Exhibitions

Whenever measurement is made it is necessary to qualify how good the measurement is in order to understand if the component being measured is in specification. In this article Dr Mike Mills - Chief Metrologist - looks at some fundamental issues regarding measurement uncertainties.

Why is uncertainty important?

These days more and more items are made from components that are themselves manufactured in different locations - sometimes even in different continents. It is important to know that each component of an assembly is of acceptable quality and dimension so that the whole assembly will be correct. This is normally controlled by applying tolerances to each component. In order to determine if the component is in specification, it is necessary to measure the component using an appropriate technique with a suitably small measurement uncertainty.

For example suppose that we measure the length of an object as 10.1mm and it is specified that the length must lie between 9.7mm and 10.2mm. At first sight it would seem that the component is within specification, but how certain are we that this is the case? Without knowing the uncertainty in the measurement we simply cannot be sure that the component is within specification. In this particular example, if the measurement "uncertainty" is less than 0.1mm then the component is acceptable. If the measurement uncertainty is greater than 0.1mm, then we cannot be so confident that this is a good component.

Figure 1 - Measurement with two different uncertainties.

This situation is shown in Figure 1 where two different uncertainty distributions are shown. In the case of the green line, the measurement is almost certainly within specification and the component should therefore be accepted. However, in the case of the red line, it is clear that there is a possibility that the measurement is outside of tolerance. In this situation it is questionable as to whether or not the component should be accepted. The course of action to be taken in this latter situation will depend on a number of factors. These factors include the cost of the component and whether there is a prior arrangement between the "supplier" and the "customer" (note that these may be roles within a single company or, in the case of sub-contract manufacture, may be separate entities). For a more detailed discussion of how to deal with this situation, the reader is referred to ISO 14253.

Estimating the Measurement Uncertainty-

It is easy to see how important it is to understand the uncertainty in a given measurement, especially where sub-contract manufacture is involved. But how do we get an estimate of that uncertainty? The usual method of arriving at this estimate is through a process known as "uncertainty budgeting". In this process each factor contributing to the measurement uncertainty is listed in a table. The likely spread in a result due to each factor is tabulated and these can be combined in some way to produce the overall uncertainty estimate. The most common way of combining these uncertainty contributions is through the simplified technique outlined in the ISO document known as "GUM" - Guide to the Expression of Uncertainty in Measurement. This method aims to model the overall uncertainty in terms of a normal distribution.

Each contribution is weighted by a factor that relates to its own probability density function. For simplicity these are normally limited to one of three distributions: normal, rectangular or U-shaped. The purpose of this weighting is to replace the actual probability function by an "equivalent" normal distribution. These weighted uncertainties are then combined using a "root-sum-of-squares" method. The result is an estimate of the standard deviation of the overall effective normal distribution. To obtain the overall uncertainty, this figure is usually multiplied by a "coverage factor", which by default is 2. For a normal distribution the interval contained in two standard deviations either side of the mean corresponds to about 95% of all values.

One major benefit of uncertainty budgeting is that it highlights the most dominant contributors to the measurement uncertainty. These can then be targeted to reduce the overall uncertainty in a cost-effective manner. There are other methods of producing an uncertainty budget. One that is well worth considering is the Monte Carlo analysis of uncertainty propagation. This more accurately models how the individual uncertainties, with their individual distributions, will combine together.

Summary:
The importance of knowing the uncertainty in a measurement has been demonstrated. Without an estimate of the measurement uncertainty it is not possible to confirm whether or not a measurement is within specification. A method for estimating measurement uncertainty has been introduced. The reader is referred to the references for a more detailed discussion of these topics.

More Information

References:
1. ISO 14253-1:1998. Parts 2 & 3 are also of interest.
2. ISO Guide to the Expression of uncertainty in Measurement, 1993.

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