THE UNCERTAINTY OF MEASUREMENTS TRUTH IN MEASUREMENT

Transcription

1 THE UNCERTAINTY OF MEASUREMENTS TRUTH IN MEASUREMENT National Association of Testing Authorities, Australia - Laboratory LA. 1

2 The concept of uncertainty of measurement is not new It has been used in calibration laboratories long before the ISO GUM It has been normal practice whenever a value is reported an uncertainty is quot What is new is the formalising of the process with the ISO GUM and the requirements in ISO 17025

3 AS ISO/IEC A calibration laboratory, or a testing laboratory, performing its own calibrations all have and shall apply a procedure to estimate the uncertainty of measurement for a librationsand types of calibrations Ex 1)

4 I USED TO BE UNCERTAIN BUT NOW I M NOT SURE

5 What is uncertainty? Does it mean we don t really know? Does it mean we should not report our result?

6 NO! Most definitely not

7 Uncertainty is a component of our measurement. It is a component that can be analysed and quantified. When quantified it provides confidence in our result.

8 HOW DO WE DETERMINE OUR UNCERTAINTY?

9 The easiest way is by creating an UNCERTAINTY BUDGET What is an uncertainty Budget?

10 An uncertainty budget is a convenient means of itemising, tabulating and calculating details of uncertainty. How does it work?

11 Uncertainty analysis is simply a means of focussing attention on the individual components that may affect the final result. The uncertainty budget is a means of capturing this information in logical steps. Such as..

12 Step 1: list the components that may affect our result. Step 2: determine the type of distribution. Step 3: determine the value of the semi range to be used. Step 4: decide on our level of confidence in the value we are using. (Ex 2)

13 TYPICAL SOURCE COMPONENTS 0-25mm External Micrometer Source Units Ref. gauge block tol. µm Ref. gauge block uncert. µm Anvil geometry µm Therm effects C Resolution/parallax (2), (3) µm Repeat./random effects (2), (3) µm

14 DISTRIBUTION TYPES A Normal Distribution - may be either Type A or a Type B uncertainty. A normal distribution may represent a random series of readings where the majority of the readings occur near the mean value. If these readings are sufficient in number and graphed, a bell shaped curve will result.

15 A Rectangular Distribution - is one in which the actual value may occur anywhere within the distribution with equal probability. It is important to define the limits of the distribution with a high degree of confidence.

16 A Triangular Distribution - is similar to the rectangular distribution with the difference being that there is a lower probability that the actual value will be at the limits of the range.

17 DISTRIBUTION EXAMPLES Source Units Dist. Ref. gauge block tol. µm Rect B Ref. gauge block uncert. µm Norm B Anvil geometry µm Rect B Therm effects C Rect B Resolution/parallax (2), (3) µm Rect B Repeat./random effects (2), (3) µm Norm A x 3)

18 UNCERTAINTY TYPES A Type A uncertainty is one which is evaluated by statistical means. This is generally only practical and meaningful with a reasonable number of repeated readings. A Type B uncertainty is one which is evaluated by other than statistical methods.

19 DETERMINING THE SEMI RANGE AND STANDARD UNCERTAINTY Normal A Distribution - The scatter of a number (n) of repeated measurements will be found to have a normal distribution. We use the population standard deviation (s) of these readings as the semi-range and divide by the square root of the number of readings to obtain the ESDM and standard uncertainty.

20 If only a small number of readings are taken during the calibration then a pre-characterisation as shown in the ISO GUM in example H may be appropriate. For example..

21 The pooled experimental standard deviation characterising a comparison was determined from th variability of 25 (n 1 ) independent repeated observations and was found to be 13µm (s). In comparison with this example 5 (n 2 ) repeated observations were taken. The standard uncertainty (u) is then..

22 u i = s n 2 or u i = 13µm 5 = 5.8µm and the degrees of freedom are based on the number of repeated observations. V i = n 1 1 i.e = 24

23 Normal B Distribution - The expanded uncertainty given on a calibration certificate is divided by the coverage factor (k) to obtain the standard uncertainty. U95 U c = k

24 If a 95% confidence level is quoted on the calibration certificate without a coverage factor then it may be assumed that the divisor is 1.96 or 2 standard deviations. i.e. k = 2 (rounded to 2)

25 If a standard deviation is quoted; for example an uncertainty of 5µm at the 3 standard deviation level, the standard uncertainty is obtained by dividing the quoted uncertainty by 3. U c = = 1.7µm

26 Rectangular B Distribution - The only information that we know about the distribution in this case will be the limits. These limits need to be selected critically with a high degree of confidence.

27 The ISO GUM advises that: There is no substitute for critical thinking, intellectual honesty, and professional skill

28 For the calculation of the standard uncertainty u for a rectangular distribution the formula is: a u = 3 Where: u is the standard uncertainty. a is the semi range of the limits of the uncertainty component.

29 STANDARD UNCERTAINTY Revision ormal Distribution: u i = s n (ESDM) ectangular Distribution: u i = a 3 riangular Distribution: u i = a 6 x 4)

30 DEGREES OF FREEDOM AND CONFIDENCE Type A Uncertainties - The number of degrees of freedom for each Type A uncertainty is generally one less than the number of readings. V i = n-1

31 RectangularType B Uncertainties - The number of degrees of freedom for each Type B uncertainty with a Rectangular Distribution is determined from the confidence in the limits. For example if the relative confidence level is 90% (a one in ten chance that the true value is outside the limits selected) then the degrees of freedom is 50.

32 The determination of the number of degrees of freedom for each Rectangular Type B uncertainty can be calculated simply by: V i = 2 (10) 2 = 50

33 Normal Type B Uncertainties - typically a calibration report, the confidence level and the k value may be provided. Referring to the Students t Distribution tables, the approximate degrees of freedom can be determined. If a k value is not provided then an infinite number of degrees of freedom may be assumed.

34 Source Units Dist. Value U or a Divisor Confidence % V i Ref. gauge block tol. µm Rect B Ref. gauge block uncert. µm Norm B Anvil geometry µm Rect B Therm effects C Rect B Resolution/parallax (2), (3) µm Rect B Repeat./random effects (2), (3) µm Norm A Ex 5)

35 SENSITIVITY COEFFICIENTS The sensitivity coefficient (c) describes how the output estimate varies with changes to the value of the input estimates. For example converting temperature in C to a common unit with other components of the budget.

36 EXPANDED UNCERTAINTY The expanded uncertainty (U)is the final estimate for the uncertainty

37 CONVERSION BETWEEN CONFIDENCE LEVELS For a normally distributed confidence level of 99% dividing the expanded uncertainty by 2.6 (k = an infinite number of degrees of freedom) will provide an approximation of the combined standard uncertainty. Spreadsheet..

38 UNCERTAINTY BUDGET l Number: 0 to 25 mm external micrometer Date: Maximum length (mm) = 25 Coeffecient of expansion (ppm / C) = 11.5 Source Units Dist. Value U or a Divisor Confidence % V i u i c i u i c i (u i c i )^2 [(u i c i )^4 gauge block tol. µm Rect B E gauge block uncert. µm Norm B E geometry µm Rect B E effects C Rect B E lution/parallax (2), (3) µm Rect B E at./random effects (2), µm Norm A E Sums E Combined Standard Uncertainty, U c Effective Degrees of Freedom, V eff Coverage factor, k = Student's t for V eff and CL 95% Expanded Uncertainty, U=ku c +/

39 EXTRACT FROM THE GUM.8. Although this Guide provides a framework for assessing uncertainty, it cannot bstitute for critical thinking, intellectual honesty, and professional skill. e evaluation of uncertainty is neither a routine task nor a purely mathematical one; it pends on detailed knowledge of the nature of the measureand and of the measuremen e quality and utility of the uncertainty quoted for the result of a measurement therefo imately depend on the understanding, critical analysis, and integrity of those who ntribute to the assignment of its value ooooooo-----