Contents Introduction GUM Basic Concepts Basic Statistics Evaluation of Measurement Uncertainty How is Measurement Uncertainty estimated? Reporting Result Conclusions and Remarks
Introduction Guide to the Expression of Uncertainty in Measurement was published by the International Organization for Standardization in 1993 in the name of 7 international organizations Corrected and reprinted in 1995 Usually referred to simply as the “GUM”
International Organisations Guide to the Expression of Uncertainty in Measurement (1993) BIPM - International Bureau of Weights and Measures http//: www.bipm.org IEC - International Electrotechnical Commision http//: www.iec.ch IFCC - International Federation of Clinical Chemistry http//: www.ifcc.org IUPAP - International Union of Pure and Applied Physics http//: www.iupap.org ISO - International Organisation for Standardisation http//: www.iso.ch IUPAC - International Union of Pure and Applied Chemistry http//: www.iupac.org OIML - International Organisation for legal metrology http//: www.oiml.org
Every measurement is subject to some uncertainty. A measurement result is incomplete without a statement of the uncertainty. When you know the uncertainty in a measurement, then you can judge its fitness for purpose. Understanding measurement uncertainty is the first step to reducing it Basic concepts
Introduction to GUM When reporting the result of a measurement of a physical quantity, it is obligatory that some quantitative indication of the quality of the result be given so that those who use it can assess its reliability. Without such an indication, measurement results can not be compared, either among themselves or with reference values given in the specification or standard. GUM 0.1
Stated Purposes Promote full information on how uncertainty statements are arrived at Provide a basis for the international comparison of measurement results
Benefits Much flexibility in the guidance Provides a conceptual framework for evaluating and expressing uncertainty Promotes the use of standard terminology and notation All of us can speak and write the same language when we discuss uncertainty
Uses of MU QC & QA in production Law enforcement and regulations Basic and applied research Calibration to achieve traceability to national standards Developing, maintaining, and comparing international and national reference standards and reference materials –GUM 1.1
R1R2 After uncertainty evaluation R1R2 No uncertainty evaluation (only precision) R1R2 10.5 11.5 11.0 12.0 12.5 mg kg -1 value Are these results different?
What is Measurement? Measurement is ‘Set of operations having the object of determining a value of a quantity.’ ( VIM 2.1 ) Note: The operations may be performed automatically.
Basic concepts Measurement –the objective of a measurement is to determine the value of the measurand, that is, the value of the particular quantity to be measured a measurement therefore begins with –an appropriate specification of the measurand –the method of measurement and –the measurement procedure GUM 3.1.1
Principles of Measurement Method of Comparison ResultDUT Standard
Basic concepts Result of a measurement –is only an estimate of a true value and only complete when accompanied by a statement of uncertainty. –is determined on the basis of series of observations obtained under repeatability conditions Variations in repeated observations are assumed to arise because influence quantities GUM 3.1.2 GUM 3.1.4 Gum 3.1.5
Influence quantity Quantity that is not the measurand but that affects the result of measurement. Example : temperature of a micrometer used to measure length. ( VIM 2.7 )
What is Measurement Uncertainty? “parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand” – GUM, VIM Examples: –A standard deviation (1 sigma) or a multiple of it (e.g., 2 or 3 sigma) –The half-width of an interval having a stated level of confidence
The uncertainty gives the limits of the range in which the “true” value of the measurand is estimated to be at a given probability.. Measurement result = Estimate ± uncertainty (22.7 ± 0.5) mg/kg The value is between 22.2 mg/kg and 23.2 mg/kg Uncertainty
Measurement Error Measured ValueTrue Value Real Number System Measurement Error Measured values are inexact observations of a true value. The difference between a measured value and a true value is known as the measurement error or observation error.
Basic concepts The error in a measurement – Measured value – True value. –This is not known because: The true value for the measurand –This is not known –The result is only an estimate of a true value and only complete when accompanied by a statement of uncertainty. GUM 2.2.4 GUM 3.2.1
Random & Systematic Errors Error can be decomposed into random and systematic parts The random error varies when a measurement is repeated under the same conditions The systematic error remains fixed when the measurement is repeated under the same conditions
Random error Result of a measurement minus the mean result of a large number of repeated measurement of the same measurand. ( VIM 3.13 )
Random Errors Random errors result from the fluctuations in observations Random errors may be positive or negative The average bias approaches 0 as more measurements are taken
Random error Presumably arises from unpredictable temporal and spatial variations gives rise to variations in repeated observations Cannot be eliminated, only reduced. GUM 3.2.2
Systematic Errors Mean result of a large number of repeated measurements of the same measurand minus a true value of the measurand. ( VIM 3.14 )
Systematic Errors A systematic error is a consistent deviation in a measurement A systematic error is also called a bias or an offset Systematic errors have the same sign and magnitude when repeated measurements are made under the same conditions Statistical analysis is generally not useful, but rather corrections must be made based on experimental conditions.
Systematic error If a systematic error arises from a recognized effect of an influence quantity –the effect can be quantified –can not be eliminated, only reduced. –if significant in size relative to required accuracy, a correction or correction factor can be applied to compensate –then it is assumed that systematic error is zero. GUM 3.2.3
Basic concepts Systematic error It is assumed that the result of a measurement has been corrected for all recognised significant systematic effects GUM 3.2.4
Measurement Error Systematic error Random error
Correcting for Systematic Error If you know that a substantial systematic error exists and you can estimate its value, include a correction (additive) or correction factor (multiplicative) in the model to account for it Correction - Value that, added algebraically to the uncorrected result of a measurement, compensates for an assumed systematic error (VIM 3.15) Correction Factor - numerical factor by which the uncorrected result of a measurement is multiplied to compensate for systematic error. [VIM 3.16]
Uncertainty The result of a measurement after correction for recognized systematic effects is still only an estimate of the value of the measurand because of the uncertainty arising; – from random effects and –from imperfect correction of the result for systematic effects GUM 3.3.1
Classification of effects and uncertainties Random effects Unpredictable variations of influence quantities Lead to variations in repeated measurements Expected value : 0 Can be reduced by making many measurement Systematic effects Recognized variations of influence quantities Lead to BIAS in repeated measurements Expected value : unknown Can be reduced by applying a correction which carries an uncertainty bunjob_ajchara33
It is important not to confuse the terms error and uncertainty ErrorError is the difference between the measured value and the “ true value ” of the thing being measured UncertaintyUncertainty is a quantification of the doubt about the measurement result In principle errors can be known and corrected But any error whose value we do not know is a source of uncertainty. Error versus uncertainty
Blunders Blunders in recording or analysing data can introduce a significant unknown error in the result of a measurement. Measures of uncertainty are not intended to account for such mistakes GUM 3.4.7
Slide 7 Population and Sample Parent Population The set of all possible measurements. Sample A subset of the population - measurements actually made. Population Bag of Marbles Handful of marbles from the bag Samples
Histograms Histograms When making many measurements, there is often variation between readings. Histogram plots give a visual interpretation of all measurements at once. The x-axis displays a given measurement and the height of each bar gives the number of measurements within the given region. Histograms indicate the variability of the data and are useful for determining if a measurement falls outside of “ specification ”.
For a large number of experiment replicates the results approach an ideal smooth curve called the GAUSSIAN or NORMAL DISTRIBUTION CURVE Characterised by: The mean value – x gives the center of the distribution The standard deviation – s measures the width of the distribution
Average The most basic statistical tool to analyze a series of measurements is the average or mean value : “Sum of” Individual measurement Number of measurements The average of the three values 10, 15and 12.5 is given by:
Deviation Need to calculate an average or “ standard ” deviation To eliminate the possibility of a zero deviation, we square d i Deviation = individual value – avg value
Standard Deviation The average amount that each measurement deviates from the average is called standard deviation (s) and is calculated for a small number of measurements as: x i = each measurement = average n = number of measurements Note this is called root mean square: square root of the mean of the squares Sum of deviation squared
For example, calculate the standard deviation of the following measurements: 10, 15 and 12.5 (avg = 12.5) 10.012.5 15.0 The values deviate on average plus or minus 2.5 :12.5 ± 2.5
Variance Relative standard deviation Percent RSD or Coefficient of Variation (CV) Other ways of expressing the precision of the data: Variance = s 2
Standard Deviation of the Mean The uncertainty in the best measurement is given by the standard deviation of the mean (SDOM)
Gaussian Distribution Given a set of repeated measurements which have random error. For the set of measurements there is a mean value. If the deviation from the mean for all the measurements follows a Gaussian probability distribution, they will form a “ bell-curve ” centered on the mean value. Sets of data which follow this distribution are said to have a normal (statistical) distribution of random data.
POPULATION DATA For an infinite set of data, n → ∞x → µ s → σ n → ∞ x → µ and s → σ population mean population std. dev. The experiment that produces a small standard deviation is more precise. Remember, greater precision does not imply greater accuracy. Experimental results are commonly expressed in the form: mean standard deviation
The Gaussian curve whose area is unity is called a normal error curve. µ = 0 and σ = 1 The Gaussian curve equation: = Normalisation factor It guarantees that the area under the curve is unity
+3 -3 +2 -2 +1 -1 Normal Error Curve 68.3% of measurements will fall within ± of the mean. xixi Relative frequency, dN / N 95.5% of measurements will fall within ± 2 of the mean. 99.7% of measurements will fall within ± 3 of the mean.
EXAMPLE Replicate results were obtained for the measurement of a resistor. Calculate the mean and the standard deviation of this set of data. Replicateohms 1752 2756 3752 4751 5760
Replicateohms 1752 2756 3752 4751 5760 NB DON ’ T round a std dev. calc until the very end.
Student's t-Distribution If the sample size is not large enough, say n ≤ 30. Then the distribution of is not normal. It has a distribution called Student’s t- distribution. t = ( x – )/(s/ n).
Student's t-Distribution The Student's t-distribution was discovered by W. S. Gosset in 1908. He used the pseudonym ‘Student’ to avoid getting fired for doing statistics on the job!!
Student's t-Distribution The shape of the Student's t-distribution is very similar to the shape of the standard normal distribution. The Student's t-distribution has a (slightly) different shape for each possible sample size. They are all symmetric and unimodal. They are all centered at 0.
Student's t-Distribution They are somewhat broader than normal distribution, reflecting the additional uncertainty resulting from using s in place of . As n gets larger and larger, the shape of the t-distribution approaches the standard normal.
Degrees of Freedom If the sample size is n, then t is said to have n – 1 degrees of freedom. We use df to denote degrees of freedom.
Student's t-Distribution for 95% Confident level
Student's t-Distribution When is estimated from the sample standard deviation, s The distribution for the mean follows a t- distribution with degrees of freedom, n − 1
CONFIDENCE INTERVAL The confidence interval is given by: Where t is the value of student ’ s t taken from the table The confidence interval is the expression stating that the true mean, µ, is likely to lie within a certain distance from the measured mean,
Degrees of Freedom 13.07776.31412.70631.82163.657 21.88562.92004.30276.96459.9250............ 101.37221.81252.22812.76383.1693............ 1001.29011.66041.98402.36422.6259 1.2821.64491.96002.32632.5758 0.80 0.90 0.950.980.99 Use of t-Table 95% confidence interval; n = 11
Example: The mercury content in fish samples were determined as follows: 1.80, 1.58, 1.64, 1.49 ppm Hg. Calculate the 50% and 90% confidence intervals for the mercury content. 50% confidence: t =0.765 for n-1 = 3 There is a 50% chance that the true mean lies between 1.58 and 1.68 ppm Hg. Find x = 1.63 s = 0.131
90% confidence: t = 2.353 for n-1 = 3 There is a 90% chance that the true mean lies between 1.48 and 1.78 ppm 1.63 1.68 1.48 1.58 1.78 90% 50%
Evaluation of Measurement Uncertainty bunjob_ajchara70
Terms specific to the GUM Standard uncertainty, – the uncertainty of the result of a measurement expressed as a standard deviation Type A evaluation (of uncertainty) – method of evaluation of uncertainty by the statistical analysis of a series of observations Type B evaluation (of uncertainty) – method of evaluation of uncertainty by means other than the statistical analysis of series of observations GUM 2.3.1 GUM 2.3.2 GUM 3.2.3
Terms specific to the GUM Combined standard uncertainty –the standard deviation of the result of a measurement when the result is obtained from the values of a number of other quantities. –It is obtained by combining the individual standard uncertainties (and covariances as appropriate), using the law of propagation of uncertainties, commonly called the "root-sum-of-squares" or "RSS method. GUM 2.3.4
Terms specific to the GUM expanded uncertainty –quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand. coverage factor, k –numerical factor used as a multiplier of combined standard uncertainty in order to obtain expanded uncertainty GUM 3.2.5 GUM 3.2.6
Process of Uncertainty Estimation Specify Measurand Identify all Uncertainty Sources Quantify Uncertainty Components Calculate Combined Uncertainty
The measurand? GUM 1.2 Measurand = particular quantity subject to measurement [VIM 2.6 / GUM B.2.9] Example: the conventional mass of a 1kg weight.
Measurement Model Define the measurand – the quantity subject to measurement Determine a mathematical model, with input quantities, X 1,X 2,…,X N, and (at least) one output quantity,Y. The values determined for the input quantities are called input estimates and are denoted by x 1,x 2,…,x N. The value calculated for the output quantity is called the output estimate and denoted by y.
78 Identify all Uncertainty Sources 2. How is MU estimated?
ISO/IEC 17025 188.8.131.52 –attempt to identify all the components of uncertainty 184.108.40.206 –All uncertainty components which are of importance shall be taken into account
Sources of uncertainty ISO/IEC 17025 220.127.116.11 Note 1: Some sources contributing to the uncertainty: – reference standards – reference materials – methods – equipment – environmental conditions – properties and condition of the item to be tested – the operator
81 Sources of MU Incomplete definition of the measurand Imperfect realisation of the definition of the measurand Non-representative sampling Effects of environmental conditions on the measurement Personal bias in reading analogue instruments Finite instrument resolution or discrimination threshold Inexact values of measurement standards Inexact values of constants obtained from external sources Approximations incorporated into the measurement Variations in repeated observations under apparently identical conditions GUM 3.3.2 2. How is MU estimated?
Causes for uncertainty Measurement results Measuring instrument Measurement standard Measuring methods Measurer Number of measurements Measurement environment Calibration certificate Secular change Manufacturer’s specification Resolution Dispersions in repetition Peculiarities in readout
Sources of error and uncertainty in dimensional calibrations Reference standards and instrumentation Thermal effects Elastic compression Cosine errors Geometric errors UKAS M3003 Dec 1999 bunjob_ajchara83
Sources of error and uncertainty in electrical calibrations Instrument Calibration Secular Stability Measurement Conditions Interpolation of calibration data Resolution Layout of apparatus Thermal emfs Loading and lead impedance RF mismatch errors and uncertainty Directivity Test port match RF Connector repeatability UKAS M3003 Dec 1999 bunjob_ajchara84
Sources of error and uncertainty in mass calibrations bunjob_ajchara85 Reference weight calibration Secular stability of reference weights Weighing machine / weighing process Air buoyancy effects Environment UKAS M3003 Dec 1997
86 Quantify Uncertainty Components 2. How is MU estimated?
The Measurement Model Usually the final result of a measurement is not measured directly, but is calculated from other measured quantities through a functional relationship This is called function a “measurement model” The model might involve several equations, but we’ll follow the GUM and represent it abstractly as a single equation:
Input and Output Quantities In the generic model Y = f(X 1,…,X N ), the measurand is denoted by Y Also called the output quantity The quantities X 1,…,X N are called input quantities The value of the output quantity (measurand) is calculated from the values of the input quantities using the measurement model
Input and Output Estimates When one performs a measurement, one obtains estimated values x 1,x 2,…,x N for the input quantities X 1,X 2,…,X N These estimated values may be called input estimates The calculated value for the output quantity may be called an output estimate
Measurement model A measurand Y can be determined from N inputs quantities X1, X2, X3 … XN The model is written abstractly as Y=f(X1,X2,…,XN) where X1,X2,…,XN are input quantities and Y is the output quantity
Developing a Measurement model Decide what is the measurand Y –the quantity subject to measurement Decide what are the quantities X 1, …, X N influencing the measurement –observed quantities, applied corrections, material properties, etc Decide the relationship between Y and X 1, …, X N –the model of the measurement bunjob_ajchara91
Example: CALBRATION OF A HAND-HELD DIGITAL MULTIMETER AT 100 V DC The error of indication E X of the DMM to be calibrated is obtained from where V i X - voltage, indicated by the DMM (index i means indication), V S - voltage generated by the calibrator, δ V I X - correction of the indicated voltage due to the finite resolution of the DMM, δ V S - correction of the calibrator voltage due to (1) drift since its last calibration, (2) deviations resulting from the combined effect of offset, non-linearity and differences in gain, (3) deviations in the ambient temperature, (4) deviations in mains power, (5) loading effects resulting from the finite input resistance of the DMM to be calibrated. EA-4/02:1999
bunjob_ajchara93 An estimate of Y, denoted by y, is obtained from x 1, x 2, x 3 … x N, the estimates of the input quantities X 1, X 2, X 3 … X N, Measurement model Represent each input quantity X i by 1. Best estimate x i as mean of distribution, and 2. Standard uncertainty u(x i ) as s.d. of distribution
bunjob_ajchara94 Measurement Model For each input quantity 1.Obtain knowledge of that quantity 2.Assign a probability distribution to each quantity consistent with that knowledge Often a Gaussian (normal) or a rectangular distribution
Classification of uncertainty components Type A components: those that are evaluated by statistical analysis of a series of observations Type B components: those that are evaluated by other means –Both based on probability distributions –standard uncertainty of each input estimate is obtained from a distribution of possible values of input quantity: both based on the state of our knowledge –Type A founded on frequency distributions –Type B founded on a priori distributions
96 Type A evaluations of uncertainty are based on the statistical analysis of a series of measurements. Type A evaluations of uncertainty
bunjob_ajchara97 Type A Evaluation of Standard Uncertainty For component of uncertainty arising from random effect Applied when multiple independent observations are made under the same conditions Data can be from repeated measurements, control charts, curve fit by least-squares method etc Obtained from a probability density function derived from an observed frequency distribution (usually Gaussian
Type A Evaluation Arithmetic mean Best estimate of the expected value of a input quantity -
Type A Evaluation Experimental standard deviation Distribution of the quantity
Type A Evaluation Experimental standard deviation of the mean spread of the distribution of the means -
Type A Evaluation Type A standard uncertainty degrees of freedom
Example A digital multimeter is used to measure a high value resistor and the following readings are recorded. The standard uncertainty, u, is therefore 0.008 83 kΩ.
Type A Evaluation For a well-characterized measurement under statistical control, a pooled experimental standard deviation S p that characterizes the measurement may be available. –The value of a measurand q is determined from n independent observations and – The standard uncertainty is Pooled Experimental Standard Deviation
A previous evaluation of the repeatability of measurement process (10 comparisons between standard and unknown) gave an experimental standard deviation If 3 comparisons between standard and unknown were made this time (using 3 readings on the unknown weight), this is the value of n that is used to calculate the standard uncertainty of the measurand Type A Evaluation Example:
Type B Evaluation of Standard Uncertainty Evaluation of standard uncertainty is usually based on scientific judgment using all relevant information available, which may include: –previous measurement data, –experience with, or general knowledge of the behavior and property of relevant materials and instruments, –manufacturer's specifications, –data provided in calibration and other reports, and –uncertainties assigned to reference data taken from handbooks. GUM 4.3.1
March 2006Slide 106 Type B Evaluations Normal distribution: Examples: –expanded uncertainties from a calibration certificate where U i is the expanded uncertainty of the contribution and k is the coverage factor (k = 2 for 95% confidence).
Type B Evaluations Normal distribution Example A calibration certificate reports the measured value of a nominal 1kg OIML weight class F2 at approximately 95% confidence level as:
“ It is likely that the value is somewhere in that range” Rectangular distribution is usually described in terms of: the average value and the range (±a)Certificates or other specification give limits where the value could be,without specifying a level of confidence (or degree of freedom). Rectangular distribution The value is between the limits The expectation
aa AB P=1/ 2a Range = 2a, Semi-range = Range /2 = a Rectangular distribution
Example From the previous example, if the Maximum Permissible Error (MPE) according to OIML class F2 (±16 mg) is used; then Example of Rectangular distribution
A Handbook gives the value of coefficient of linear thermal expansion of pure copper at 20 and the error in this value should not exceed, assuming rectangular distribution the standard uncertainty is: Handbook
Example of Rectangular distribution Manufacturer ’ s Specifications A voltmeter used in the measurement process has the accuracy of ± 1 % of full scale on 100 V. range semi - range ( a ) = 1 V
Example of Rectangular distribution If the resolution of the digital device is δx, the value of X can lie with equal probability anywhere in the interval X - δx /2 to X + δx /2 and thus described by a rectangular probability distribution with the width δx 1234 5 6 4 Resolution of a digital indication
Example of Rectangular distribution Digital indication A digital balance having capacity of 210g and the least significant digit 10 mg. The standard uncertainty contributed by this balance is:
Example of Rectangular distribution Hysteresis The indication of instrument may differ by a fixed and known amount according to whether successive reading are rising or falling. If the range of possible readings from that is dx
March 2006Slide 117 U-shaped distribution When the measurement result has a higher likelihood of being some value above or below the median than being at the median. Examples: –Mismatch (VSWR) –Distribution of a sine wave
Example of U-Shaped distribution A mismatch uncertainty associated with the calibration of an RF power sensor has been evaluated as having semi-range limits of 1.3%. Thus the corresponding standard uncertainty will be bunjob_ajchara118 UKAS M3003
Distribution used when it is suggested that values near the centre of range are more likely than near to the extremes Assumed standard deviation: 2a (= a) 1/a X Triangular distribution
Valu es close to x are more likely than near the boundaries Example: A tensile testing machine is used in a testing laboratory where the air temperature can vary randomly but does not depart from the nominal value by more than 3°C. The machine has a large thermal mass and is therefore most likely to be at the mean air temperature, with no probability of being outside the 3°C limits. It is reasonable to assume a triangular distribution, therefore the standard uncertainty for its temperature is : Example of Triangular distribution In case of doubt, use the rectangular distribution UKAS M3003
Which is better A or B? It should be recognized that a Type B evaluation of a standard uncertainty can be as reliable as a Type A evaluation, especially in a measurement situation where a Type A evaluation is based on a comparatively small number of statistically independent observation. GUM 4.3.2
bunjob_ajchara123 Components of standard uncertainty of measurand y=f(x 1, x 2,x 3 ……x N) are combined using the “ Law of Propagation of Uncertainty” or “Root Sum of Square :RSS” combined standard uncertainty
124 The relationship between the measurand, Y, and A, B and C is written most generally as Y = f(A,B,C). Combined Standard Uncertainty, u c u(a), u(b) and u(c) are the standard uncertainties of best estimates a, b and c respectively obtained through Type A or Type B evaluations.
bunjob_ajchara126 sensitivity coefficient Partial derivative with respect to input quantities Xi of functional relationship between measurand Y and input quantities Xi on which Y depends sensitivity coefficient formula
bunjob_ajchara127 Example The value of the resistance Rt, at the temperature t, is obtained from equation: Where: α is the temperature coefficient of the resistor in Ω / °c t is the temperature in °c, and R 0 is the resistance in ohms at the reference temperature, The partial differentiation of R t with respect to t is:
bunjob_ajchara128 Correlation of Input Quantities Ref UUT S Ref S UUT S corr Difference (Correction Ref- UUT)
bunjob_ajchara130 correlation coefficient correlation coefficient, r(x i, x j ) - degree of correlation between
bunjob_ajchara131 Uncorrelated input quantities For uncorrelated input quantities r (x i, x j ) = 0 For c i =1
Combinations of Uncertainties Addition/Subtraction For independent variables, we have,
Combinations of Uncertainties Multiplication/Division For independent variables, we have, Similar arguments would apply to the expression
134 Worked example The mass, m, of a wire is found to be 2.255 g with a standard uncertainty of 0.032 g. The length, l, of the wire is 0.2365 m with a standard uncertainty of 0.0035 m. The mass per unit length, , is given by: Determine the, a) best estimate of , b) standard uncertainty in .
135 Worked example continued The partial differentiation of µ with respect to m and l
bunjob_ajchara136 For the very special case where all input estimates are correlated The combined standard uncertainty correlated input quantities
bunjob_ajchara137 Correlated input quantities Example 1)R i (R 1,R 2,R 3,……,R 10 ) each has nominal value 1000 ohms 2)Each has been calibrated by direct comparison with negligible uncertainty 3)Standard uncertainty of R s is u(R s ) = 100 mohms Model equation : R1R1 R2R2 R3R3 R 10 R ref 10 k
Expanded Uncertainty expanded uncertainty –quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand. GUM 3.2.5
140 Expanded Uncertainty, U Y = y U The Expanded Uncertainty, U, is a simple multiple of the standard uncertainty, given by U = ku c (y) k is referred to as the coverage factor. So we can write:
coverage factor, k –numerical factor used as a multiplier of combined standard uncertainty in order to obtain expanded uncertainty GUM 3.2.6
Coverage factor Most cal labs adopt 95.45% which gives k 2 for effective degrees of freedom 30 Coverage Factor - k Confidence Interval 1.0068.27% 2.0095.45% 2.5899.% 3.0099.73%
bunjob_ajchara143 Coverage Factor of Combined Uncertainty Effective Degree of Freedom –to determine the coverage factor of combined uncertainty, the effective degree of freedom must be first calculated from the Welch-Satterthwaite formula: Based on the calculated v eff, obtain the t-factor tp(v eff ) for the required level of confidence p from the t-distribution table The coverage factor will be: k p = tp(v eff )
bunjob_ajchara144 Example -- A steel rod was measured 4 times. The calculated. The effective degree of freedom: For @ 95% confidence level and from “student’s t” table, we get k = 2.52 Effective number of degrees of freedom
bunjob_ajchara145 Effective number of degrees of freedom Therefore, the expanded uncertainty U is:
bunjob_ajchara146 Relative standard uncertainty Relative standard uncertainty of input estimate, Relative combined standard uncertainty, y then
bunjob_ajchara147 Relative standard uncertainty Example The measurand: DescriptionValue,xStandard uncertainty, Relative standard uncertainty, rep Repeatability 1,00,0005 Weight of KHP 0,3888 g0,00013g0,00033 Purity of KHP 1,00,00029 Molar mass of KHP204,2212 gmol -1 0,0038gmol -1 0,000019 Volume of NaOH for KHP titration 18,64 ml0,013ml0,0007
bunjob_ajchara148 Relative standard uncertainty 1)Value of the measurand = 0,10214 mol l -1 2) Combined relative standard uncertainty u c (C NaOH ) = 0,00097 X 0.10214 mol l -1 = 0,00010 mol l -1
Reporting should include –result of measurement –expanded uncertainty with coverage factor and level of confidence specified –description of measurement method and reference standard used –uncertainty budget example of uncertainty statement e.g.The expanded uncertainty of measurement is ± ____, estimated at a level of confidence of approximately 95% with a coverage factor k = ____.
Reporting Result It usually suffices to quote u c (y) and U [as well as the standard uncertainties u(x i ) of the input estimates x i ] to at most two significant digits, although in some cases it may be necessary to retain additional digits to avoid round-off errors in subsequent calculations. In reporting final results, it may sometimes be appropriate to round uncertainties up rather than to the nearest digit. For example, u c (y) = 10,47 m might be rounded up to 11 m. However, common sense should prevail and a value such as u(x i ) = 28,05 kHz should be rounded down to 28 kHz. Output and input estimates should be rounded to be consistent with their uncertainties. bunjob_ajchara151 GUM 7.2.6
Reporting Conventions 1000 (30) mL –Defines the result and the (combined) standard uncertainty 1000 +/- 60 mL –Defines the result and the expanded uncertainty (k=2) 1000 +/- 60 mL at 95% confidence level. –Defines the expanded uncertainty at the specified confidence interval
The 9-steps GUM Sequence 1. Define the measurand 2. Build the model equation 3. Identify the sources of uncertainty 4. (If necessary) Modify the model 5. Evaluate of the input quantities and calculate the value of the result 6. Calculate the value of the measurand (using the equation model) 7.Calculate the combined standard uncertainty of the result 8. Calculate the expanded uncertainty (with a selected k) 9. Report result bunjob_ajchara153
Some Important Practical Consequences … or a little common sense with errors! 1.When several (independent) errors are to be added, addition in quadrature is much more realistic than addition. 2.If one error ie less than one quarter of another error in the addition then the smaller error may be realistically ignored. 3.There is little point in spending much time estimating small errors – concentrate on the large errors! 4.The experimental procedure should minimise the dominant errors, This implies that these must be identified and estimated (usually in a pilot run) before the final data is taken. 5.Try to bring the precision of each variable to a common level, if possible, by repeated measurements.
Basic concepts “… The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailed knowledge of the nature of the measurand and of measurement …” GUM 3.4.8
158 ISO (1993) Guide to the Expression of Uncertainty in Measurement (Geneva, Switzerland: International Organisation for Standardisation). NIST Technical Note 1297 (1994) Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. M 3003, The Expression of Uncertainty and Confidence in Measurement, published by UKAS EA-4/02 - December 1999 Expression of the Uncertainty of Measurement in Calibration EURACHEM / CITAC Guide: Traceability in Chemical Measurement - A guide to achieving comparable results in chemical measurement 2003 Assessment of Uncertainties of Measurement for Calibration and Testing Laboratories - Second Edition, c R R Cook 2002 Published by National Association of Testing Authorities, Australia ACN 004 379 748 ISBN 0-909307-46-6 Bibliography and acknowledgement