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LA. 1National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
TRUTH IN MEASUREMENT
LA. 2National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q The concept of uncertainty of measurement is not new
q It has been used in calibration laboratories long before the ISO GUM
q It has been normal practice whenever a value is reported an uncertainty is quoted
qWhat is new is the formalising of the process with the ISO GUM and………
q the requirements in ISO 17025
LA. 3National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
AS ISO/IEC 17025
5.4.6.1 A calibration laboratory, or a testing laboratory, performing its own calibrationsshall have and shall apply a procedure to estimate the uncertainty of measurement for allcalibrations and types of calibrations
(Ex 1)
LA. 4National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
I USED TO BE UNCERTAIN
BUT
NOW I’M NOT SURE
LA. 5National Association of Testing Authorities, Australia - Laboratory
q What is uncertainty?
q Does it mean we don’t really know?
q Does it mean we should not report our result?
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
LA. 6National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
NO!Most definitely not
LA. 7National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q Uncertainty is a component of our measurement.
q It is a component that can be analysed and quantified.
q When quantified it provides confidence in our result.
LA. 8National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
HOW DO WE DETERMINE OUR UNCERTAINTY?
LA. 9National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
The easiest way is by creating an
UNCERTAINTY BUDGET
q What is an uncertainty Budget?
LA. 10National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q An uncertainty budget is a convenient means of itemising, tabulating and calculating details of uncertainty.
q How does it work?
LA. 11National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q Uncertainty analysis is simply a means of focussing attention on the individual components that may affect the final result.
q The uncertainty budget is a means of capturing this information in logical steps.
q Such as ……..
LA. 12National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q Step 1: list the components that may affect our result.
q Step 2: determine the type of distribution.
q Step 3: determine the value of the semi range to be used.
q Step 4: decide on our level of confidence in the value we are using.
(Ex 2)
LA. 13National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
0-25mm External MicrometerSource Units
Ref. gauge block tol. µmRef. gauge block uncert. µmAnvil geometry µmTherm effects °CResolution/parallax(2), (3) µmRepeat./random effects(2), (3) µm
TYPICAL SOURCE COMPONENTS
LA. 14National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
DISTRIBUTION TYPES
LA. 15National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
LA. 16National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
LA. 17National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
Source Units Dist.Ref. gauge block tol. µm Rect BRef. gauge block uncert. µm Norm BAnvil geometry µm Rect BTherm effects °C Rect BResolution/parallax (2), (3) µm Rect BRepeat./random effects(2), (3) µm Norm A
DISTRIBUTION EXAMPLES
(Ex 3)
LA. 18National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
q A Type B uncertainty is one which is evaluated by other than statistical methods.
UNCERTAINTY TYPES
LA. 19National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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 rootof the number of readings to obtain the ESDM and standard uncertainty.
DETERMINING THE SEMI RANGEAND STANDARD UNCERTAINTY
LA. 20National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q If only a small number of readings are taken during the calibration then a pre-characterisationas shown in the ISO GUM in example H.1.3.2 may be appropriate.
q For example……..
LA. 21National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q The pooled experimental standard deviation characterising a comparison was determined from the
variability of 25 (n1) independent repeated observations and was found to be 13µm (s). In comparison with this example 5 (n2) repeated observations were taken.
q The standard uncertainty (u) is then……..
LA. 22National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
or = 5.8µm
and the degrees of freedom are based on the number of repeated observations.
i.e. 25-1 = 24
513µm
ui =
11 −= nVi
2ns
ui =
LA. 23National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q Normal B Distribution - The expanded uncertainty given on a calibration certificate is divided by the coverage factor (k) to obtain the standard uncertainty.
kU
U c95=
LA. 24National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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)
LA. 25National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
= 1.7µm3005.0
=cU
LA. 26National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
LA. 27National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q The ISO GUM advises that:
“There is no substitute for critical thinking, intellectual honesty, and professional skill”
LA. 28National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q For the calculation of the standard uncertainty u for a rectangular distribution the formula is:
q Where:
u is the standard uncertainty. a is the semi range of the limits of the
uncertainty component.
3a
u =
LA. 29National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
STANDARD UNCERTAINTYRevision
Normal Distribution: (ESDM)
Rectangular Distribution:
Triangular Distribution:
ns
ui =
3a
ui =
6a
ui =
(Ex 4)
LA. 30National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q Type A Uncertainties - The number of degrees of freedom for each Type A uncertainty is generally one less than the number of readings.
Vi = n-1
DEGREES OF FREEDOM AND CONFIDENCE
LA. 31National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
LA. 32National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q The determination of the number of degrees of freedom for each Rectangular Type B uncertainty can be calculated simply by:
Vi = 2)10( 2
= 50
LA. 33National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q Normal Type B Uncertainties - typically a calibration report, the confidence level and the kvalue 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.
LA. 34National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
Source Units Dist.ValueU or a Divisor Confidence % V i
Ref. gauge block tol. µm Rect B 0.30 1.7321 95 200Ref. gauge block uncert. µm Norm B 0.24 2.0000 95 30Anvil geometry µm Rect B 0.25 1.7321 95 200Therm effects °C Rect B 2.00 1.7321 80 12Resolution/parallax(2), (3) µm Rect B 1.00 1.7321 95 200Repeat./random effects(2), (3) µm Norm A 0.12 1.7321 95 9
(Ex 5)
LA. 35National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
SENSITIVITY COEFFICIENTS
LA. 36National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q The expanded uncertainty (U)is the final estimate for the uncertainty
EXPANDED UNCERTAINTY
LA. 37National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
q 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.
q Spreadsheet……..
CONVERSION BETWEEN CONFIDENCELEVELS
LA. 38National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
UNCERTAINTY BUDGET
Item: 0 to 25 mm external micrometer Date:Serial Number:
Maximum length (mm) = 25 Coeffecient of expansion (ppm / °C) = 11.5
Source Units Dist.ValueU or a Divisor
Confidence % V i u i c i u i c i (u i c i )̂ 2 [(u i c i )^4]/V i
Ref. gauge block tol. µm Rect B 0.30 1.7321 95 200 0.1732 1 0.1732 0.0300 4.5000E-06Ref. gauge block uncert. µm Norm B 0.24 2.0 95 30 0.1200 1 0.1200 0.0144 6.9120E-06Anvil geometry µm Rect B 0.25 1.7321 95 200 0.1443 1 0.1443 0.0208 2.1701E-06Therm effects °C Rect B 2.00 1.7321 80 12 1.1547 0.2875 0.3320 0.1102 1.0122E-03Resolution/parallax (2), (3) µm Rect B 1.00 1.7321 95 200 0.5774 1 0.5774 0.3333 5.5556E-04Repeat./random effects(2), (3) µm Norm A 0.12 1.7321 95 9 0.0693 1 0.0693 0.0048 2.5600E-06
Sums 0.5136 1.5839E-03
Combined Standard Uncertainty, U c 0.7166
Effective Degrees of Freedom, V eff 166.5300
Coverage factor, k = Student's t for V eff and CL 95% 1.9744
Expanded Uncertainty, U=kuc +/- 1.4149
LA. 39National Association of Testing Authorities, Australia - Laboratory
THE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTSTHE UNCERTAINTY OF MEASUREMENTS
EXTRACT FROM THE GUM
3.4.8. Although this Guide provides a framework for assessing uncertainty, it cannotsubstitute for critical thinking, intellectual honesty, and professional skill.
The evaluation of uncertainty is neither a routine task nor a purely mathematical one; itdepends on detailed knowledge of the nature of the measureand and of the measurement.
The quality and utility of the uncertainty quoted for the result of a measurement thereforeultimately depend on the understanding, critical analysis, and integrity of those who contribute to the assignment of its value.
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