximal or minimal at the same time is very small. Therefore to simply add up the uncertainties would be overly pessimistic. Instead the component uncertainties are combined statistically to give a combined uncertainty.

The measurement result is given by y=f(x) where x1, x2 etc are inputs such as the true length and the various errors. Each input has an associated uncertainty u(xi). The combined uncertainty is then given by:-

 U_{c} ^{} \left(y\right)=\sqrt{\sum _{i=1}^{N}\left[\frac{\partial f}{\partial x_{i} } \right]^{2} u^{2} \left(x_{i} \right) }

For simple cases, such as our example of the callipers measuring the bolt, where
y = x1 + x22 … xn
The partial derivatives will all be equal to one so that

 \begin{array}{l} {\frac{\partial f}{\partial x_{i} } =1\quad \Rightarrow } \\ {U_{c} ^{} \left(y\right)=\sqrt{\sum _{i=1}^{N}\left[\frac{\partial f}{\partial x_{i} } \right]^{2} u^{2} \left(x_{i} \right) } =\sqrt{\sum _{i=1}^{N}u^{2} \left(x_{i} \right) } } \end{array}

Applying this to the example

y=X+ΔxT+ΔxC

The true length (X) has no uncertainty, leaving two sources of uncertainty; the uncertainty in the error due to thermal expansion u(ΔxT); and the uncertainty in the error due to the calliper measurement itself u(ΔxC). The combined uncertainty is therefore simply

U_{c} ^{} \left(y\right)=\sqrt{u^{2} \left(\Delta x_{T} \right)+u^{2} \left(\Delta x_{C} \right)}

This information can be used to create a simple uncertainty budget. First the standard uncertainty for each source of uncertainty is estimated.

 

A table listing the standard uncertainty for each source of uncertainty of measurement

In this simple uncertainty budget for a calliper measurement each source of uncertainty is first estimated

There is a simple functional relationship where the errors are simply added to the true value to give the measurement result. The combined uncertainty is therefore simply the square root of the sum of each component uncertainty squared (RSS). The combined uncertainty is multiplied by a coverage factor to give the uncertainty at a required confidence level (the expanded uncertainty).

The simple uncertainty budget is completed by calculating the combined standard uncertainty and expanded uncertainty

The simple uncertainty budget is completed by calculating the combined standard uncertainty and expanded uncertainty

This example made a lot of assumptions and simplifications in order to introduce the concepts. A full explanation of how to calculate an uncertainty budget is also provided.

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