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Uncertainty & Confidence in Measurements

An Introduction for Beginners

This is one of the most challenging aspects of calibration -- but are we sure??? If you are alsouncertain (or lack confidence!) about some of the special language used in metrology, you maywish to review our terminology guide before continuing with this more detailed explanation.

Terminology Guide

But before we scare ourselves with the mathematics involved, let's consider the fundamentalquestions... who cares about uncertainty and why?

Significance of Uncertainty to User and Calibration Lab

Specifications define any product's performance capability so that its adequacy for a certain taskmay be determined. In order to have confidence that outgoing product meets its specification,good practice is for the "standard" to be several times more accurate than the item being tested.A rule-of-thumb for this so-called test accuracy ratio (TAR and also known as test uncertaintyratio, TUR) is for it to be greater or equal to 4:1 (as indicated in ANSI/NCSL-Z540).

Where a measurement involves more than simple comparison which means that the overallaccuracy of the test is less evident, perhaps because several items of test equipment areinvolved or environmental factors (including test method) influence the result, an uncertaintybudget should be developed. This is also termed an error budget but this is not encouragedsince, by definition, errors are known and can usually be taken into account by correctingmeasured values, whereas uncertainty merely defines the limits of potential inaccuracy.

Until 1993 and the publication of the ISO Guide to Expression of Uncertainty in Measurement,there was no international consensus on the method for calculating uncertainty. This is also areason that manufacturers' specifications lack consistent definition. Of course, compliance isn'tmandatory but at least standardization may be encouraged by it. A statistical approach isrecommended, including the combination of contributions by quadratic summation and reportingthe final value at a 95% confidence level with Gaussian distribution. Where the confidence levelor distribution of a contributor is unknown, such as is often the case with instrumentspecifications, the "worst-case" rectangular distribution form is assumed and the equivalent 95%confidence, normal distribution error-limit calculated. The attributes of the item under test mustalso be considered in the budget.

Uncertainty, Test Limits & Risk

It is also common industry practice to use the specification as the acceptance (or test) limit whendeciding compliance of the tested item. Except that uncertainty shall be taken into account whenthe TAR falls below the prescribed minimum, the referenced standards do not stipulate how itshould be done.

Acceptance of this practice establishes a maximum consumer risk for the tested item beingincorrectly determined as within tolerance. Assuming both the specification and uncertainty haveGaussian distribution at 95% confidence and that the TAR is 4:1, a test result at the specificationlimit means that there would be a 0.8% chance that it was, in reality, out-of-tolerance. There is anassociated chance of conforming product being falsely declared non-compliant, resulting inunnecessary corrective action. In this example, the producer risk is 1.5%. As the TAR reduces,these risks increase but by setting a test limit which is tighter than the specification, thepossibility of incorrect acceptance can be maintained at the same level as the de facto standard4:1. Rather complex mathematics is necessary to determine the actual test limit for a particularTAR but this guardbanding provides a mechanism to comply with the standards' need to accountfor uncertainty, thus protecting the customer from undue risk of unknowingly usingnonconforming product, while limiting the supplier's commercial exposure.

An alternative method, often demanded under accreditation schemes based on ISO/IEC17025,particularly in Europe, is to guardband to the full extent of the uncertainty, whatever the TAR.That is:-

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Agilent | Uncertainty for Beginners http://www.agilent.com/metrology/uncert.shtml

1 sur 2 13/01/2012 07:29

Test Limit = Specification - Uncertainty

This provides attractive minimal risk to the consumer, for instance 0.02% at 4:1 and only 0.03%at 2:1, but significant producer risk of 10% at 4:1 (33% at 2:1). The economic impact to thesupplier (and inevitably the customer) may therefore outweigh the benefit of this practice.Opponents argue that such a conservative approach is unnecessarily pessimistic and isinconsistent with the established statistical basis for error (uncertainty) propagation. To explain,the specification will contribute to the user's uncertainty budget by quadratic summation and,therefore, the setting of the acceptance limit by simple arithmetic is inappropriate. The calculationshould be a quadratic difference:-

Test Limit = Squareroot [Spec 2 - Uncertainty 2]

This provides a fairly constant chance of under 0.7% of false acceptance for TARs from 4:1 to1.5:1 although the chance of incorrect rejection is 2% at 4:1 rising to 8.2% at 2:1.

This latter method is useful because of its simplicity in application, generally acceptableconsumer-risk and commercial viability. It's argued that a statement on a calibration certificate,such as follows, could succinctly address the uncertainty and measurement adequacy criteria ofstandards such as ISO9001 and ANSI-Z540.

"Our calibration procedures are designed to provide a measurement uncertainty of less than aquarter of the specification of the unit-under-test. In these conditions and at 95% confidence levelfor specification and uncertainty, the chance of incorrect declaration of conformance tospecification is 0.8%. Where the objective cannot be achieved, tightened test limits are used tomaintain equivalent confidence in the product's compliance to specification."

Agilent | Uncertainty for Beginners http://www.agilent.com/metrology/uncert.shtml

2 sur 2 13/01/2012 07:29